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How computers use binary numbers

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hi in this video let's talk about how

computers store data we'll talk about

binary numbers [Music]

[Music]

binary numbers are the systems that use

ones and zeroes to store data using

computer circuits so you likely know

that computer circuits have either an on

or an off value which are represented in

numbers as a one or a zero let's go back

to your school days do you remember when

your teacher taught you how to count by

numbers that were greater than a hundred

we used things like expanded notation so

8473 can be written like this 8 000 plus

400 plus 70 plus 3.

the idea here is that decimal numbers

have place values

so first of all place values in decimals

look like this

each place has a 10 as a base

and then there's an exponent so we could

write this 8 473

473

in a different form

we could say that 8 is the thousands

place which is represented as 10 to the

third power

the four is in the hundreds place which

is represented as four times ten to the

second power

seven is in the tens place which of

course is ten to the first power and

and

the ones place is really a disguise of

10 to the zero power remember in

exponents any number raised to a zero

gives you the number one as an answer so

understanding place value is critical to

understand how binary numbers work

let's look at an example here so i have

a binary number written on the screen

and i have the place values listed below

so each number that you see either one

or zero falls into a specific column so

the first column is the ones place it is

represented as two to the zero power

so when we say that binary is a base two

number system we're talking about the

base of the exponential problem

so when let's take a look at how we

would compute this answer

so we see that in this example there is

a one in the second place

the second place is represented as the

is the two to the first power

and the next number over we find is in

the two to the third power so what is

two to the third power it is two times

two times two so we'll call that the

eighths place so really we have a first

place which is the

ones place the two's place the fours place

place

what we have here is the eighths place

look ahead

we have another one coming up it's at

two to the sixth power

so what is two to the sixth

the answer is 64.

so this number is in the 64 place

what's the last one then it's 2 to the

seventh so double of 64

that's going to be 128.

so to compute the answer here we have to

take all of the places and add them together

together

so 128 plus 64 plus 8 plus 2.

if you add all those up and if i did my

math correctly we come up with 202 is

our number

so using place value is a way that we can

can

calculate a number even if it is not a

base 10 system

so why did we pick a binary number

system it's because we have these

gadgets that are binary computers in the

early days of computing we had vacuum

tubes which would store a value in the

computer's memory so a vacuum tube is

essentially like a light bulb except it

has a charge in it so it is either a

high charge or a low charge high voltage

or low voltage on or off and in the

early days these took an immense amount

of space so you can see here that we

have eight vacuum tubes in a row

each of these represents either one or a

zero a bit

let's switch over into a program written

in c sharp that will give us an example

of how this looks

so i am going to translate numbers into

their binary value i'm going to use this

statement here which is called convert

dot to string

and the numbers that i put in here are

the integer value which is going to run

from 0 to 255 in a loop and then the

second is a base number so we are using

a base 2 system so this will give us a

count of all of the binary numbers let's

see what it looks like

so here's the results of the program you

can see that we start our for loop at

zero so it says the decimal value is

zero and the binary storage is zero so

far so good you can see the pattern

starts to move along here when we get

along to

like three bits

so the number one one one represents the

number seven so if you were to add these

up here we have the ones place

the two's place and the fours place

that's four plus two plus one so that

gives us seven

let's scroll down to see where we have

the next bump up in place value so that

occurs down here at number fifteen and

sixteen so you can see that fifteen has

the value of 1 1 1 1. so add those

together you have 8 plus 4 plus 2 plus

1. so that gives you 15. if we want to

represent 16 then we have to go to the

next value down which is

1 with 4 zeros so the place that we're

looking at is a sixteenths place that's

ones twos fours eights

sixteen place so that's why we get a one

followed by four zeros

where's the next break going to occur or

where does the next

significant digit come in so that would

be double of 16. let's scroll down to 31.

31.

31 gives you the 1 1 1 1 1 and then we have

have

32 which takes a new place value so

let's double that again let's go to 64

and see what comes up so i'm scrolling

down to 64 and sure enough 64 begins a

new place value

the number 128 should be next in our

list let's scroll to that and you can

see that the difference between 127 and

128 is a significant value and then we

get to the end here and what was the

last item so i got up to 254 had i

gotten to 255 we would have included all

ones here

so this program here as you can see its

purpose is to convert decimals into a

base two value so we can see what those

look like in ones and zeros now let's

talk about storage units that are in

computers memory or on their disk drive

so we start off with the smallest here

which is one bit

that's a zero one one byte is the value

of eight bits

and then you can see we jump into

kilobytes then which is a thousand

twenty four bytes one megabyte one

gigabyte one terabyte petabyte and exabyte

exabyte

round off the title of the uh all these

different measurements so

1024 where did that number come from

well that is another one of those

doubled values so two to some power i

think it's nine maybe we'll give you uh

the value of 1024.

