Hey there! Ready to crack the code on IEEE 754 floating-point numbers? In this video, I break down how to convert decimals to binary and back for 32-bit and 64-bit floats with clear, easy-to-follow steps. From scientific notation to biasing exponents, I’ve got you covered with examples (like 45.84375!). Perfect for computer science students, programmers, or anyone curious about how computers store fractions. Pause, rewind, and learn at your pace! Subscribe for more, check my other videos, and drop a comment with your thoughts!

Introduction to IEEE 754 00:00:00
Prerequisites for Understanding 00:01:03
Steps to Convert Decimal to IEEE 754 00:02:31
Bias and Exponent Explanation 00:03:17
32-bit Float Layout 00:04:09
Sign Bit and Fraction Bits 00:05:44
Example: Decimal to 32-bit Float 00:07:56
Converting 45.84375 to Binary 00:08:34
Scientific Notation for Binary 00:09:05
Biasing the Exponent 00:10:26
Placing Bits in Layout 00:11:17
Final 32-bit Float Representation 00:13:32
Example: IEEE 754 to Decimal 00:14:28
Extracting Sign, Exponent, Fraction 00:15:36
Unbiasing Exponent and Reconstructing 00:16:23
Converting Binary to Decimal 00:18:53
Introduction to 64-bit Floats 00:19:47
64-bit Float Layout and Bias 00:20:05
Example: 64-bit Float to Decimal 00:21:09
Processing 64-bit Float Bits 00:22:05
Reconstructing 64-bit Scientific Notation 00:23:26
Final Decimal Conversion for 64-bit 00:25:00
Conclusion and Recap 00:25:54
Call to Subscribe and Outro 00:26:36

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Hey there! Let’s talk about IEEE 754 floating point numbers stored inside of your machine.

What the heck am I even talking about? Well, uh, there’s a standard called IEEE 754 which governs

how floating point numbers are stored inside of your computer on x8664 machines. So, you know,

of watching this video you probably already know how to convert binary back and forth you know

between binary and decimal and hopefully you understand how to do binary with fractions

if you don’t i have other videos i’ll talk about that in a second and you also understand

that in the machine there is like a slightly different format that’s my doggy slightly

different format for storing signed integers in the machine okay so floating point numbers have

an even different uh format that we have to follow and the hardware is designed that way

hardware is designed that way so before we can really benefit you’re all gonna

see how bad I spell okay before we can benefit from this video your prerequisite

knowledge should be that for starters you should be able to convert back and

forth between binary and decimal including numbers with fractions so if

if you don’t know how to do that yet this video is is not for you just yet you

should go check out my other videos I explain in other videos exactly how to

do that go to the other videos come back and then check this one out after you

know how to convert back and forth between binary and decimal with

fractions the second thing you should know how to do before watching this

video is convert binary numbers with fractions to scientific notation and

and back again. Meaning you should understand scientific notation and you should also understand

how to take a binary number with a fraction and convert it to scientific notation and then

convert it or deconvert it back to like regular form. Again, I have another video for this,

so if you don’t know how to do that yet, you should probably pause this video and then go

Let’s see. For converting from a regular number to IEEE floating point number in the machine using just binary alone is the following.

So I’m going to say basic steps. Step one, convert the decimal number to binary if needed.

And I say if needed because I don’t know, maybe you started with a binary number so you don’t actually need to convert it.

it but I’m just going to assume that you have a decimal number so you’re going to convert it to

binary first then convert the binary number to scientific notation I don’t know if convert is

the right word here because we’re not actually changing the value of the number we’re just

changing the representation so convert it to a scientific notation form maybe would be better

step three we’ll add a bias to the exponents of the scientific notation form what do I mean by

notation form. What do I mean by that? You know, if you have like a binary number and it’s in

scientific notation format, it’s going to look like this. I liked when I’m doing IEEE 754,

I like to keep the right side of the number in decimal. I’m sorry if this confuses you, but

it’s not really necessary to convert this to binary when we’re just trying to convert back

and forth between IEEE. So I always say two to the something power rather than one zero to the

something power because I want to stay in decimal. So I’ll say, you know, two to the seventh power,

right? What this means is that seven is the exponent and I have to bias the exponent before

I put it into the machine. So maybe it’s worth understanding or maybe copy pasting right now

that the bias for 32-bit floating point numbers, I should probably, I’ll paste the whole thing here.

Okay. So here’s like the format for a 32-bit floating point number. We’re good about this

number. We’re good about this just for a second. The bias for 32-bit floating

point number in IEEE 754 is 127. So that means I pretty much have to just take

the number 7 and bias it by adding 127 to it. After I’ve done that I can convert

the biased exponent to unsigned binary whole number. To an unsigned binary whole

number. So step four here is just regular decimal to binary conversion. So I would take, you know,

seven plus 127. The result of that, which I think is like 134, maybe we’ll just be converted into

binary and unsigned binary integer, not two’s compliment or anything like that. Then step five

is respect the layout. So I just, I just copy pasted the layout up above and the layout is this.

you can see that here let me see if I just maybe paste this down at the bottom

you can see that we actually have 32 bits here if I count them for you real

fast we’ve got one for the sign and then we’ve got I can never remember this I

think it’s eight for the exponent one two three four five six seven eight got

eight bits for the exponent that means I’ve already used nine of 32 bits so

that means I should have like 23 bits here so one two three four five six seven

11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, just to double check.

So we know what the sine bit is.

Actually, we don’t yet know in this video.

So in this video, the sine bit, it’s always going to be a zero if the number is positive

and a one if the number is negative.

So that’s the sine bit.

The biased exponent bits go there.

And then these Fs, those are just the fractional bits.

just the, you know, like the mantissa or the mantissa part of the fraction. So imagine if we

have a number like this, all I’m saying is we would copy paste those numbers right here.

Notice how I’m ignoring that first one dot something, because when you put binary in

scientific notation, the first number is always going to be a one. So it can be assumed. There’s

no point in storing it inside the machine. And then if you don’t have enough digits to

you will literally just pad with zeros on the right side because padding with zeros on the right side of a fraction

doesn’t actually change the value. One, two, three, four.

If I put the zeros on the left side of the fraction, it would make the fractional part smaller and smaller.

So that would be bad. I don’t want to change the value.

And this is the opposite of how you pad a whole number.

If I put numbers on the left side of a whole number part, I’m not changing the value.

But if I put numbers on the right side of a whole number part,

which in this case would be the biased exponent, then I would be changing the value.

So we wouldn’t want to do that.

changing the value so we wouldn’t want to do that. Anyway I’m going to remove this and just stick

with step five so let’s see 5a is place the sine bits step 5b is and I’m going to do an example

for you don’t worry place the exponent bits and then 5c is place the fraction bits everything

the first one dot from scientific notation form.

Again, because the first one is implied, it will never be a zero if you’ve done it correctly.

If you end up with a binary number that looks like this in scientific notation, then you’ve

done something wrong because the first number is supposed to be a one.

Okay, so we have those basic steps.

And now let’s do an example.

1, 2, 3 example. Let’s convert decimal to IEEE 754 32-bit float format.

AKA single precision floating point number. Double precision floating point number is 64-bit.

We’ll do an example for that at the very end of this, but I’m just going to stick with 32 and do

number 45.84375, just double checking, and then we’ll convert to binary. Again, this knowledge

is contained in a different video. So if you don’t know how to do this yet, you’ll need to go see my

other video. And so you can see it’s, you know, 45.84375 in binary is just going to be this.

I’ve padded it with two zeros to the left just because, I don’t know, I have a habit of wanting

wanting everything to look like it’s inside of 8-bit chunks.

You can see there’s 8 bits here.

But really, I can just remove those first two numbers.

It doesn’t matter.

Convert the number to binary.

Then, I have to convert the binary number to scientific notation.

I’m going to put maybe pseudo-scientific notation

because I’m not going to convert the right part

where it’s the exponent into binary.

I’m going to keep it in decimal.

