Want to conquer negative numbers in binary? This fun, beginner-friendly tutorial breaks down two’s complement with step-by-step examples (-109, -29) and shows how to convert, pad, and subtract in binary. Learn sign bits, carry bits, and avoid common mistakes. Subscribe for more coding tips and tech tutorials that make learning a blast! Scan the QR code for more resources and join our community!

Introduction to Two’s Complement 00:00:00
Signed vs. Unsigned Integers 00:00:28
Sign Bit Explanation 00:01:55
Positive and Negative Representation 00:02:06
Range of Signed Integers 00:02:48
Padding Signed Integers 00:05:36
Converting to Negative (Example: -109) 00:07:00
Binary Addition and Carry Bits 00:10:16
Correcting Conversion Mistakes 00:16:38
Converting Negative 29 00:13:32
Subtraction Using Two’s Complement 00:18:21
Adding Binary Numbers (109 – 29) 00:20:41
Verifying Results 00:23:56
Conclusion and Call to Action 00:25:40

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Hey there! Let’s talk about representing negative numbers in binary.

We’re going to use a system called 2’s complement which is going to allow us to represent signed integers,

which means there will be a positive sign or a negative sign on the integer

and we’ll still be able to do it in pure binary.

So let me talk a little bit about what I mean first here.

signed and unsigned okay so if i just type the number 14 by itself that is uh that’s an unsigned

integer we don’t really know if it’s positive or negative like do we know that it’s negative 14 or

do we know that it’s positive 14 right so the sign is what lets us know if something is positive or

negative okay so if i if i type a number in binary let’s just do a pure binary number one two three

if you know how to convert binary to decimal you probably recognize right away that this is just

the number zero even if we add a couple of bits here that’s just like the number three

and so this is the number three but so far if if you haven’t learned signed integers in twos

complement or in binary then you don’t really know that there’s a sign you just kind of assume

that the number is positive by default if you’re not using twos complement and you’re just saying

and you’re just saying like well let’s just do a bunch of binary you know digits then yeah it’s a

safe assumption that the the sign is positive but we’ll use two’s complement which starts with the

idea that the highest bit the leftmost bit the bit with the height you know the most power that’s

going to turn into the sign that’s that’s going to turn into the plus or the minus so like you

the sign.

So this means that we have to decide, you know, does a zero mean positive or does it

mean negative or what means what?

So in two’s complement, we’ll say that zero is positive and one is negative.

So you can see right here that we’re still actually representing the number three, but

it’s positive three for sure because this sign bit right here is zero.

That’s positive.

Once we decide to represent two’s complement, then we can say that for sure.

On the other hand, if we put a one here,

then we definitely know that this number is now negative.

However, we don’t actually know that it’s a three

because the numbers don’t work out the same anymore

in two’s complement.

So the positive numbers, they will mostly look the same.

The negative numbers, they’ll look a lot different,

but they’ll still be valid in order,

you know, in terms of us being able to add them together,

subtract them from each other and things like that.

So two’s complement is pretty cool.

Let’s do, let’s see, what else can I tell you?

Oh, one thing to understand is that

One thing to understand is that in an 8-bit unsigned integer in binary, let’s say 1, 2,

3, 4, 5, 6, 7, 8, let’s say unsigned bits, maybe I’ll just put like, maybe just some

like V for value, all the bits are values, that means the range is 0 to 255 with 256

total combinations.

But if we want to use a signed number, then we’ll actually have the signed bit be the first bit,

and all the rest will be value bits.

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

In this case, the range kind of goes down.

Because if you think about it, every single bit, you know, represents like, you know,

it helps towards the highest possible value that you can represent.

