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

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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

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