Just to clarify again,

convert a binary number to scientific notation and you were actually going to publish it somewhere

or give it to someone, then you should convert the right side. It’s just that I’m not going to

because I don’t really need to convert the right side. I’m not interested in having the whole

number in scientific notation in binary. I’m just interested in what the fractional bits are and

then what the exponent is going to do. Like how many positions to the left or right is it actually

going to move the decimal point? And that’s just going to be the number five, five times to the

five you know five times to the left or to the right so I’m gonna go pseudo

scientific notation you pretty much you know as I covered in the different video

all we’re gonna be doing is just moving the decimal point over one two three

four five until there’s a one in the leading spot and then account for it by

saying times two to the fifth power to say the number is a lot bigger than it

looks so that’s that and let’s see if there are any trailing zeros that’s

Let’s see if there are any trailing zeros, that’s okay on the right side of the fraction.

They don’t actually matter.

You can also just delete them if you want.

Then we need to bias the exponents.

So how do we bias the exponent?

The exponent is just the number five.

That’s see, I was telling you like I don’t really care to convert it to binary because

I’m just going to bias it first and then after that I’ll buy it or I’ll convert it to binary.

So biasing it is just basically adding the bias.

just to make sure that it’s on this page and it’s easy to see.

So I’ll do that.

So in the 32-bit float layout, the bias is 127.

So I’m going to add 127 to 5.

The final number is going to be 132.

So that’s the biased exponent now.

Then I convert the biased exponent to binary.

And that’s going to be this.

10000100.

Then set up the layout so you don’t get confused.

This step is crucial.

Everyone always gets confused by the layout because like we’re human beings.

I can’t count 32 zeros that are all like in the same area and like, you know, and not

get confused every single time.

I mean, I guess I won’t get confused half the time, but that’s not good enough.

So step one is place the layout for yourself so it’s easier for you to place the bits.

place the bits and I wrote this at the top 8 bits and 23 bits just as a reminder double check

yourself 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 that’s 23 bits 1 2 3 4 1 2 3 4 that’s

8 bits we don’t need to remember that the sign bit is one bit for now it’s not too hard so now

let’s place it so the first thing to note is that this number was actually positive positive is

the negative sign there, but if it makes you feel better, we’ll just put a positive sign there just to clarify and be specific.

So because the number is positive, the sign bit is going to be zero. If it were negative, it would be one.

Then we will put the biased exponent bits. So we already converted up there. So I’m just going to copy paste it here.

We’re pretty lucky that the bits are just as big as the number of biased exponent bits that are available.

But suppose for the sake of argument that we had,

the sake of argument that we had just like you know this small number here for the biased exponent

what would you do with the remaining spaces you would just pad with zeros to the left

because remember in whole numbers in binary and decimal you pad to the left you’re not actually

changing the value you definitely would not want to pad to the right but you got to put

a bit in every position it’s got to be something so i’m just going to pad with zeros

then we take the fractional part from the scientifically notated form so that’s just

form so that’s just this right here notice how I’m copy pasting every bit

except for the one that starts the number I’m not going to do the one dot

part because that’s always assumed so we don’t need to store it it wouldn’t make

sense I’ll copy paste the fractional part and I’ll just stick it down there

and now we have a bunch of leftover bits that we need to fill out with the

fractional part you pad to the right not to the left so I’m just going to put

zeros to the right because that won’t change the value of the fractional part

Now we’ve got it. I’m just going to double check my work here.

Oh gosh, it’s too hard to read and then a bunch of zeros. Okay.

Then remove these spaces because now that you’ve done your layout,

you actually want to bear in mind that the machine doesn’t have spaces.

The machine is just, you know, a bunch of ones and zeros everywhere.

So I’m just going to carefully remove the two spaces.

And this is it.

for floating point representation of the number of the decimal number positive 45.84375

seems like a lot but it’s not too bad once you kind of get used to the format

and the bias and you already know the other steps of you know converting decimal to binary and back

again so let’s do another number let’s do it backwards so i’m going to open up a new tab here

example, IEEE 754 32-bit float 2 decimal.

Okay, so suppose we started with the following number.

Start with this number.

Oh my gosh, right?

Remember the layout.

So in the layout we have, use the layout.

Maybe I should copy paste the layout again one more time just to make sure that it’s

easy to see.

it’s easy to see. So I’m going to paste it up at the top and well, no, maybe I want this because

this is like a little bit better. I want that version. Okay. So 32 bits, a bias of 127. So

we’re going to use the layout. I’m just going to copy paste the original number and then add

spaces because I know, well, I guess I can copy paste the layout itself right above that number.

I know I need a space right there that’s the sine bit and then I need a space

after that and notice how the numbers all line up so there’s nothing missing

just to double-check yourself okay so I have the layout I know right away that

the number is going to be positive because the sine bit is zero sign is

positive okay so then I need to pull the biased exponent will be biased

the exponent bits only so I’m just going to copy paste those bits and then I’ll

pull out the fractional part first pull out the fractional part I’m going to

copy paste all those bits and maybe I’ll just say zero point those bits because

that’s what I originally grabbed and I can delete all the zeros on the right

side because again that won’t change the value of the fractional part so I’m

going to delete make sure you’re only deleting zeros though delete that now I

I need to unbias the biased exponents.

First, actually, I should, well, let’s convert the biased exponent to decimal.

A little easier.

I guess you don’t have to do that, but it’s a little easier for me.

Pull the biased exponent.

So this number is going to be 132, which I think is what we had last night.

Let me just double check that.

132.

Yeah.

132, yeah so the biased exponent that we pulled out is 132 then we unbiased the

biased exponent so just subtract the bias again in 32-bit format here the bias

is 127 so I’m going to subtract 127 instead of adding 127 so now I know the

real bias or sorry I know the real exponent is 5 then I can recreate the

I’ll put sudo here just because it’s not, you know, I’m not, I’m not converting binary for,

for the entire thing. So that means I can take the fractional part here and I can say it’s going to be

one dot the fractional part. Maybe I should remove that zero here to make it less.

I don’t know. Cause if I put a one there, it kind of feels like I’m saying the number is

that it’s one dot something.

I don’t like that I’m putting a zero there.

I’m just gonna erase it

because it’s really gonna end up being a one.

And if I say pull the fractional part

and I put a one there,

then it doesn’t it kind of sound like

I’m pulling the one from somewhere?

No, the one is implied.

Anyway, so now I can recreate the number

in pseudo scientific notation.

I can say times two to the something power.

We know that the exponent is five.

Again, it’s pseudo scientific notation

because if five we’re gonna do the complete,

complete, you know, true scientific notation. The right side should also be in binary, but I’m not

going to do it. I’m a little bit lazy because we don’t need that IEEE. So it’s going to be two to

the fifth power. Then if I undo the scientific notation, I’m really just going to take this

scientifically notated version and I’m going to say, well, I’m going to move the decimal point to

the right five times because that’s what the part on the right says. So I’m going to go one, two,

1, 2, 3, 4, 5, stick it there, remove the original decimal point, and then I can remove

the exponent part. So then let me just double check my work. 1, 0, 1, 1, 0, 1, 1, 1, 0, 1. Okay.

So we’ve got that now. Then all we have to do is convert this number from binary to decimal.

The final number here is just, you know, it’s going to be the same number that we worked with

number that we worked with previously so I’m just going to copy paste it there is

you know of course this video is not about converting back and forth from

binary to decimal with fractions check out my other video if you want to learn

how to do that but once we do that you know this number becomes that number

and and now we know how to convert back and forth between a decimal number and

for 32-bit floats in the machine.

No problem.

A lot of steps, but no problem once you understand them.

Okay, now let’s do an example

with 64-bit floating point numbers.

Honestly, this is not gonna be more difficult.

In 64-bit floats, the idea and the format and the steps,

it’s all gonna be the exact same thing.

It’s just that we have more bits

for the biased exponent and the fraction,

and then the bias itself will be a little bit higher.

So let me copy paste the format for you to see.

at for you to see. So in IEEE 64-bit floats, IEEE 64-bit float, well IEEE 754, 64-bit float,

you know we refer to these floats as doubles right double precision floating point numbers.

Notice how instead of 8 bits for the biased exponent we have 11 so we can represent a much

notice how for the uh the fraction part instead of having 23 bits we have 52 bits so we can also

represent much more precisely that’s really the only difference let’s uh oh sorry the other

difference is also that the bias is now equal to 1023 instead of 127 so what is that like

10 times larger of a number that we can represent or something like that so mostly we get more

If you don’t like it, sorry, it’s the way it is.

