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

Neural Lantern

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 × 10?), 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!  Find us on other social media here: - https://www.NeuralLantern.com/social  Please help support us!  - Subscribing + Sharing on Social Media - Leaving a comment or suggestion - Subscribing to our Blog - Watching the main "pinned" video of this channel for offers and extras

Transcript

00:00hey there let's talk about representing binary numbers with fractions in scientific notation

00:12why would you need to do this probably the best use that i can think of off the top of my head

00:17is being able to represent binary numbers with fractions inside your machine in a format known

00:24as ieee754 this video is not about ieee754 it's just a crucial step before you can represent

00:31numbers inside your machine using ieee754 okay so first off let me just show you a little bit here

00:40about scientific notation so you probably have seen something like this before where it's like

00:448.3873 and then you'll see like a multiplier times 10 to the fifth power right i can't really

00:54i can't really type this out very well so maybe i could just draw it for a second you know

00:588.54 times 10 to the fifth power right that's what i'm trying to convey but anyway so about

01:10scientific notation itself it's standardized so that it's easier to use and that is you know it's just

01:15like faster for everyone to understand and there's less confusion part of the standard is that you

01:20always want to have a number on the left side of the multiplier that is between one and nine

01:27inclusive you don't ever actually want to have a zero there that would be bad you don't want to have

01:31a 10 or anything greater that would also be bad you just want to have one two three four five six seven

01:36eight nine on the left side and then on the fractional side you want to have a number uh that just kind

01:42of like helps you represent the entire original number without losing precision and then on the

01:49right side you want a number let's say like x to the y power where x is the base of the number system

01:58you're working in so this is decimal the base for decimal is 10 so we're going to say 10 to the

02:03something power what is the power the power here helps you understand how big or small the number

02:10on the left really is that's kind of one of the benefits of scientific notation it seems to help you

02:16understand a little bit more of how big or how small a number is rather than exactly down to you know

02:22the last digit what what is the number precisely so we could say for now it focuses more on largeness

02:29or smallness than preciseness what number are we actually representing uh with this in scientific

02:36notation well times 10 to the fifth power just means move the decimal point over a certain number of

02:42times so uh you know that's why we have 10 to the something power because we're in base 10 every time you

02:48go left or right in a base 10 number you're looking at a different number that has a factor of 10 for

02:54its strength in either direction like multiply by 10 multiply by 10 multiply by 10 or divide by 10

02:59divide by 10 divide by 10 in the other direction so this means we want to move the decimal point

03:04five times to the right to increase the strength of the number times five so one two three four five

03:12if we put the decimal number there then this is the number we were going to represent originally

03:18so if somebody says hey give me this number eight three eight seven three zero and put it in

03:24scientific notation then you your first instinct is to say all right let's uh type that number out

03:29and we'll put like a dot zero there and it will just we'll move the decimal point over until there's only

03:35one digit um and it's a you know somewhere between a one and a nine inclusive how many times do we have

03:42to move this over one two three four five times over in order to get the decimal point there so that

03:49means it's going to be this times 10 to the fifth power because we moved it over five times and you can

03:54see that's the original number that i showed you these zeros at the very end they don't actually mean

03:59anything so we can omit them probably a smarter idea to omit them and that's why we see numbers that way

04:06okay so keep that in mind there's only one digit let's uh maybe do like another practice number

04:12here i have a couple practice numbers written down already uh let's see so we'll start with this number

04:17a huge gigantic number just to practice if we copy paste this down to the next line

04:23and then we decide all right how many times do we need to move the decimal point to get the decimal

04:27point right there uh so that the two is the first number remember one to nine inclusive

04:34so i'll just i'll use two decimal points so i can count more easily one two three four five six

04:39seven eight nine ten eleven twelve thirteen so that means i did uh 13 moves i'll put 13 right here so i

04:46don't forget times 10 to the 13th power and the 13 is positive because when we're looking at the

04:55scientifically notated uh format of the same number you know two point something is way smaller than the

05:01original number so we want the scientifically notated uh format or form to get bigger uh in

05:08order to reach this number so that means 10 times uh sorry times 10 to a positive number positive means

05:15it'll be bigger in its original form okay so now let's do another practice number let's do a number