If we’re using the leftmost bit, then for an 8-bit integer, that’s 128.

integer that’s 128 that’s a value of 128 so we lose a lot off of the maximum

integer that we can represent so the range here is I think it’s negative 20

128 to positive 127 just keep in mind you’ll have to trust me in terms of why

is the negative 128 bigger and the positive 127 you know smaller but that’s

just the way it is so that means we only get these value bits here and if you if

that um let’s see if i can do this quickly enough without screwing it up we’ll say 127 minus um

minus negative 128 i guess that’s 255 and then also a zero but uh in terms of positive and

negative the way this is going to work out is the zero will probably show up twice because uh

255 possible combinations with seven bits.

Normally you would have a range from negative, sorry, from zero to positive 255.

And in this case, we’re just, you know, losing our range, but we can represent negative numbers.

Anyway. Okay.

So let’s talk about doing an example real fast. Let’s see.

I’ve got like a little notes to tell me what I should do. Okay.

It’s important to understand that normally when you have, let’s say, let’s say you have

like an 8-bit number and you go 1, 2, 3, 4, 5, 6, 7, 8.

And then you have 2 bits right there and so this is like, you know, positive 3.

If you wanted to copy that number into more bytes, like for example, if you wanted to

take a 2-byte integer and have it copy the value of a 1-byte integer, then it’s pretty

easy.

zeros to the left you’ll say one two three four five six seven eight that would work perfectly

however if you did this with a negative number let’s say that we have a negative number

I’m going to put some of the random patterns so that you don’t think it’s the three let’s see

one two three four five six seven eight that’s eight total so I’m going to make that a one and

then get rid of that so we have like eight negative number if we were going to copy that

integer then we would copy paste it to start but then you’d have to pad with

ones to the left one two three four five six seven eight same negative number

more bits so be very careful about how you pad if you’re padding an unsigned

integer then yeah you’ll always pad zeros to the left no matter what is

happening but if you’re padding a signed integer then you have to pad differently

signed integer you have to pad with whatever the highest bit is so in this case the highest bit was

a zero so we do pad with zeros but then in this case the highest bit was a one so we have to pad

with ones if you don’t do that you’re going to end up with a number that doesn’t actually make

sense okay so now let’s work on actually converting a number to a negative number or representing a

negative number in twos compliment okay so i’m going to write twos compliment here and then uh

Let’s start off with the number negative 109.

Okay, how do we do this?

The first thing is convert it to its positive form.

Take the absolute value. Okay.

So we’re really just, you know, take positive form here,

and that’s just going to be positive 109.

Okay, no problem.

I’ll say start with negative 109, and then we’ll take the positive form 109.

Now we’ll convert it to binary.

Okay, this is not a video that teaches you how to convert to binary, so I’m just going

to try to do this in my head real fast here.

It’s going to be, let’s see, we got 8 bits, we’ll use just 8 bits to store the number,

and because it’s low enough and I don’t want to use that many bits, so it’s odd, so I can

add a 1 there, but maybe for now I should find the highest bit that is less than the

actual number.

So this is the 128 bit right here. So that’s going to be a zero. This is the 64 bits

I’m going to put a one there and I’m just going to say

Mmm, maybe like

This is like maybe not the smartest way to do it. I’ll say 109 minus 64

Because I put a one there and then 45 so now this is 32 that’s less than 45. So I’ll put a one there and

Then it’s going to be minus 40

So it goes from 45 to 13.

So then that was the 32.

Okay, so 128, 64, 32, and then 16.

Is 16 less than 13?

No, it’s not.

So we’ll put a zero here.

And then the next one is going to be just four.

Four is definitely less.

So I’ll put a one bit there.

And then I’ll subtract four.

And then one, two, four.

Oh, wait a minute.