It’s in the hardware.

So let’s do an example for a 64-bit float for a double.

64-bit example.

Let’s say we started with the following number.

Inside of the machine.

Like we looked inside of the machine and we grabbed these bits.

And I’ll get rid of the spaces.

So we have like a giant, big, huge number.

We have like a giant big huge number.

We’re trying to figure out, you know, what is this number in decimal.

So I’m going to add spaces to respect the layout.

I’m going to just copy paste the layout right here.

I’m going to say respect the layout.

I’m going to paste it and then that helps me line everything up.

So I’m going to say the sine bit is the first bit and then I’ll put a space there so I know.

All right, the sine bit is a one.

That means it’s going to be a negative number.

The biased exponent bits are there.

The fractional bits are there.

bits are there so well first thing I can just do is say that the sine is going to be negative

because the sine bit is a one

and then I’ll pull the biased exponent bits which is going to be this number right here

should pull the fraction bits next. Fraction bits are just here I’m gonna

copy paste those here and then I can remove the zeros. Be very careful you’re

not removing any ones. So there was a lot of wasted bits there. That’s okay. I’ll

convert the biased exponent to decimal. So that’s going to be…

I think and I did not do that in my head.

I have an answer sheet.

Don’t worry.

I’m not that cool.

Then we’ll unbiased or de-bias unbiased the biased exponents by just subtracting the bias.

Remember in this 64 bit format, the bias is 1023.

So that means I got to take 1031 minus 1023 and that will be equal to 8.

Okay.

Okay, now we kind of have everything we need to sort of reconstruct the scientific notation part, right?

We have the fraction bits, that’s the most important part.

Oh, and then also the sine bits.

So I can say negative something.

It’s obviously going to be one dot something because even though the number is negative,

it will still always start with a one.

That rule doesn’t change.

Negative one dot something and I’ll just paste the, whoops, paste the fraction bits.

I think I did paste the right thing.

right thing so we paste the fraction bits and then we have to use the regular exponent the

unbiased exponent the non-biased exponent two times times two to the oops times two to the eighth power

now this is our number in scientific notation um

convert to scientific notation i’ll put p for pseudo scientific notation

and then now we have to convert to just regular format okay so I’m going to

copy paste this right here and if it’s 2 to the eighth power that means the real

number is going to be bigger than it looks in scientific notation even though

if you are very familiar with scientific notation you kind of already know that

it looks big but you know if we just if we’re only talking about the bits on the

1, 2, 3, 4, 5, 6, 7, 8.

Put the decimal point there.

Remove the original decimal point.

Now we no longer need 2 to the 8th power.

And again, just as a reminder, if it was 2 to the negative 8th power,

you would be moving the decimal point in the other direction.

So now we’ve got this number 111, 010101, 0101, 0101, 0101.

Then a 0.01.

Okay, so then convert to decimal.

doesn’t talk about how to convert back and forth from binary to decimal find my other videos but

basically uh it’s going to end up being six uh negative four six nine point two five

okay we now have done a quick practice starting with a giant number of bits and slowly converting

them to the real uh floating point number in decimal that uh that they represent i’m not going

going to do another number in the reverse direction because we already did that for 32 bits

and again with 64 bits the only real difference is just you have more exponent and fraction bits

and the bias is a higher number and that’s it so like all the other steps are just the same

I just wanted to show you one time with a 64-bit number in one direction but

I think at this point we’re good thank you so much for watching this video I hope you learned

had a little bit of fun. I’ll see you in the next video.

and grow this community so we’ll be able to do more videos, longer videos, better videos,

or just I’ll be able to keep making videos in general. So please do me a kindness and subscribe.

You know, sometimes I’m sleeping in the middle of the night and I just wake up because I know

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Thank you.

The post Master IEEE 754 Floats: Binary to Decimal Guide appeared first on NeuralLantern.com.

Struggling with converting decimal fractions to binary? In this fun, beginner-friendly tutorial, we dive into how to convert numbers like 45.8046875 to binary and back to decimal, step by step! Learn the multiplying-by-2 method, handle precision loss, and understand fractional binary representation. With real examples (0.126753, 0.84375), you’ll master this key computer science skill. Perfect for students, coders, or anyone curious about how computers store numbers. Subscribe for more tech tutorials, and let me know what you want to learn next! Visit [YourWebsiteLink] for more. #Binary #DecimalToBinary #ComputerScience #TechTutorials

Introduction to Decimal-Binary Conversion 00:00:00
Understanding Fractional Binary Numbers 00:00:15
Fractional Binary Representation Basics 00:01:00
Converting Decimal Fractions to Binary 00:04:06
Example: Converting 0.126753 to Binary 00:04:52
Precision Loss in Conversions 00:05:36
Example: Converting 0.84375 to Binary 00:09:57
Converting Binary Fractions to Decimal 00:14:22
Example: Converting Complex Number 45.8046875 00:16:48
Combining Whole and Fractional Parts 00:21:00
Conclusion and Verification 00:23:25
Outro and Community Engagement 00:24:16

Thanks for watching!

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Hello there, let’s talk about how to convert decimal numbers with fractions into binary

numbers with fractions and vice versa.

So up to this point, if you’ve been watching my other videos, you probably know that you

can easily convert a decimal number to binary and binary to decimal if the decimal and binary

numbers but in order to progress towards IEEE 754 representation in other words

in order to eventually be able to represent binary numbers with fractions

like floats or doubles in the machine you have to start learning how to get

the fraction part of a decimal number into binary the way we’re going to do

it in this video it’s not actually going to be the final representation of how

your machine stores floating point numbers but it’s an important step on

on the way there. Without this step, you probably wouldn’t be able to do it.

So what am I talking about? I’m just going to open up this notepad real fast and just show you

that what I really mean is, suppose we have like a number 45 point blah, blah, blah, blah, blah,

right? So this is the whole part and this is the fractional part. Okay, no problem. Also in binary,

we have like a bunch of ones and zeros. This is a whole number, but we could put a decimal point

and then just start doing more numbers after the decimal point in order to represent a

in order to represent a fractional binary number.

Okay, so let’s see.

We know, hopefully at this point, if you watch my other videos,

that, I don’t know, I’m just going to put some random numbers here

and then like some more random numbers there.

And you know that the strength of each of these numbers

is sort of like increasing by powers of two on the left side, on the whole side.

So the strength of that first digit is one.

digit is one the next digit has a strength of two and then four and then eight and then 16 maybe i’ll

put the six up here and then 32 so three and two and then 64 and maybe i’ll do another digit i’ll

just put like a zero there and then you know the left most digit has a strength of 128 so basically

a 1 or a 0 for the actual binary number multiplied by its strength. So 1 times 1 is 1, add that to

0 times 2, add that to 1 times 4, and 0 times 8, and 1 times 16, and so forth, right? So hopefully

we know how to do that at this point. To do the same thing on the fractional side, maybe I won’t

put a decimal marker there, it’s really the same deal except as we go left, we were multiplying by

have more power more strength but as we go to the right past the decimal point they should have less

strength and we’ll just divide them by two so pretty much at this point let’s see we’ll have

one half will be the strength of this digit right here whoops one half so the strength that that

digit has is just one half the strength that the next digit has is going to be one fourth remember

1 4th remember you multiply sorry you divide by 2 each time whoops I’m getting lost oh man I

probably need to add some spaces because it’s starting to get pretty ugly so this digit has

a strength of 1 half this digit has a strength of 1 4th and then we’ll just say that the next one

has a strength of 1 8th and so forth we’ll just keep dividing by 2 each time so don’t forget that

don’t make the mistake of saying 1 4th or sorry don’t make the mistake of saying 1 half 1 3rd

one half, one third, one fourth.

Don’t increase the denominator by one.

You want to multiply, or sorry,

you want to divide by two each time.

Or if you want to say the word multiplication,

then you multiply by 0.5, I guess, if you want to.

I’m just going to write 1 16th

and then just be done with the divisions.

I hope you understand what I’m talking about.