05:22that's really really really small like you're inside of inner space or something

05:27so we start off with this number and we still want to have a number between one and nine inclusive

05:34before the decimal point so if i rewrite it here i really want to have eight point something because

05:39that's the first number that's bigger than zero that i can see so again i'm using two decimal points

05:44so that it's easy for me to count i'm going to go one two three four five six seven i had to move

05:50it seven times so it's going to be negative seven is going to be the exponent so you know raised

05:55something raised to the negative seven it's still going to be 10 to the negative seven that

05:59i multiply it by so then i'll say get rid of the decimal point and now this is the same number

06:05represented in scientific notation it should have all the same digits uh the decimal point basically

06:12should just be moved of course you know when you represent in scientific notation depending on

06:18what standards you're working with you might actually omit some of the numbers at the very

06:22end of the fraction here but that's why we say this is kind of more to impress upon you the smallness

06:28or largeness of a number rather than represent the number exactly precisely okay so we got that

06:35two practices in there how can we do this same exact concept in binary well keep in mind in binary

06:43uh binary is a base two number uh this video is not about binary conversion as a whole number or

06:50binary with fraction let's just pretend that we already know how to do that and we have a binary

06:55number to start off with so let me grab my example number here suppose we have some kind of a binary

07:01number uh with a fraction which you can do if you don't understand how to do this part yet from decimal

07:06with the fraction to binary the fraction or back uh back and forth uh see my other videos

07:13for now we'll assume you can do this so how can we get this in scientific notation so the first thing

07:18we have to understand is that is that it's going to be you know some some uh number times a base

07:27to an exponent right because that was the format we used before the number should only start with a one

07:35it should never even start with a zero remember in binary we can only use ones and zeros

07:39before i said here let me just show you this real fast again uh before i said the starting number

07:47has to be one through nine inclusive that was because in decimal we have zero one two three

07:52four five six seven eight nine and we would just ignore the zero so it would only be um only use

08:00you know one two three one five six seven eight nine so but in binary

08:04um i'll put like a character set like the available characters we can use to represent the numbers

08:11in decimal so in binary the care set that we can use is just you know a zero and a one only but the

08:18same rule applies only use everything but the zero so that means only use the one so that means the first

08:26number always has to be one it has to be always one dot something for our purposes to represent the same

08:32number in uh scientific notation so it's going to be this and obviously that one has to be it it cannot

08:41ever be a zero so i'm going to put the decimal point there uh and then i'm just going to count

08:46like how much did i actually move the decimal point i moved it by one two three four five six seven

08:52eight nine ten eleven twelve thirteen fourteen fifteen six just fifteen just fifteen not sixteen

08:57so i'm going to put times something to the fifteen power and uh remove that other decimal point

09:05and then the base is two so it's going to be two to the fifteenth power

09:12so now maybe i should move that up a little bit

09:15we know how to represent this gigantic number uh in scientific notation it's going to look a little

09:24smaller but then the times base to the fifteenth power is going to help us understand how big it is

09:29oh it's like pretty big let's do the same thing backwards let's say that we wanted to start off with

09:36a very very small number so it's like you know a zero point something in binary so you can imagine if

09:42this is like one half one fourth one eighth 16 32 64 128 uh 256 that's probably going to be a number

09:50that's no bigger than um or just like slightly bigger than uh 256 so it's going to be like kind of

09:57a small number right well we'll do the same thing just copy to another line and then make sure that the

10:05decimal point sits in a place where there's always a one at the start

10:09and then just count the number of times you moved

10:18the number uh oh i guess before we we would have possibly deleted numbers on the right if if we were

10:24going to reduce precision in this case after we count the numbers we're going to remove everything

10:29to the left of the one so that the one is in the first position and i'll just go ahead and do it okay

10:34so how many times do we move it one two three four five six seven eight that's eight times so i'm

10:40going to put an eight there just to remind myself that there will be an eight i'll remove all the

10:44stuff at the beginning that doesn't matter anymore and it's going to be times two because that's our

10:49base to the eighth power but the original number is a lot smaller than the scientifically notated number

10:57looks so that means we have to put a negative eight because remember when you say times let me just