One, two, four, eight.

wait a minute one two four eight sorry that was supposed to be subtracted uh subtracting eight

because that was the eight bit uh then i want to get five so here is the four bit so i put a one

there and then i’ll just put a one here because four plus five is equal to uh i’m sorry four plus

one is equal to five so let’s see um it is let me just double check here one

one let me just double check my conversion real fast it’s going to be

the one bit plus two four plus eight six thirty two plus thirty two plus sixty

four did I get one on nine yeah okay so I guess I did it right so convert to

binary and that’s going to be this the next thing we’ll do is we will invert the

bits I made a little edit jump here because I inverted the bits incorrectly

on the first try which is sad but hey it happens the next thing we’ll do is

but hey it happens the next thing we’ll do is we’ll invert the bits so basically you know we

take this original sequence here and i’ll just turn every single bit uh i’ll flip it i’ll flip

ones to zeros and zeros to ones so i’m going to go one zero zero one zero zero one zero okay so now

we have this number invert the bits then we just have to add one so we’ll add positive one to that

that it’s just going to be a one there at the end.

But sometimes that might not be the case

because what if we already had a one there

and we had to add one to that?

Well, we’d add one to the right side

and it would turn into a zero

and then it would carry a bit to the left.

That would become a zero

and then the carry bit would show up all the way over there.

So I just want you to be aware of the fact that

when you add two binary numbers together,

you have to be careful.

You have to add them the same way you would add decimal numbers.

For each digit’s position,

you have to add the two numbers together and then if they overflow then you just

kind of wrap around to the lowest number again subtracting you know the highest

value or sorry subtracting the base like in decimal if you add 9 and 9 the answer

is 18 but you’re not going to write 18 in that one position you’re going to

subtract the base which is 10 so it’ll actually be 8 and then you’ll carry the

plus nine is equal to eight carry the one right so we’ll do the same thing in binary we’ll say

if we ever get a one plus one when we’re adding the answer will be two but then we subtract the

base which is two so the answer is actually zero carry a one bit keep that in mind i’m just gonna

write it all out for you so you can kind of get a little bit of practice it’s important to to

practice this because it’s easy to get wrong i’m going to put a bunch of dashes up at the top to

and I’m going to say we’re going to add, you know, one number plus another number,

put a little plus symbol over there just to try and make sure we do it the right way.

Okay, so how do we add these?

Well, we’ll just go to the right side.

Zero plus one is one, no carry bit.

One plus zero is one, no carry bit.

Then we got a couple zeros here, no carry bit, of course.

One plus zero is one, no carry bit.

And then a couple zeros and then another one with no carry bit.

Okay, so, you know, we could have done that pretty easily, but

but well now we’re getting a taste for binary addition that might be harder later.

So we’ll just do that for now.

And now this is the two’s complement representation of negative 109.

I’ll say now we have negative 109 is equal to that.

leftmost number the most powerful sorry leftmost bit the most powerful bit is a one remember one

always indicates negative so when you’re looking at it if your leftmost bit turns out to be a zero

then you probably did something wrong or you had an overflow maybe um and again if we were going

to try to you know send this number into a two byte number or an eight byte number or whatever

uh then we would just have to pad with the sign bit so one two three four five six seven eight

help my brain so this was this is the way it would look in a two-byte number this is the way it would

look in a three-byte a four-byte and five six seven how many one two three four five six seven

okay one more this is what it would look like as an eight-byte number or a quad word 64-bit number

yeah okay so now we know how to do negative 109 okay so now let’s do a number that’s a little bit

negative 29. So let’s say convert, or how about represent negative 29 into

2’s complement. Okay. So first we, you know, first get the absolute value.

So just 29. And then we have to invert the, sorry, we have to get that into binary. So

next convert to binary. And I’m going to start with zeros, one, two, three, four, five, six,

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

For this video, remember, we’re choosing to use one byte integers.

But if you wanted to do a bigger one or you had to do a bigger one,

then just, you know, keep that in mind.

Okay, so 128 is not smaller than 29.

64 is not.

32 is not.

16 is, though.

So I’ll put a 16 bit there.

And I’ll just subtract 16 from 29.

29 minus 16.

Now we’ve got 13 left.