We’re going to do some examples right now.

Okay, so let’s calculate.

Well, actually, let’s just, just to emphasize, just to make sure everybody understands, what

we would do to convert such a number from binary to decimal would basically be, you

know, compute the left side as if it were a whole number.

I’m going to put W for whole number.

And then separately convert the fractional part as if it were just, you know, zero point

something.

And then just combine the two things when you’re in decimal.

And you’ll do the same thing when converting a number from decimal to binary.

decimal to binary you convert the whole part first put that into ones and zeros then convert the fact

the fractional part into a fractional binary part and then just combine them across the decimal point

so let’s do an example let’s start with converting 0.126753 i don’t think that this number is going

to resolve easily it’s kind of like a good lesson here when you’re converting between binary and

there are always going to be some numbers in in in decimal that you can’t represent in binary

and vice versa there are always going to be some numbers in binary that you can’t represent in

decimal so it’s not that decimal or binary or you know superior or inferior i mean all technology

is pretty much running on binary so it can’t be that bad but um just keep that in mind some numbers

that you type randomly are just never going to resolve and you’ll have to give up at some point

have to give up at some point and say, well, we’re just going to call this a precision loss

because we just, we just have to like give up, especially if you only have so many bits

to put the fraction in. Okay. So what am I going to do? What’s the, what’s the basic algorithm

here? There are lots of different ways to do it. The way that I’m going to show you is just

multiplying by two. So if we take, you know, 0.126753 and we multiply it by two,

We end up with, let’s see, times 2.

Whoops, hello.

Oh, I got to do that.

Times 2.

We’re going to end up with a different number.

If the number is equal to or greater than 1,

then we will say that we have achieved a number of 1 for the converted binary number.

And if not, we’ll say we have achieved a number of 0.

If the number is 1, we’ll also subtract 1 from the remaining number here in decimal.

The result we have is just, what’s going on?

Oh, I somehow turned on my annotator.

Okay.

In this case, we just have, you know, this number right here.

It’s not greater than one or equal to one.

So I’m just going to say this gave us a binary number of zero.

And because it gave us a zero, we’re not actually going to change it at all.

We’re not going to subtract one from it.

We’re just going to leave it as is.

We will then multiply by two again.

by 2 again. So I’m just going to say, do it one more time. We’ll say this multiplied by 2,

and it’s going to be this number. Again, it did not exceed or equal 1, so the bit in that position

is going to be a 0, and we’ll just copy and carry it over just as is. Do it again, and we’ll say

Now we have a number that equals or exceeds one.

So we’ve achieved a one in the binary position.

So it’s going to be a one here.

Then we got to subtract one.

So we’re just going to be left with 0.014, 024 and whatever.

And then we just continue.

Maybe I’ll do it a couple more times,

but I’m going to give up because I don’t think

that this number will translate completely.

So I’m going to say multiply by two again,

and it’s going to give us,

hang on a second here.

What was that?

Hang on a second here. What was that? That was one. Oh, I forgot to subtract one. So I got to

subtract one. And then I’m going to multiply by two. Now we get this number right there.

This is not equal to or greater than one. So the bit is going to be zero.

And then that means the remaining number is just going to be the same thing.

We’ll do another. Actually, let me let me clean the calculator for a second here.

going to select this whole thing and just paste that and I’m going to say

multiplied by two that’s going to be that new number and I think you’re

starting to understand why I want to give up because we’ll just be kind of

doing this forever this might end up being an irrational number in binary I

don’t really know I haven’t gone that far but it’s definitely gonna take a long

time so I’m gonna give up and I’m just gonna say all right the final binary

number is gonna be 0.00100 and then just say maybe there’s some more stuff at

maybe there’s some more stuff at the end we’ll just call this a precision loss

you can see well maybe I should do one more number actually because it it’s kind of a mirror there’s

like two zeros and then a one and then two zeros on the other side let me multiply this by two

real fast just to show you without symmetry what would happen so I’m gonna multiply that by two

and it’s gonna be this number right here so that’s gonna be zero because it didn’t exceed

or meet one and so the number we have left over is just that okay so now

because there’s no symmetry you can see that the number is going to be point

zero zero one zero zero zero we go from top to bottom we don’t do any kind of

reversal like you might do with whole number binary conversion I don’t know

maybe it’ll be zero for a long long long long time and never or yeah they’ll

probably be some ones in there at least I don’t know if it’s ever going to

resolve you’ll know you’re finished when the

Here is a zero.

So again, I’m just going to give up on this because I don’t really know if it’s going to work.

Let’s do a number that I know is going to resolve.

I have this one prepared in advance.

So let’s do 0.84375.

Okay.

So how do we convert this?

Again, just multiply by two.

Every single time you meet or exceed one, then you’ll say that we have a one in the binary number.

binary number and then subtract one after that and if not we will say that

we have a zero in the binary position and then we won’t subtract one okay so

that number multiplied by two is going to be point one six eight seven five so

that’s definitely greater than or equal to one so I’m going to say we have a one

in the binary position and then the next number is going to be just be zero point

now we take that number 6875 you could also just you know in your calculator you could say minus

one and then we’ll multiply it by two to get the next bit the number is going to be 1.375 so i’m

going to write 1.375 here and then it’s greater than or equal to one so it’s going to be another

one here and then after we subtract one it’s going to be 375 no problem okay so let me just

And then multiply by 2 to get to the next bit position.

It’s going to be 0.75.

All right, so 0.75, that number is less than 1, so we have a 0 in that bit position.

We also don’t subtract anything because we had a 0 in that bit position.

Then we just multiply by 2 again.

Multiply by 2, now we have 1.5 because that’s greater than or equal to 1.

We have a 1 in that bit’s position, then we subtract 1.

one it’s just going to be 0.5 that should be pretty easy right because that’s that sounds

like there’s a one in a bit and then it’s just done after that so i’m going to do minus one

and then multiply by two again notice how that is exactly a one so it’s going to be 1.0

which means we have a one in the position because it is equal to or greater than one

and then uh well what we have left over is zero

Zero multiplied by two is going to be zero.

So that means if we kept doing this forever,

then all the numbers down here are just going to be zeros forever.

And I said in another video,

when you realize what’s going on with the zeros,

like where are they?

Are they on the left or the right?

When you’re converting a whole number from binary to decimal or decimal to binary,

the zeros would be on the left side.

So that’s why we would reverse

because if you just add infinity zeros on the left of a whole number,

you’re not changing the value.

number you’re not changing the value but if you added them to the right you’d be increasing the

value when it comes to the fractions the right side of the fraction won’t change the number for

example if we had you know one point and then some like random binary numbers if we just kept adding

zeros on the right side that’s not changing the value of the number because this is the fractional

part however if we started adding numbers on the left side then we would make the fractional part

smaller and smaller and smaller so that means the zeros have to be on the right side and you can see

and you can see here the right side is the bottom so that means we’re going to take these numbers

I’ll just put etc here just so you can see etc that means we’re going from top to bottom so the

final number is 0.11011 and then a bunch of zeros after that are just you know nothing

and that’s the final answer we now have 0.84375 is equal to binary that number and we’ll just say

that number and we’ll just say OB to to indicate that the following is a binary

number because again you could have a decimal number and then have like a

bunch of like ones and zeros in the fraction part I guess so to make sure

that the reader understands what they’re seeing you’ll say OB to indicate this is

a binary number on exams or quizzes if you’re out there doing this for an exam

or a quiz keep in mind you probably want to talk to professor to make sure that

ob is supposed to be part of your answer they might just want this it depends anyway so 0.11011

okay then let’s convert it back again let’s let’s convert this number to the original decimal number

there okay so how can we do that pretty much just start adding the fractions up so remember the

Remember, the first fraction that we see is going to be one half.

One half.

And the second fraction we see is going to be one fourth.

And then we just keep multiplying.

One fourth, one half.

Maybe I should write this in a notepad here.

We’ll see one half plus one fourth plus one eighth.

And some of these bits aren’t going to count.

Like, for instance, the one eighth, it’s not going to count towards the fraction.

But for me personally, if I forget to type one eighth,

I’m probably going to accidentally use it for the next position.

So I’m just going to type everything one at a time.