11:04show you this on a calculator when we say let's let's go back to decimal for a second if we say ten

11:11uh to the fifth power then you know that we're just basically adding four zeros right so like

11:17we have uh five total zeros so we're adding four zeros to the ten but if we did to the negative five

11:24power uh we're going to be like dividing it by ten a bunch of times so instead of multiplying it by

11:29ten for a total of five times we're going to divide by ten so then the number gets really really really

11:34small so that means when we say uh two to the negative eight power we're going to be dividing it

11:41by two that many times and so we end up with a really really really small number um isn't that what

11:50i kind of said let's see uh i mean like not exactly but uh you know it's like point zero zero three and

11:55then some numbers after that didn't i say one divided by 256 it's point zero zero three and then some

12:00numbers so this number is just a little bit bigger than point zero zero three uh nine let's see how much

12:08bigger it is point three six two five what

12:18oh because i'm not i'm not including the the part on the left that we will multiply it by so if i you

12:23know if i did some like binary up here and i was in binary mode then it would probably make more sense

12:28it ended up being exactly the same exact number that's why i was confused because if we just type

12:33that part on the right side then it really is going to be one over 256. anyway long story short

12:39we have this number here the fractional part and then we're going to multiply it by 2 to the something

12:44power uh notice something in particular that i'm doing which is probably my mistake but um i like i

12:52kind of like doing it this way notice how the left part is in binary and the right part is in decimal

12:57there's no number two in binary or no number eight in binary so when it comes to converting numbers

13:04like this to scientific notation so that you can convert a binary number to ieee floating point

13:09number this is as far as you really need to go but if you truly want to represent a binary number in

13:14scientific notation then you should also convert all of the relevant parts so how do we represent

13:19the number two in binary it's going to be one zero how do we represent the number eight in binary

13:24um it's going to be one two four eight it's going to be that so uh you know this big number times

13:34ten in binary is still the number two to the something power the negative one thousand in

13:38binary power is going to be you know eight the negative eight power so this is great if you just

13:43want to write an entire number in scientific notation but uh you know in probably my next video when we

13:49talk about ieee 754 notation this is as far as you really need to go because we'll basically just

13:58be converting this uh this negative eight number into a a whole number in binary and then putting that

14:04somewhere but so just forget about this for now keep in mind this is how far you have to go if you

14:10want to go to ieee if you only want to be in pure binary then this is what it would look like

14:14okay that's it uh i think that's all the example i have for you today in this video thank you so

14:21much for watching i hope you learned a little bit and had a little bit of fun see you in the next video

14:30hey everybody thanks for watching this video again from the bottom of my heart i really appreciate it

14:34i do hope you did learn something and have some fun uh if you could do me a please a small little favor

14:40could you please subscribe and follow this channel or these videos or whatever it is you do on the

14:46current social media website that you're looking at right now um it would really mean the world to me

14:50and it'll help make more videos and grow this community so we'll be able to do more videos longer

14:55videos better videos or just i'll be able to keep making videos in general so please do do me a kindness

15:02and uh and subscribe you know sometimes i'm sleeping in the middle of the night and i just wake up because

15:07i know somebody subscribed or followed it just wakes me up and i get filled with joy that's exactly what

15:12happens every single time so you could do it as a nice favor to me or you could you control me if

15:16you want to just wake me up in the middle of the night just subscribe and then i'll i'll just wake up

15:20i promise that's what will happen also uh if you look at the middle of the screen right now you should

15:26see a qr code which you can scan in order to go to the website which i think is also named somewhere at

15:31the bottom of this video and it'll take you to my main website where you can just kind of like see

15:36all the videos i published and the services and tutorials and things that i offer and all that

15:40good stuff and uh if you have a suggestion for uh uh clarifications or errata or just future videos

15:50that you want to see please leave a comment or if you just want to say hey what's up what's going on

15:54you know just send me a comment whatever i also wake up for those in the middle of the night i get

15:59i wake up in a cold sweat and i'm like it would really it really mean the world to me i would really

16:04appreciate it so again thank you so much for watching this video and um enjoy the cool music

16:11the cool music as as i fade into the darkness which is coming for us all

16:23so

16:35so

Category

Transcript

Recommended