So 64, 32, 16.

six four thirty two sixteen eight okay so now I’m gonna put a one bit there and

I’m gonna subtract 8 from the remainder and then we got a five which is pretty

easy to do eight four and then a one so now we have well zero zero zero one one

one zero one that’s the binary number the positive or unsigned representation

representation. So now we’ll add one. Positive one. And let’s try to do this the right way so

that we can practice carry bits with addition a little bit. Notice how this one is already there

on the right side. So it’s going to, we’re going to have at least one carry bit for sure.

So then I’m going to go doop like that. And then I’m going to say that we have like, you know,

what is the result? Put a positive sign there. And then I’m going to put a bunch of dashes

my carry bits because I can I can forget that pretty easily. So the first thing is we add one

and one. The answer is two, but we can’t put the number two here because it’s binary. Instead we

need to subtract the base which is two. So two minus two is equal to zero, but then we have a

carry bit of one. So I’m going to put a one there. The first carry bit will stay as a dash for this

whole you know exercise because you’re not going to carry on to the first digit. So now we have

what would have been just zero plus zero now we have one plus zero plus zero so

that means this is going to be one and then the carry bit is just going to be

zero because we don’t actually carry anything so then we have zero plus one

plus zero so that’s going to be a one and then zero carry bit and then zero

one zero is just going to be one and then there’s going to be no carry bit

and then zero one zero again a one no carry bit because we didn’t actually

zero zero zero and then I’m just gonna put zeros here okay so now we have

successfully added I blew it totally blew it I always forget steps don’t

forget the steps this is a good lesson I’m gonna leave this in the video

because I want you to see that everybody makes mistakes and you got to practice

practice practice especially before you have to actually do this in real life or

or something like that.

Next, convert to binary.

Before you add one, I’m gonna just remove this.

Oh my gosh.

Next, flip the bits,

which is gonna be 1110010.

Okay.

So now we take this bit flipped number

and we will add one.

So that’s gonna be zero, zero, zero, zero, zero, zero, zero,

one, oh, it’s too easy.

Maybe I got excited and I thought,

excited and I thought, oh, it’s carry bit time.

But even though the last edition that I did was wrong because I forgot to carry

the or flip the bits, you still at least saw a little bit about how to carry the bits.

Right. OK, so it’s just going to be one.

Let me start from the right side.

One one zero zero zero one one one one one one.

Let me just double check here.

One one one zero zero zero.

OK, so I got that.

Now we have.

negative 29 in twos compliment. Again notice if we actually tried to add those numbers up to be

like an unsigned binary number they’re not really going to make sense because this is like 64 plus

32 plus 3. So what would that end up being? Let’s just double check here. 64 plus 32 plus 3, 99.

That’s not actually the number but the number is 29. So keep in mind you can’t just look at this

unless you’re like really, really practiced.

Okay, and again, notice that the number is 1 at the very left,

indicating that it’s a negative number.

Now, let’s look at how to subtract one number from another

using 2’s complement.

Okay, so what I want to do is I want to subtract,

let’s say, 29 from 109.

Okay, so let’s subtract, and I’ll just say 109 minus 29.

109 minus 29.

And how I would do that is basically I’ll start by just taking 109.

Let’s see, convert to binary.

So I’m just going to copy paste that number.

So 109 is this and then 29, take the positive version.

It is just, let’s see, before we flip the bits or anything,

let me make sure that I grab the right one.

Invert the bits, okay.

So positive 29 is this number.