So let’s see, we have one, two, three, four, five, one, two, three, four.

So I got to do one more plus one 32nd and then multiply by the bits.

So say like, you know, one times that and then one times that and then zero times that

to basically, you know, cancel it out.

And then one times that and then one times that.

plug it all into the calculator

let’s see if I got that right

0.84375

so that’s how you convert

back from binary to decimal

pretty easy and then also

of course when you’re practicing this you want to try

with a bunch of different random numbers to start off with

you just want to be

careful again this process

could take forever if you have like

you know the wrong number that you

start with but I guess at least

at least when we’re converting from binary to decimal it’s going to end up resolving to something

uh it won’t necessarily be something that you can’t represent but you know if you start

up with a random number in decimal there’s a chance maybe it’s not going to work with binary

okay well you have to give up and call it a precision loss okay so then now let’s uh let’s

ask what if we had a complex number because we know how to do this now with just the fractional

now with just the fractional part but what if we had a number that was um a little bit more

difficult let’s say we have wait a minute you know for you know for 37 5.

oh i think i just accidentally did a totally different number than i wanted to on my plan

luckily i got the right answer okay

Anyway, what if we had a more complicated number?

Let’s see, 804, 6875.

Okay, so 804, 6875, but with a 45 in front of that.

Remember we said before that this is the whole part

and this is the fractional part,

and you just wanna do them separately

and then combine them afterwards.

So the whole part, let’s see, what is that gonna be?

I’m just gonna say that this is not a video

for whole number binary conversion.

This is just dealing with fractions.

just dealing with fractions. Find my other videos if you want to know how to convert the whole part,

but I’m going to say that 45 is just this number in binary. And you can double check real fast.

You can say 1 plus, here let’s do it again, let’s do it just to be sure. 1 plus not a 2 because

there’s a 0 there, and then we’ll say 4 plus 8 plus not a 16 because there’s a 0 there, and then

plus 32. Add that together, that’s 45. So the whole number part is pretty easy if you already know how

number part is pretty easy if you already know how binary conversion but now let’s convert point uh

zero eight zero point eight zero four whoops screenshot no thank you zero point eight oh four

six eight seven five so i’m gonna have to do this from scratch since this is a different number than

i just worked with we’ll do that multiplied by two and it’s going to be this number

is that really not the same number?

Yeah, I guess I really just did a different number.

Okay, so that means we’re going to have a 1 in the binary position

and the remaining fractional part is just going to be this.

Whoops, zero point that.

So then we take that number and bring it down.

Multiply it by 2.

Whoops, not supposed to actually bring that part down.

We’ll bring it into the calculator.

We’ll multiply it by 2.

actually let me get that again and subtract one just to double check myself it’s always good to

double check yourself then we’ll multiply this by two and then the number is going to be point

1.28 so we have that and then since it’s either equal to or greater than one it’s going to be

you know a one in the binary position so 0.21875 is going to be the remainder there

so I’m going to do minus 1 and then times 2 to make sure the calculator agrees with me whoops

21875 okay then I’ll do a multiply by 2 to get the next number 4 3 7 5

and that’s not equal to or greater than 1 so we’ll put a 0 there and we won’t subtract anything

so it’s just going to be 0.4375 again then we’ll multiply that by 2 to get the next number

times two it’s going to be 0.875 so 0.875 again this is not one or greater so we have zero in

the binary position and we don’t subtract anything 0.875 multiply by two again times two so it’s

going to be 1.75 1.75 and then of course we’ll have a one in the binary position and the leftover

0.75 so let me just go back to the calculator and I’ll say minus 1 times 2 it’ll give us 1.5

so 1.5 over here means we’ll have a 1 in the binary position because it was equal to or greater

than 1 the leftover is going to be 0.5 now you know we’re about to finish because 0.5 times 2

is just going to be 1.0 which will give us a 1 for the binary part and then the remainder is going

remainder is going to be 0.0 and then you know there’s no need to multiply 0 by 2 because it is

forever going to just be zeros uh maybe i’ll just write it out one more time just so you know 000

right remember that so that means of course the zeros are on the right side of the fractional

part and on the left side is going to be 0.110011

1100111 and then now we just have to combine those two numbers so maybe just I’ll put something

here indicating that this is the conversion of just that number like that now we’ll combine

both of those parts right so 45 was this so I’ll say maybe therefore this big number right here

combine this number for 45 and then I’ll just put a decimal point and then I’ll put the fractional

part on the right side and then let me just double check my work real fast it should be

10110111 wait wait what no no 101101 and then 1100111 okay so that’s it we’ve converted

converted a complicated decimal number into binary.

And we can do it in the reverse, just using the same thing.

You know, step one, convert this number into a whole number of decimal.

And then step two, convert this number into a whole number,

sorry, a just only fractional number for decimal

by just getting the one half plus one fourth and so forth.

put them around a decimal point just for practice okay I’ll go ahead and do it

some of you are probably like why won’t you do it I’ll do it okay so just only

looking at the fractional part because that this is not a video to convert

whole numbers in binary and decimal but just to convert the fractional part only

I’m gonna do let’s see say whoops whoops whoops whoops whoops whoops maybe I need

Okay, so starting with just this one right here,

it’s going to be 1 half plus 1 fourth plus not 1 eighth, not 1 16th, a 32.

1 32 plus 1 over 64 plus 1 over 128.

0.8046875.

So it looks like we succeeded.

Okay, so now we know how to convert

a decimal number with a fraction

into a binary number with a fraction.

And we also know how to convert

a binary number with a fraction

to a decimal number with a fraction.

That feels like a long video.

Let’s see what it is after I cut this.

Thank you so much for watching.

I hope you learned a little bit of stuff

and you had a little bit of fun.

I will see you in the next video.

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The post How to Convert Decimal Fractions to Binary (and Back!) – Easy Step-by-Step Guide appeared first on NeuralLantern.com.

Confused by hexadecimal numbers? Don’t worry! This fun, step-by-step guide makes converting hex to decimal super easy, even for beginners. With clear examples and a chill vibe, you’ll master this computer science skill in no time. Perfect for students, coders, or anyone curious about number systems. Hit subscribe for more coding tutorials, and check out our site for extra resources! #HexToDecimal #CodingForBeginners #ComputerScience #LearnToCode

Introduction to Hexadecimal Conversion 00:00:00
Purpose of Conversion 00:00:11
Number System Basics 00:00:39
Hexadecimal Explanation 00:01:12
Converting Hex to Decimal 00:01:59
First Example Setup 00:02:35
Decimal Place Value Recap 00:03:02
Hexadecimal Place Value 00:05:14
Formula for Conversion 00:05:43
Translating Hex Letters 00:08:17
First Example Calculation 00:09:51
Second Example Introduction 00:11:14
Second Example Conversion 00:11:58
Second Example Result 00:13:11
Conclusion and Call to Action 00:14:00

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Hello there. Let’s talk about converting hexadecimal to decimal.

So why would you do this? Of course, sometimes in computer science and other realms, you may be

faced with a number that looks like this, and it’s got some letters in there, and it’s weird,

and you’re thinking, oops, not a V. That wouldn’t work. Not a G. And you’re thinking like, what does

what does this number mean in decimal well this is a number that is real and we can just change

the way it’s represented uh to decimal so that we can understand what it is so just like a quick

recap if you haven’t watched my other videos yet uh in decimal we have a base 10 system which just

basically means we have these characters zero one two three four five six seven eight nine there

base is 10. In binary, of course, we have base two, which is just, you know, we have a zero and a one

because in the computer, which is why we learn binary, we just have on and off basically for

every single, you know, part of the computer. There’s just like a certain voltage or there’s

not a certain voltage and that’s it. And then for hexadecimal, which is a way to represent the same

numbers, but just in a more compact way. It’s a little bit more compact than decimal. It’s a lot

than binary we can say that this is base 16 which means we have 16 total characters that we can work

with zero one two three four five six seven eight nine and once we run out of numbers we just start

using letters so a b c d e f so there are 16 total characters if we include the zero which means we