so that kind of tells you what the

number value is of each of these

measurements but let's get a concept of

how big each of these are so a bit is a

true or false value that's zero one one

byte is one letter from your keyboard so

the letter a b or c

then we have a kilobyte which is one

page of unformatted text

one megabyte is about a minute was worth

of music or maybe a photo that you take

a gigabyte is maybe an hour of video not

the highest quality video but a medium

quality or a word document that has a

bunch of pictures in it that's about a megabyte

megabyte

one terabyte is the size of a kind of a

typical hard drive in a computer and so

you could put a lot of things on one

terabyte you could fill up your hard

drive with all of the episodes from

famous tv shows such as law and order

the simpsons family guy friends the

office take all those together and you

have a terabyte worth of video a

petabyte is the next guy up and this is

starting to get into some major large

storage at least in our opinion so you

could take a video camera and record it

24 7 at pretty high quality

video quality and you could do that for

three and a half years and you would

fill up your one petabyte hard drive or

you could take 4 000 pictures every day

and do that for your entire life and

that would be a petabyte an exabyte is

about 11 million movies if they were in

high resolution uh the entire internet

would handle hundreds of exabytes of

data per day

and then finally at the very top is a

zettabyte so a zettabyte

is um a lot obviously

it's about estimated that we have more

than 30 exabytes of data stored on every

device in the world so that includes

every phone and every computer and every

server and every data storage facility

so we're in the astronomical range

that's really hard for us to conceptualize

conceptualize

so as a programmer you're going to be

required to understand the differences

between these types of storage units in

your programming language so that's what

i teach is computer programming and we

have to understand the difference

between integers and longs and floats

and doubles so let's take a look at how

these are categorized so in the first

column you find what's called integer

values so these are whole numbers so int

so uent is an unsigned integer so only

positive numbers can be stored in a uint

so you can see the rest of them are long

shorts and bytes and things like that

the floating points are where you would

have fractions or decimal points so

decimals doubles and floats are

different sized items for taking care of

decimal numbers

the object

data format is completely different than

any of these so we'll leave that for a

future video what is an object what are

objects how do you program with them

in characters we have two different uh

types of data storage a char is a single

letter a string is a bunch of chars put

together into a sentence and then lastly

we have logic which is a boolean

variable now essentially you could store

a boolean as a single bit true or false

in reality i think they use maybe a byte

let's take a look at two floating point

numbers so here's an example of a

floating point number it's a decimal so

this rounds off to the nearest penny you

might say it's nine cents but if you

were to look at a more precise value if

you wanted to go out to the tens of

zillions of digits

this is going to require a lot of data

storage so many many bits are going to

be required if you want to have very

precise numbers so if you're working

with currency in a program you're going

to want to have a pretty good

accuracy here you don't want to be

rounding off things

transactions are you know kind of

sensitive and if you obviously are into

scientific instruments then this is

where it really makes sense to have more precision

precision

so compare the different types of

storage and how you can use a maximum

and minimum range for each of them so a

boolean is the first row in our table

and it is simply a byte it's going to be

either a true or false so not much of a

range there a char and a byte are both

ways to represent an 8-bit number or a

byte and so we can go from 0 to 255 in

that range so a char is

a letter

so in the ascii character symbol table

65 has been assigned to the letter a

so whoever got to assign that first

picked 65. first for a lowercase a

the number is 97. so there is a key that

corresponds between letters and numbers

so chars and bytes are really the same

thing but are interpreted differently by

the computer

a short is a double length thing it's

two bytes and you can see that the range

here goes from a very narrow range

really negative thirty two thousand

seven sixty eight all the way up to

positive thirty two thousand seven sixty seven

seven

so the negative sign takes one of the

bits and then the remainders are used

for the numbers you can see ins are much bigger

bigger

longs are huge floats are like massive

so with a precision comes a cost so

really the cost is the memory usage so

if you're using

longs for everything you're wasting

memory because if your values of your

game or your program never exceed the

limits of an integer then there's no

there's no reason to use a long

so let's switch back into visual studio

i'm going to show you some examples of using

using

these numbers such as

an integer and a short and a float so

the first thing i see here is a variable

called score and i've set it to be

32768 which we're going to see why is

significant in a minute but the range of

this uh integer is quite large so we

could we could do up to 10 to the ninth

power of uh

place values so that's in the billions

however the short value you can see here

is only up to 32 000 so shorts have a

pretty small limitation and then we're

going to work with the difficulty level

here in the game which is going to be a

decimal of 4.2

so let's scroll down a bit now and we're

going to see that when we try to convert

from one to the other we get a problem

so if i say short score is now going to

be equal to score

this doesn't work however the reverse

should work i'm going to comment out the other

other

so we can convert score

from short score so we can go from the

smaller value into the larger without

any trouble however we cannot go the

reverse so

we will lose some significant values of

what's going on here so in other words

we have a limitation of storage now we

can fix this problem if we do something

called casting so i'm going to put a

parenthesis in here and i'm going to put

in the word short

which is going to say that we're going

to take score and we're going to convert

it into a short value which then allows

us to assign it to the type called shortscore