Whoops.

is this number whoops let me put parentheses around that so it’s easy to tell and then I’ll

put a positive sign there like that maybe like that nope nope nope nope how about this okay

so now we have both of these numbers in positive form so now if we added 109 plus 29 that wouldn’t

negative positive 29 that would be what we wanted right because really if you’re subtracting

i’ll say aka

positive 29 plus negative 29 all we really need to do is um invert the 29 and then add the result

to 109 so that means we’ll we’ll turn positive 29 into negative 29 using two’s complement

through the steps again but basically put it there but basically you know that’s negative 29

so say positive 129 is equal to what i just put up here and then negative 29 is equal to

maybe i should do the parentheses again for clarity uh is equal to this okay so you can tell

that positive 29 is pretty pretty different from negative 29 but now we have

29 but now we have both of those numbers so let’s see 0 1 1 0 and then we’re ready to add okay all

we got to do is add them together next add them together maybe I should write the steps up here

negative 29 using twos complement and then next add them together so then I’m

going to copy paste the bits here and it’s going to be this plus this do a

positive plus sign just to remind ourselves that we are actually adding

and then I’m going to put a bunch of placeholders for sign bits up at the

top and now we’ll have a little bit more fun adding numbers together maybe I’ll

drag this down

Oh my god. Oh, there we go. Okay. So I’ll start with the one on the right, the position on the

right. That’s going to be one plus one equals two, but then that’s an overflow. So I’m going to

subtract the base. So it’s going to be zero. And then don’t forget to carry the one. Oh,

cool. More interesting. So we have one plus zero plus one. That’s going to be another two carry

two carry the one so it’s going to be zero and then carry the one again so i’m going to put the

carry bit up there and then again we have one plus one is equal to two so it’s going to be zero

carry the one again zero carry the one and then finally we don’t really have a carry bit

um so we’ll just have like a one and there’s there’s no carry so it’s going to be you know

carry a zero and then we add these two together so it’s going to be a zero carry the one and then

Now we have a three.

Oh, that’s kind of nice.

So this is an interesting edge case kind of.

One plus one plus one is three.

But if we subtract two, the base from it, you know, three minus two, it’s going to be one, not zero.

So it actually is going to be a one and then carry the one on top of that.

Then for here, let me space this over a little bit so that I can illustrate what’s going on a little bit better.

We’re going to have one plus zero plus one.

definitely going to be zero and then carry the one but there’s no bit where that carried one can

can show up on right so that one overflows it falls off the edge if this was a bigger number

then okay we you know if we had more bits to this number then sure we would just keep carrying over

over to the left but remember we said before that when we have a very big number let’s see

bits it’s just ones all the way to the side that will actually help us make sure that if our final

number is actually going to end up being positive that everything kind of like dominoes like carry

the one carry the one carry the one carry the one carry the one all the way until one of the ones

falls off think about it so anyway this one just is gone we don’t really care about it anymore

the result is going to be just only eight bits because that’s the number that we started with

bunch of zeros what is you know the final answer let’s just compute this

real fast to decimal so this is 128 and then 64 so it’s gonna be 64 plus not 32

but 16 so 64 plus 16 that’s gonna be oops 16 that’s gonna be 80 and now we

just have to ask ourselves again as like a final step to double check yourself to

what you’re doing and that you got it right is just punch up 109 minus 29 just to make sure

109 minus 29 whoops what happened here 109 minus

oh i think i stole my subtraction key for the annotator 109 minus 29 is 80

so again you know if you’re if you’re trying to like you know write something down to do some

you know taking an exam or something you definitely want to double check yourself in several ways

As you can tell from this video alone

I got one of these things wrong because I forgot to input the bits as a step before adding one

So you know your final step should be actually trying to add two numbers together or subtract numbers or whatever

You’re doing to make sure that you got the binary correct

So let’s see

Hmm

I guess maybe your first indication that the result was going to be positive would be that

there’s a zero there. And just, you know, as a sanity check, you look at the top and you’re like,

well, I was going to subtract a small number from a larger number. So the result should probably be

positive, right? Like 29 is like way lower than 109. So it should be positive, which means the

final result should have a zero at that leftmost position. Okay, so that’s two’s compliment,

two’s complement how to convert numbers from positive to negative in two’s complement you

know what the sign bit means and all that stuff and how to perform subtraction via two’s complement

i hope you enjoyed this video thank you for watching i hope you learned a little bit of

stuff and had a little bit of fun 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

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The post Master Two’s Complement: Convert & Subtract Negative Binary Numbers appeared first on NeuralLantern.com.