So that’s the basics of you know the number basis let’s uh let’s work out how to convert a hex

number to decimal. So I’m going to say ox because in hex when you write down a hex number you should

usually put ox in front of it unless the program you’re working with doesn’t understand that but

usually especially for a human reader you should do it this way. So I’m going to just type like a

few random maybe I have to I have to keep the number small because hex will explode it’ll be

not careful. So I’m going to put like a I don’t know, maybe a few decimal numbers there or a few

like digits that are zero to nine, then I’ll just to make it more interesting, I’ll put some letters

in there. How about like an E there? And then I don’t know, like, we’ll do a B there. Okay, so

how many do I have? 12341234? How about we get rid of? Now, let’s go for it. This is going to be an

So this is our starting number hexadecimal. What we need to do is understand that, well,

a recap if you haven’t watched my other videos yet. In decimal we have, let’s say, 0, 1, 2, 3,

4, 5, 6. You know that the first digit has a strength of 1. You multiply 6 by 1 to understand

the real power of that 6. For the 5, you know that it has a power of 10 or a strength of 10

10 because it’s the next digit over you know to the left same thing for the four it’s like got a

strength of 100 and so every time you move to the left the strength of the digit increases by a

factor of 10. why does it increase by a factor of 10 because decimal is base 10 because we have 10

possible digits we can represent for each character okay so by the time we get to that

left, we have a strength of 100,000. So, you know, if you wanted to really understand 100,000,

if you really wanted to understand, you know, how powerful, let me, let me give some spacing here.

So this feels better. Okay. If you really wanted to understand like, you know, what is the strength?

What is like the actual value of this digit right here? You’re imagining this is like 100,000.

this digit here you’re imagining that this is two of ten thousand you know two times ten thousand

and so forth so uh uh just like another recap the first digit is really 10 to the zero power

like in terms of the strength because if you multiply um let me pin this to the top

if you multiply you know 10 to the something power and you start at zero with the most

the weakest digit 10 to the zero power gives a strength of one and then every time you move over

to the left you just increase the exponent there by one so 10 to the first power is going to be 10

that’s what that five has got so it’ll be five times 10 to give us an understanding of how you

know powerful that digit is move one over to the left it’s 10 to the second power so that’s 100

move one over to the left it’s 10 to the third power and so forth until we get to 10 to the fifth

1 times 10 to the fifth power to understand how powerful that is so we can do the same exact thing

with hexadecimal except instead of raising 10 to a power we raise 16 to a power because hexadecimal

is base 16 whereas decimal is base 10. so that means this e is is uh you know 10 to the or sorry

16 to the zero power and then that 3 is 16 to the 1 power and that f is 16 to the 2 power and so

16 to the 2 power and so forth so first what we should do is let’s write out a

formula which won’t work in a calculator because we’re going to put some letters

in there and then later we’ll translate the letters to actual values so we’ll

start off with the e we’ll say e times 16 to the something power it’s going to be

16 to the zero power okay so then the next digit is 3 so 3 times 16 to the

And then the next thing is going to be F times 16 to the second power.

And just as a reminder here, I’m starting with zero with the smallest digit.

And I’m just working my way up in steps of one.

So it’s zero power or raised to zero and raised to one and raised to two and just so forth all the way to the left.

That number will just keep increasing by one.

The exponent will increase by one.

times 16 to the third power do it again um one times 16 to the fourth power do it again

um d times 16 to the fourth power oh what did i do did i erase something

0, 1, 2, 3, 4, 5.

Oh, that was supposed to be a 5.

Okay, maybe it’s good that I thought I made a mistake

because that drew my attention to the bad exponent.

Okay, so after the D is a B.

B times 16 to the 6th power.

And then one more.

We’ll say 6 times 16 to the 7th power.

And then another one.

A times 16 to the, whoops, to the eighth power.

And another one.

One times 16 to the ninth power.

Ninth power.

Okay.

So just double check your work real fast.

You know, because I make typos all the time and I get things wrong all the time.

So just double check.

Zero, one, two, three, four, five, six, seven, eight, nine.

It’s sequential.

Double check the digits.

6 b d 1 3 f 3 e okay so i got that right i’m not going to erase my work because if i end up screwing

up the next part oh man is it going to be a hassle to correct so i’m just going to copy paste it here

and then i’m going to start translating the letters to numbers anytime you see a number here like one

time something it’s just one but every time you see a letter you have to translate that into

in decimal. Remember in hex, we’ll say, what can I do?

I can say A, B, C, D, E, F.

And I can say that the A is worth 10.

The B is worth 11.

The C is worth 12.

And the D is worth 13.

The E is worth 14 and the F is worth 15.

Well, maybe I should do the other numbers too.

five, six, seven, eight, nine,

just so we have a visual reminder

of like what we’re even looking at.

And I won’t write down what the digits are worth

because they’re worth themselves, right?

So like zero through nine, it’s just worth zero through nine.

So now that we have this little translation table up here,

anytime we see a letter,

we can just translate it very quickly to the decimal value.

So A is worth 10, we’ll put a 10 there.

Maybe I’ll add some spacing so that this continues to line up.

I see a B here, so the B is gonna be worth 11,

11 add another space so it lines up i’m running out of room but i’ll try the d is worth 13 add

another space so it lines up and then the f is worth 15 add another space so it lines up the e

is worth 14 so i’ll add another space so it lines up okay let me just double check my work here a b d

So now I’ve got like a big formula that I wrote out.

I can literally now, I mean, you can do this in your head if you’re like a crazy genius,

but I’m just going to paste this into a calculator and hit enter.

And this is the number that we had originally in hex.

Maybe I’ll put commas here to make things more fun.

I don’t know.

You don’t really need to do that, but I’m going to.

So for me, it’s easier to read.

this is uh like 113 trillion 478 no wait that’s a million and that’s a bill okay so 113 mil a

billion 478 million 25 022. let me punch up my personal calculator here to make sure that i’m

getting this right i’m not going to show this on the screen because i’ve just got this up

on my host machine bet you didn’t know i’m inside of vm right now surprised you didn’t know that

decimal and the number is supposed to be 113478025022 okay so we did this right we now know how

to convert from hexadecimal to decimal and it’s pretty awesome right okay let’s do another number

one that is not quite as hard let’s see how many digits do we have here one two three i think we

5, 6, 7, 8, 9, 10.

Yeah, okay, we had 10.

Let’s do a five-digit hex number.

Okay, maybe I’ll copy…

I’ll copy just this table at the top

since you don’t need it anymore,

but I’m going to need it to do my calculations.

So we’ll do 1, 2, 3, 4, 5,

and I’ll just start randomly changing

some of these numbers.

Like an 8 over here, and how about like a 2?

So I’ve got five numbers.

This is OX.

I’m going to say this is a hexadecimal number.

Kind of ambiguous if we tell the reader this is a hex number,

but then we put OX.

Kind of don’t really need to tell them that it’s hex

because OX tells you it’s hex.

It’s not even part of the value.

So let’s get on with it.

I’ll start by just doing 8 times 16 to the something power,

which is going to be 0 for that first position.

the power as we go to the left so it’s going to be 1 times 16 to the first power and then it’s

going to be f times 16 to the second power and then it’s going to be 2 times 16 to the third

power oh no my thoughts are wandering i think i’m getting bored of recording this video i’m starting

to the fourth power but honestly why couldn’t they end up together so and

then I think one two three four five okay so that’s five digits a to f one

eight a to f one eight all right zero one two three four just to make sure

that I got my exponents right copy paste it so I don’t have to repeat my work if

I get something wrong I’m gonna translate the letters into numbers so a

becomes 10 f becomes 15 and the other numbers are fine as is I can just copy

fine as is I can just copy paste this whole thing stick it into a calculator and now I know that

this number is actually 667 416 with a little comma in there don’t put commas if you are taking an

exam somewhere if you’re watching my video to help with your exam because most most likely the exam

that you’re taking will not accept a comma it’s not been pre-programmed for a comma I don’t know

If you start typing numbers and a comma just shows up, then it probably was programmed for a comma.

But don’t assume it might be a string match and not a numeric match.

So be careful out there.

Be careful.

So 667-416.

Let me punch this into my personal calculator just to make sure I got this right and I don’t have to issue an errata.

667-416.

All right.

We’ve done it.