shortscore

now we could have some problems with

that and we'll find out in just a jiffy

what those problems are so let's let's

change score to something like

9000. so it's a fairly typical number

it's not outside of any range of

anything and then we're going to print

both of these so i'm going to say score

equals and now short score equals

something they probably look a lot like

each other they should be identical value

value

so when i run the program sure enough i

get the value it says score equals 9 000

and short score equals 9 000. no problem

now the problem will occur if i try to

go beyond the range of what the short

score is so let's go right up to the

edge of it 32 000 767

and run the program this should still work

work

so we can see the results are that the

score and the short score are both the

same thing so no problems with that

conversion we will encounter problem if

we try to go outside the range so

remember the maximum value of short

score goes up to seven six seven what

happens if we push it just one beyond

that we're going to go to 768. so now

we're out of bounds

will the program crash

let's find out

uh-huh so you can see that the program

did not crash

instead it switched this into a negative number

number

so the way that the numbers are stored

you says that the the most significant

bit in the value is a one or a zero

which is for the sign of the integer and

so by pushing it beyond the boundaries

we have turned the positive number into

a negative so this wraps around to the

opposite end of the number line and so

you can get some pretty strange results

if you go outside the boundaries of a short

short

now let's try another conversion i'm

going to take the float value difficulty

and try to store it into the score which

is an integer

obviously we've got a red line here

we've got a problem it says you're not

allowed to do that so did you want to

try a cast so i'm going to say show

potential fixes add the explicit cast

and now it's doing a integer so

can you convert a float to an integer

well you can if you make some

assumptions like we're going to chop off

the value

if the decimal point and just give you

the integer part let's see if that's

what happens

so let's put a print statement in first

of all that says hey we're going to

print the difficulty first and then

we'll show you the score and the short

score as a matter of fact i'm not

interested in short score anymore so i'm

just going to comment him out let's go

ahead and run the app

so you can see the results at the top it

says difficulty was 4.2 and when we

converted it into score it was an

integer value of 4.

what happens if i try rounding it to a

different number so let's go to 4.9 so

will it round it up to 5

as you can see the result is still 4. so

doing a cast into an integer is called a

floor operation in other words it always

rounds it down to the nearest integer so

it is possible to convert from a

floating number into an integer value

however you're going to lose the

decimals and you're going to round off

to the nearest lowest integer

so now that you've seen some ways of how

numbers are stored let's talk about how

storage has actually changed over the

years so here's an example of a memory

card you could have bought in 2005 and

one that was in 2014

probably for about the same price each

way one is a

and twenty four times larger than the

other so gigabytes and megabytes are

obviously different scales and we've

grown accustomed to streaming videos and

downloading huge files let's compare the

different storage items that are in a

binary system so way back in the

beginning as i mentioned we have these

vacuum tubes then later in the 50s we

transitioned into transistors which are

much smaller but they still hold charges

ones and zeros then we've gone become a

next generation up which is integrated

circuits and then microchips which have

grown smaller and smaller more complex

using less power and more efficiency so

there's different ways to arrange

microchips together so we're all

familiar hopefully with the cpu idea cpu

is the largest and most complex of the

memory chips that are in your computer

however there are others and they're all

designed to do different things but the

idea of integrating everything on one

circuit and making a gadget that looks

like a small cockroach is the idea of making

making

simpler and more cheaply when you're

trying to manufacture a computer

so we've shrunk the size of our storage

over time so the number of transistors

that were in the 1950s were somewhat

better than the vacuum tubes was greatly

improved but we've kept up that same

improvement on generation after

generation after what's known as moore's law

law

so moore's law is the idea that we

noticed or mr moore noticed that the

intel corporation

was cramming more things into smaller

spaces about every two years or so they

would double the number of transistors

that they could put on a microchip

so that is possible because they have

smaller and smaller gaps between the

wires and the pathways and the

transistors that are in their integrated

circuits so way back in 2002 they had a 180

180

nanometers of space between items on a

chip so a nanometer is a one billionth

of a meter very small you need

microscopes to be able to detect this

kind of thing we're getting down to the

size of almost molecules and atoms on a

chip and as you can see we've gotten

smaller and smaller however you get to

the top of the graph and you can see

that the

the exponential growth has leveled off

we're still getting smaller but the

physics are getting harder and harder to overcome

overcome

and so in 2015 there was a 16 nanometer

manufacturing process

currently 2021 is when i'm making this

video a 5 nanometer process is

considered state of the art and so

they've shrunk it even further so we're

getting to the point where instead of

using tools to do the etchings or the

making of the chips we're into light

waves and x-rays and gamma rays and small

small

little uh changes in the actual

molecules of the substance that we're

working on so very very small means less power

power

and faster response times

now if you're one of the students in my

programming classes i'm going to

eventually assign you this the

temperature conversion app it'll be a

very simple app and it will use integers

and floats and the conversion between

the two using temperatures such as

fahrenheit celsius and calvin and so

that's a different video but i invite

you to stick around to join me for class

if you're not one of my students please

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How computers use binary numbers