Ready to conquer binary scientific notation? Let’s make it fun and simple! In this video, I walk you through how to represent binary numbers with fractions in scientific notation—a must-know skill for computer science, programming, and understanding IEEE 754 floating-point representation. We start with the basics of scientific notation in decimal (think 8.54 × 10⁵), then dive into binary with clear, step-by-step examples. You’ll learn how to handle large and small binary numbers, move decimal points, and use base-2 like a champ. Whether you’re a student, coder, or just curious about how computers process numbers, this video has you covered!

I’ll show you practical examples, like converting huge binary numbers and tiny fractions, plus tips to avoid common mistakes (like mixing decimal and binary notation). By the end, you’ll be ready to tackle binary in IEEE 754 or impress your friends with your number-crunching skills. Subscribe for more tech tutorials, and hit that bell to stay updated! Visit my website (link below) for more resources, and leave a comment with your questions or video suggestions—I read every one! Let’s keep learning and having fun with tech together!

Introduction to Binary Scientific Notation 00:00:00
Purpose of Binary Representation 00:00:12
Overview of Scientific Notation 00:00:41
Rules for Scientific Notation 00:01:12
Decimal Scientific Notation Example 00:02:26
Practice with Large Decimal Number 00:04:12
Practice with Small Decimal Number 00:05:21
Binary Scientific Notation Concept 00:06:32
Binary Number Representation Rules 00:07:28
Large Binary Number Example 00:08:24
Small Binary Number Example 00:09:31
Mixing Binary and Decimal Notation 00:12:54
Pure Binary Scientific Notation 00:13:04
Connection to IEEE 754 00:13:48
Conclusion and Call to Action 00:14:21
Engagement and Website Promotion 00:15:32

Thanks for watching!

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Please help support us!

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Hey there! Let’s talk about representing binary numbers with fractions in scientific notation.

Why would you need to do this? Probably the best use that I can think of off the top of my head

is being able to represent binary numbers with fractions inside your machine in a format known

It’s just a crucial step before you can represent numbers inside your machine using IEEE 754.

Okay, so first off, let me just show you a little bit here about scientific notation.

So you probably have seen something like this before where it’s like 8.3873,

and then you’ll see like a multiplier times 10 to the fifth power, right?

I can’t really type this out very well, so maybe I could just draw it for a second, you know.

well so maybe I could just draw it for a second you know 8.54 times 10 to the fifth power right

that’s what I’m trying to convey but anyway so about scientific notation itself it’s standardized

so that it’s easier to use and that is you know it’s just like faster for everyone to understand

and there’s less confusion part of the standard is that you always want to have a number on the

that is between 1 and 9 inclusive. You don’t ever actually want to have a 0 there that would be bad.

You don’t want to have a 10 or anything greater that would also be bad. You just want to have

1, 2, 3, 4, 5, 6, 7, 8, 9 on the left side. And then on the fractional side,

you want to have a number that just kind of like helps you represent the entire

original number without losing precision. And then on the right side, you want a number,

a number, let’s say like x to the y power, where x is the base of the number system you’re working

in. So this is decimal. The base for decimal is 10. So we’re going to say 10 to the something

power. What is the power? The power here helps you understand how big or small the number on

the left really is. That’s kind of one of the benefits of scientific notation. It seems to

help you understand a little bit more of how big or how small a number is rather than exactly

than exactly down to you know the last digit what what is the number precisely so we could say for

now it focuses more on largeness or smallness than preciseness what number are we actually

representing with this in scientific notation well times 10 to the fifth power just means move the

decimal point over a certain number of times so uh you know that’s why we have 10 to the something

go left or right in a base 10 number you’re looking at a different number that has a factor