We know how to convert hexadecimal numbers into decimal numbers.

Thank you so much for watching this video. I hope you enjoyed it and had a little bit of fun and

learned a little bit of stuff. I’ll see you in the next video. Hey everybody, thanks for watching

this video again from the bottom of my heart. I really appreciate it. I do hope you did learn

something and have some fun. If you could do me a please, a small little favor, could you please

subscribe and follow this channel or these videos or whatever it is you do on the current social

It would really mean the world to me and it’ll help make more videos and grow this community.

So we’ll be able to do more videos, longer videos, better videos, or just I’ll be able to keep making videos in general.

So please do me a kindness and subscribe.

You know, sometimes I’m sleeping in the middle of the night and I just wake up because I know somebody subscribed or followed.

It just wakes me up and I get filled with joy.

That’s exactly what happens every single time.

So you could do it as a nice favor to me or you could troll me if you want to just wake me up in the middle of the night.

Just subscribe.

just wake me up in the middle of the night just subscribe and then I’ll just

wake up I promise that’s what will happen also if you look at the middle of

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to the website which I think is also named somewhere at the bottom of this

video and it’ll take you to my main website where you can just kind of like

see all the videos I published and the services and tutorials and things that I

for clarifications or errata or just future videos that you want to see please leave a comment or if

you just want to say hey what’s up what’s going on you know just send me a comment whatever I also

wake up for those in the middle of the night I get I wake up in a cold sweat and I’m like

it would really it really mean the world to me I would really appreciate it so again thank you so

much for watching this video and enjoy the cool music as as I fade into the darkness which is

the darkness which is coming for us all.

Thank you.

The post Hex to Decimal Made Easy: Fun & Simple Conversion Guide! appeared first on NeuralLantern.com.

Binary got you baffled? Let’s break it down! In this fun crash course, I’ll show you how to convert binary to decimal step-by-step—super easy, no stress. We’ll go from 765,432 in decimal to 10101111 in binary (that’s 335, btw), with tricks to eyeball it fast. Perfect for beginners or anyone who loves a good tech challenge. Hit subscribe—I wake up in a cold sweat when you do, and it means the world! More vids coming your way!

Introduction to Binary and Decimal 00:00:00
Understanding Decimal System 00:00:44
Decimal Position Strengths 00:01:39
Decimal Formula Explanation 00:03:48
Transition to Binary System 00:06:31
Binary Position Strengths 00:07:19
Binary Formula Breakdown 00:10:46
Calculating Binary Example (335) 00:12:55
Quick Binary Conversion Trick 00:15:24
Memorizing Binary Positions 00:13:52
Small Binary Example (19) 00:15:43
Closing and Subscription Request 00:16:48

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Hello there. Let’s talk about converting binary to decimal.

Just a quick recap for those of you who don’t know. Watch my other videos if you’re not sure

about number bases, but basically, you know, decimal is a number system that has 10 characters

to represent a number. So 0, 1, 2, 4, 5, 6, 7, 8, 9. That’s 10 characters in decimal.

In binary, we just have two characters.

So it’s base two, whereas decimal is base 10.

How can we convert between the two?

Well, first, before we can try to convert between the two,

we should try to understand what’s really going on with normal decimal representations.

So if I have a decimal number here, and I say, I don’t know,

let’s just type a couple of random numbers.

How do we really know what this value is?

How do we kind of understand what is the meaning?

Okay, so you know that, you know,

if you just kind of look at this for a second,

you’ll realize this is 765,000, right?

765,432.

How do we know that this is 700,000?

Because it’s in a certain position.

It’s like, you know, several digits to the left.

How do we know that the next one represents 65,000?

Well, it’s one less to the left.

uh it’s a tenth of the like power of the seven digit same thing for the five how do we know

that’s a thousand same thing for the four and the three and the two what we really do is we end up

deciding okay you know what the uh the first digit here on the right side that two uh the strength of

of one. So like, you know, two times one is just two. So whatever, whatever that digit is, if it

was a five, it would just be a five, five times one is five, right? So the strength of the very

first digit on the right side is just one. Every time we move one digit to the left, we multiply

the strength by 10. The reason we multiply the strength by 10 is because there are 10 possible

digits, or 10 possible characters that we can use in decimal. Makes sense, right? So that means the

Makes sense, right?

So that means the three, we know that represents, you know, an addition of 30 because that’s

the 10th strength place.

Same thing for the four.

We multiply 10 by 10 again to get 100 in that position.

So this one has a strength of 100.

I’m writing this vertically.

Hopefully this is not too messy, but I’m hoping you’ll understand.

Well, I don’t know.

Maybe if I space this out, does that make it better or worse?

maybe it makes it slightly better i don’t know i’m going to roll with it

what can i say i’m a rebel okay so i live dangerously the five we know that’s worth

five thousand because again the four position over here you know that third digit it had a

strength of 100 so we multiply that strength by 10 going to the left to to end up with a thousand

to end up with a thousand.

So same thing with the six.

I’m not going to explain that any further.

You know, it’s got a strength of, you know, 10,000.

Okay.

And then the seven, it’s got a strength of 100,000.

And it just keeps going and going and going to millions and billions and trillions

and however far you actually want to go.

That’s how we understand the decimal numbers

that we’re looking at.

How could you imagine this in terms of a formula?

Well, we’re really raising something to the 10th power.

If you think about this, the one,

we could imagine that the value one

is actually 10 to the zero power

because anything raised to the zero power is just a one.

Let me pull up this calculator real fast.

I got to stick that on my desktop in the future.

If we say 10 to the zero power, it’s just one.

Any number to the zero power is just one.

Okay, so then we would multiply that strength

by the actual digit that we see,

the character that we see.

It’s a two.

Okay, so that’s going to be two times 10 to the zero power.

I’m going to put that in parentheses

to keep it a little bit clean or somewhat cleaner.

So now when we go one to the left

to look at that three digit,

to look at that three digit, we end up with three times something.

Let me get a space here for my brain.

Times 10 to the something power.

Well, every time we go to the left, we just really have to increase the power by one.

It’s 10 to the first power because we’re looking for actually just the number 10.

So if we say 10 to the first power on that calculator, it’s 10.

So then, you know, just keep repeating the process.

10 to the second power. And that should be 100. So if I say 10 to the second power, it’s 100.

All right. And, you know, we’re adding all the positions together, even though we’re multiplying

the digit by its strength. So I’m going to say five times 10 to the third power. And then again,

just to double check here, 10 to the third power is 1000. So you can see the five was supposed to

and then again we’ll say six times ten to the fourth power and then just double check here

the fourth power should be a thousand sorry ten thousand and then we’ll do it one last time we’ll

say seven times ten to the fifth power which should be a hundred thousand all right so now

that we’ve written this all out you know this is kind of madness right what you can do now

do now is you can put this into the calculator and it should give you the exact same number that

we started with. We should see 765432, unless there’s a typo. 765432. Nice. Okay, that might

have seemed like it was a little bit of a waste of time, but it’s not because now we kind of

understand the breakdown of the different positions of the digits in decimal, which means

now we can do the same thing in binary, basically the exact same thing, except just use a power of,

use a two to some power rather than a 10 to some power because the only reason we use 10 down here

it says we were in decimal that has a base of 10 now let’s go into binary

which is um a base of two so i’m going to just i guess maybe what did i do wrong

i hit something and it like did a space i don’t even want to know i don’t even want to know

okay so let’s do a binary number i’m just going to type a few random digits

a few random digits. I don’t know what number this is yet, but let’s work it out slowly

in the same way. You know what, maybe instead of doing the formula first and only, let’s do

both parts like we did with decimal. So what’s the position, what’s the strength of the position

for that first character? Well, I said before the first character is always just going to be,

sorry, the first digit is always just going to be a one, right? That’s going to be true

So I’m going to say this has a strength of 1.

How do we know it’s a strength of 1?

Because we’ll take 2 to the something power.

We start at 0 for that first position, and it’s going to give us a 1.

Okay.

So the strength of this one, I already know binary.

So I just know off the top of my head that to go to the left,

the strength just multiplies by 2.

And that’s pretty easy after you start memorizing it.