of 10 for its strength in either direction like multiply by 10 multiply by 10 multiply by 10

or divide by 10 divide by 10 divide by 10 in the other direction so this means we want to move the

decimal point five times to the right to increase the strength of the number times five so one two

three four five if we put the decimal number there then this is the number we were going to represent

number we were going to represent originally so if somebody says hey give me this number 838730

and put it in scientific notation then you your first instinct is to say all right let’s uh type

that number out and we’ll put like a dot zero there and it will just we’ll move the decimal

point over until there’s only one digit um and it’s a you know somewhere between a one and a nine

three four five times over in order to get the decimal point there so that

means it’s going to be this times 10 to the fifth power because we moved it over

five times and you can see that’s the original number that I showed you these

zeros at the very end they don’t actually mean anything so we can omit

them probably a smarter idea to omit them and that’s why we see numbers that

way okay so keep that in mind there’s only one digit let’s maybe do like

practice number here I have a couple practice numbers written down already

let’s see so we’ll start with this number a huge gigantic number just to

practice if we copy paste this down to the next line and then we decide all

right how many times do we need to move the decimal point to get the decimal

point right there so that the two is the first number remember one to nine

inclusive so I’ll just I’ll use two decimal points so I can count more

9, 10, 11, 12, 13.

So that means I did 13 moves.

I’ll put 13 right here so I don’t forget.

Times 10 to the 13th power.

And the 13 is positive because when we’re looking at the scientifically notated format

of the same number, you know, 2 point something is way smaller than the original number.

So we want the scientifically notated format or form to get bigger in order to reach this

number.

in order to reach this number so that means 10 times sorry times 10 to a positive number positive

means it’ll be bigger in its original form okay so now let’s do another practice number

let’s do a number that’s really really really small like you’re inside of inner space or

something so we start up with this number and we still want to have a number between

rewrite it here I really want to have eight point something because that’s the first number that’s

bigger than zero that I can see so again I’m using two decimal points so that it’s easy for me to

count I’m going to go one two three four five six seven I had to move it seven times so it’s going

to be negative seven is going to be the exponent so you know raised something raised to the negative

seven it’s still going to be 10 to the negative seven that I multiply it by so then I’ll say get

And now this is the same number represented in scientific notation.

It should have all the same digits.

The decimal point basically should just be moved.

Of course, you know, when you represent in scientific notation,

depending on what standard you’re working with,

you might actually omit some of the numbers at the very end of the fraction here.

But that’s why we say this is kind of more to impress upon you the smallness or largeness of a number

rather than represent the number exactly precisely.

okay so we got that two practices in there how can we do this same exact concept in binary

well keep in mind in binary binary is a base two number this video is not about binary conversion

as a whole number or binary with fraction let’s just pretend that we already know how to do that

and we have a binary number to start off with so let me grab my example number here

have some kind of a binary number with a fraction, which you can do if you don’t understand how to do

this part yet from decimal with a fraction to binary with a fraction or back and forth.

See my other videos. For now, we’ll assume you can do this. So how can we get this in scientific

notation? So the first thing we have to understand is that it’s going to be, you know, some number

right because that was the format we used before the number should only start with a one it should

never even start with a zero remember in binary we can only use ones and zeros before i said here

let me just show you this real fast again before i said the starting number has to be one through

nine inclusive that was because in decimal we have zero one two three four five six seven eight nine

be only use you know one two three five six seven eight nine so but in binary um i’ll put like

a character set like the available characters we can use to represent the numbers in decimal

so in binary the care set that we can use is just you know a zero and a one only but the same rule

the one so that means the first number always has to be one it has to be always one dot something

for our purposes to represent the same number in scientific notation so it’s going to be

this and obviously that one has to be it it cannot ever be a zero so i’m going to put the

decimal point there uh and then i’m just going to count like how much did i actually move the

seven eight nine ten eleven twelve thirteen fourteen fifteen six just fifteen just fifteen not

sixteen so i’m going to put times something to the fifteen power and remove that other decimal

point and then the base is two so it’s going to be two to the fifteenth power

so now maybe i should move that up a little bit

gigantic number uh in scientific notation it’s going to look a little smaller but then the times

base to the 15th power is going to help us understand how big it is oh it’s like pretty big

let’s do the same thing backwards let’s say that we wanted to start off with a very very small

number so it’s like you know a zero point something in binary so you can imagine if this is like one

256 that’s probably going to be a number that’s no bigger than or just like slightly bigger than

256 so it’s going to be like kind of a small number right well we’ll do the same thing just

copy to another line and then make sure that the decimal point sits in a place where there’s always

a one at the start

and then just count the number of times you moved

number of times you moved the number oh I guess before we would have possibly

deleted numbers on the right if we were gonna reduce precision in this case

after we count the numbers we’re gonna remove everything to the left of the one

so that the one is in the first position and I’ll just go ahead and do it okay so

how many times do we move it one two three four five six seven eight that’s

eight times so I’m gonna put an eight there just to remind myself that there

will be an eight I’ll remove all the stuff at the beginning that doesn’t

all the stuff at the beginning that doesn’t matter anymore and it’s going to be times two because

that’s our base to the eighth power but the original number is a lot smaller than the

scientifically notated number looks so that means we have to put a negative eight because remember

when you say times let me just show you this on a calculator when we say let’s let’s go back to

fifth power then you know that we’re just basically adding four zeros right so

like we have five total zeros so we’re adding four zeros to the ten but if we

did to the negative five power we’re gonna be like dividing it by ten a bunch

of times so instead of multiplying it by ten for a total of five times we’re

gonna divide by ten so then the number gets really really really small so that

means when we say two to the negative eight power we’re gonna be dividing it

We’re going to be dividing it by two that many times.

And so we end up with a really, really, really small number.

Isn’t that what I kind of said?

Let’s see.

I mean, like, not exactly, but, you know, it’s like 0.003 and then some numbers after that.

Didn’t I say one divided by 256?

It’s 0.003 and then some numbers.

So this number is just a little bit bigger than 0.003.

Nine.

Let’s see how much bigger it is.

bigger it is point three six two five what

oh because i’m not i’m not including the the part on the left that we will multiply it by so if i

you know if i did some like binary up here and i was in binary mode then it would probably make

more sense it ended up being exactly the same exact number that’s why i was confused because

if we just type that part on the right side then it really is going to be one over 256.

Anyway, long story short, we have this number here, the fractional part, and then we’re

going to multiply it by two to the something power.

Notice something in particular that I’m doing, which is probably my mistake, but I kind of

like doing it this way.

Notice how the left part is in binary and the right part is in decimal.

There’s no number two in binary or no number eight in binary.

numbers like this to scientific notation so that you can convert a binary number to i triple e

floating point number this is as far as you really need to go but if you truly want to represent a

binary number in scientific notation then you should also convert all of the relevant parts

so how do we represent uh the number two in binary it’s going to be one zero how do we represent the

number eight in binary it’s going to be one two four eight it’s going to be that so uh you know

big number times 10 in binary is still the number two to the something power the negative 1000 in

binary power is going to be you know eight the negative eight power so this is great if you just

want to write an entire number in scientific notation but uh you know in probably my next

video when we talk about ieee 754 notation this is as far as you really need to go

eight number into a a whole number in binary and then putting that somewhere but so just

forget about this for now keep in mind this is how far you have to go if you want to go to ieee

if you only want to be in pure binary then this is what it would look like

okay that’s it uh i think that’s all the example i have for you today in this video thank you so

much for watching i hope you learned a little bit and had a little bit of fun 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.

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this channel or these videos or whatever it is you do on the current social media website

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Thank you.

The post How to Master Binary Scientific Notation (with Fun Examples!) 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

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

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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.