I haven’t quite memorized hexadecimal yet, but maybe you will in another video.

So I’m going to multiply one by two and I’m going to end up with two.

Double check over here in the calculator.

Two to the first power is two.

So then the strength of this next digit should be four.

Two times two is four, right?

So let’s do two to the second power.

That’s four.

Multiply by two again.

It’s going to be eight.

Double check over here.

Two to the third power is eight.

The next digit is going to be 16 of its strength.

So I’m going to write this vertically again.

So it’s going to be 16.

Double check over here with the calculator.

Double check over here with the calculator, 2 to the 4th power is 16.

Next digit is going to be 32.

Double check with the calculator, 2 to the 5th power.

And things are starting to get messy, so I think I’m going to like space everything out probably.

Just to make it easier to read.

Tell me if you think this makes it actually easier to read or if I’m making it way harder.

I think I’m making it easier.

Okay, so we got 32.

two. Next one up is going to be 64. Just multiply it by two. Double check two to the sixth power.

That’s going to be 64. And then the biggest one that we’ve written down is going to be 128.

Double check it. Six to the seventh power is 128. We could go on and on and on, but I’m just going

to leave it here because we, I hope we have a pretty good idea of, you know, what this means.

delineation or like a delimiter showing us that these are just representing the strengths and

this is the actual number. Okay, so how can we write this out in a formula?

Whoa, what did I do wrong? Did you see that? Oh no. Hang on a second.

I think I missed it. How many digits are there? If there are eight digits, then I definitely forgot

something. No, no. Okay. There are nine digits, so the last one should be 256. Okay.

256. Okay. So I got it all lined up. At some point I must have not lined it up. My apologies,

but hey, maybe I’m making these mistakes on purpose to make sure that you’re paying attention.

You never know. I want you to think. So 256 is going to be the next number. Double check it with

the calculator. Two to the eighth power, 256. Cool. By the way, a quick trick in binary that

the actual highest number that you can represent in an unsigned binary integer is basically the

strength of the highest digit, you know, this 256 here, multiplied by two and then subtract one

from it. So 256 multiplied by two is going to be 512. So it’s going to be 511. So I could

represent a number between zero and 511 or 512 possible combinations. Okay, so now let’s work

let’s work out the formula.

See 16, 30, 16, 40, okay, I did it okay.

I probably should have rehearsed this.

So let’s do each position.

So it’s either always gonna be one times something

or zero times something, right?

Because binary, these characters can only be one or a zero.

So let’s do on the, starting from the right,

we’ll say one times two to the something power.

It’s gonna be two to the zero power

You know, just going to be a one.

Working our way over to the left, it’s going to be one times two to the something power

to the first power because it just increases every time the power increases.

We have four ones in a row here.

I got to try to remember that.

This is where I’m going to start making lots and lots of typos.

Two to the second power.

And then we have another one.

One, two, three, four, one times two to the third power.

And then again, we are going to hit a zero.

So it’s going to be zero times two to the something power.

You might be tempted to omit the zeros.

You can if you want to.

But for me personally, it helps me quickly visually see that I’m getting the powers in

the right order.

I can see two to the zero power, first power, second power, third power, fourth power.

Sometimes when I omit the zeros, I end up kind of like messing up the order of the powers

and or the order of the exponents.

of the exponents and I have to redo everything all over again.

So I just keep it this way.

Okay, so it’s one, one, one, one, one, one, one, one, zero.

So there’s another zero that we need

times two to the fifth power.

So we got both of those zeros now.

And then we need another one times two to the sixth power.

And then we need

0 times 2 to the 7th power.

Okay.

And then we have another 1 times 2 to the 8th power.

And I know we’re supposed to be done on 8

because that’s what we were doing before.

The 256 strength.

So unless I made some mistakes here,

this is probably the number that we can punch up into the calculator

to see what this binary number is.

So I’m going to punch it up.

Huge.

It says that it’s the number 335.

Let’s see if that’s actually right.

I’m going to punch this up in my personal calculator real fast.

I’m going to say 10101111.

And the expression is decimal 335.

Yep.

So that’s it.

We know how to convert from binary to decimal.

And just again, like as a quick shorthand,

it’s probably a good idea if you’re involved in computer science,

to memorize these positions up to maybe

6, 5, 5, 3, 6.

That might sound a little extreme sometimes,

but I don’t know.

Personally, I’m not like the most advanced

binary reader at all times,

but I can remember up to that much.

And what do I mean when I’m saying that?

I’m saying, you know, start with a 1, 2, 4, 8,

8, 2, 56, 5, 12, 1, 2, 4, 2, 0, 4, 8, 4, 0, 9, 6, 8, 1, 9, 2, 1, 6, 3, 8, 4.

Took me a while to remember that one.

3, 2, 7, 6, 8, 6, 5, 5, 3, 6.

So if you think about it, how many bits is this?

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.

Right here, 16.

That’ll give you two bytes worth of memorization or a word on many systems.

on many systems. For me personally, I just, I like to go up to 6, 5, 5, 3, 5 because

remember I was saying, you know, what is the highest number that you can go up to

with a certain number of bits? It’s always the power or the strength of the highest bit

multiplied by two and then subtract one. So when I remember 6, 5, 5, 3, 6, that tells me that if I

65535. This is very, very useful if you’re going to be working with binary numbers a lot,

or if you’re taking exams or whatever it is that you’re doing, I would recommend everyone do this.

Okay. What do you say? What do you say we do another binary number, but we’ll just do something

a little bit smaller to make it easy. Okay. So by the way, when you start to memorize these positions,

start to eyeball it, which is really, really fast and a good idea. Like I can see those two

numbers right there. I know the first one is a one and the second one is a two. So that means the

one and the one are just going to be a value of three. Then I just quickly go, all right, one,

two, four, eight, one, two, four, eight, 16. So it’s going to be 16 plus three. So it’s going to

be 19. I guarantee it. Let’s, let’s double check this real fast. We’ll say one times two to the

and then we’ll say 1 times 2 to the first power

and then we’ll say 0 times 2 to the second power

and then we’ll say 0 times 2 to the third power

and then we’ll say 1 times 2 to the fourth power

and if I didn’t go too fast and make a bunch of typos

it should be the number 19

So there is a lot of benefit in memorizing the strength of these different positions.

I personally never remember very much beyond 256 when I’m actually trying to work out a number

conversion, but when I’m just thinking of how to compute things, it’s faster if I can go up to

65536. Okay, I hope you enjoyed this video. I hope you learned a little bit of stuff. I hope

you had a little bit of fun. I’ll see you in the next one.

Hey everybody! Thanks for watching this video again from the bottom of my heart.

I really appreciate it. I do hope you did learn something and have some fun.

If you could do me a please, a small little favor, could you please subscribe

and follow this channel or these videos or whatever it is you do on the current

social media website that you’re looking at right now. It would really mean the

world to me and it’ll help make more videos and grow this community so we’ll

be able to do more videos, longer videos, better videos or just I’ll be able to

just I’ll be able to keep making videos in general so please do do me a kindness and uh and subscribe

you know sometimes I’m sleeping in the middle of the night and I just wake up because I know

somebody subscribed or followed it just wakes me up and I get filled with joy that’s exactly what

happens every single time so you could do it as a nice favor to me or you could you could troll me

if you want to just wake me up in the middle of the night just subscribe and then I’ll I’ll just

wake up I promise that’s what will happen also uh if you look at the middle of the screen right now

screen right now you should see a QR code which you can scan in order to go to the website which

I think is also named somewhere at the bottom of this video and it’ll take you to my main website

where you can just kind of like see all the videos I published and the services and tutorials and

things that I offer and all that good stuff and if you have a suggestion for clarifications or

errata or just future videos that you want to see please leave a comment or if you just want to say

what’s going on? You know, just send me a comment, whatever. I also wake up for those in the middle

of the night. I get, I wake up in a cold sweat and I’m like, it would really, it really mean the

world to me. I would really appreciate it. So again, thank you so much for watching this video

and enjoy the cool music as, as I fade into the darkness, which is coming for us all.

Thank you.

The post Binary to Decimal Made Easy: Fun Crash Course for Beginners appeared first on NeuralLantern.com.