Mathematics for Data Science: Natural Numbers, Integers, and Primes

[Music] so welcome to the first week of mathematics one for data science so we

are going to start with some very basic things which you probably know right from the beginning we are going to start

talking about numbers so in this first module what we are going to talk about is natural numbers and integers so as

you probably remember from as young as you were in school when you first came across numbers we use numbers mainly for

counting so for instance if we see seven balls like this and then we see seven pencils like this then we need to know

that these are the same number of things and for this we use this number seven so seven represents what is common to these

two objects that there are seven balls and seven pencils so seven is an abstract concept in that sense and it

refers to a quantity so we all of course know the numbers 1 2 3 4 and all that so when we see a number of things we can

count them but perhaps the most important number of all which is of Indian origin is 0 so it's quite

important to have a way to represent something when there is nothing to count because without a 0 we cannot use our

place numbering system that we use to manipulate numbers so these numbers starting with 0 I what are often called

the natural numbers now there is some confusion in some books and many books will actually use only 1 2 3 4 to

represent the natural numbers so we use this kind of in with a double line across it could represent the set of

natural numbers and in case there is any confusion whether 0 is included in this set or not now sometimes people will not

include 0 in the set of natural numbers so sometimes to emphasize that we are using 0 we will actually put this

subscript 0 below the end right so we will write either N or n 0 but whenever we are talking about natural numbers it

always includes a 0 now what can we do with natural numbers well we can add them we can subtract them we can

multiply them we can divide them so these are the normal arithmetic operations which you have studied in

school but what is really interesting from a mathematics perspective is when we take natural numbers and we

perform an operation on them do we always get a natural number so if we add two natural numbers do we get a natural

number if we subtract a number from another we get a natural number if we multiply them do we get a natural number

if we divide one by another do we get a natural number so the first operation which fails this test is subtraction

because if we subtract a larger number from a smaller number so supposing we take 6 and subtract it from 5 then we go

below zero right if you have five things and we take away 6 things well we cannot take away 6 things that word subtraction

means so we need to expand the scope of our numbers to allow these operations to work sensibly and this is how we get the

negative numbers so we had the positive numbers 0 1 2 the non negative technically because 0 is neither

positive nor negative so we had the positive numbers 1 2 3 4 we added a zero to account for the fact that we are

counting nothing and now we add symmetrically on the other side negative numbers minus 1 minus 2 minus 3 so this

is just to illustrate why we get them of course this is something that you should know from school so this set which is

the natural numbers extended with the negative numbers is what we call the integers and we use Z with a bar across

it to indicate the set of integers so we have n the set of natural numbers which starts at 0 and goes forward 0 1 2 3 4

and we have the integers which start at no at minus infinity and go to plus infinity so these are both infinite sets

but the natural numbers have a starting point 0 and the integers extend to infinity in both directions so it's very

convenient mentally to think of the integers as forming this kind of a sequence where on the left you have the

very small ones and on the right you have the very long ones and this is normally called the number line so as

you go from left to right the numbers are increasing and this is how the integers are arranged so we said that

subtraction takes us away from natural numbers and we brought the integers so now let's look at the other two

operations that we talked about multiplication and division so let's start with multiplication

so when we say seven times four what we are really saying is take seven objects and make four copies of them so for

instance on the right we have those seven balls that we started with and then we have made four copies of them so

if we want to know how many balls are here then we have seven from the first group seven from the second group and so

on so we have four groups of seven and this is if we add it up going to be 28 so in general this is how we multiply

when we take a number M and multiply it by n what we are doing is we are making n copies of M so we are taking M plus M

plus M n times so in this sense multiplication is repeated addition so we often use this

time sign the X sign for multiplication but this is often cumbersome when we write out the equations so sometimes we

replace this time sign by a dot and sometimes we write nothing at all so if we just write two symbols together we

don't write this normally for numbers because imagine that if I write seven four like this then you don't know

whether it's the number 74 or it's seven times four so if we have numbers we will normally write a dot explicitly between

them like seven times three but when we have names like M or n standing for numbers then if we write m n we assume

that it is one number M multiplied by another number n now we have mintages and integers have signs they are

positive and negative numbers so we have to remember that when we multiply numbers with signs the resulting number

also has a sign and there is a sign rule which basically says that if we have one negative number multiplied by one

positive number then the result is a negative number so minus M so let us assume that M is a positive number so

minus M is a negative number so say -7 times 4 would be minus 28 on the other hand if I take minus 7 and x minus 4

then the two negations will cancel and I will get plus 28 so if you have an even number of minus signs you get a positive

number if you have an odd number of minus signs you get a negative number now just like we have repeated addition

we can also do repeated multiplication so instead of M plus M we can take M times M and this is called M squared and

the reason that is called M Squared is in the picture here so we have now here six balls and six balls so we have six

times six right so this means that we can arrange these six times six balls in a square and this is why we call this

square so this notation M to the power two stands to the fact that M is multiplied by itself twice now if you

multiply it by self three times then we get a cube so here for instance we have three balls by three balls and then we

have a height a stack of three such balls so we have a square of three by three nine balls and we have three

stacks of these one on top of the other so this naturally forms a cube so M times M times M is written M cubed now

unfortunately we live in a three-dimensional world and we can't imagine objects which have more than

three dimensions so our vocabulary star stops with cube so in general if we have M to the power K then we write M times M

times M K times and we just say it's m to the power K we don't have a fancy name for it we just say it is the K

power of M so to emphasize multiplication is repeated addition and exponentiation as we have seen is

repeated multiplication so now let's come to division so you would have seen this familiar problem in school you have

a certain number of objects and you want to divide them among certain number of people so for example supposing you have

20 mangoes and you want to give them to five friends so how many mangoes does each friend get so here on the right we

have this picture and then what you do is well you start by distributing one mango to each friend right so you take

out five mangos and you give them to each of your friends so now you have given away five mangos and you have only

15 mangos left so you repeat the process among the 15 mangos you give away five to your friends one each and now your

fifteen mangos have become ten and do it one more time and your 10 mangos have become five do it a third time a fourth

time rather and the five mangos are now gone so after four rounds of distributing mangos each time giving one

mango each so five mangos per round you have got rid of your twenty mangos so 20 divided by 5 is 4 so here as we

have illustrated division is actually repeated subtraction you keep

subtracting by the number you are trying to divide and finally if you hit zero then you have divided it exactly well

what if you had only 19 mangoes now you know very well that 19 mangoes cannot be evenly divided into 4 into 5 groups so

if you start distributing like we had above the first 3 rounds would go fine you would come from 9 20 19 to 14 from

14 to 9 and then you will come from 9 to 4 and now you have only 4 mangos left and you have 5 friends so you can't give

one each so we have managed to distribute 3 times and we have 4 left over so formally this is written as

saying that the quotient the number of times you can actually divide without getting into a fractional part is 3 and

the remainder that is after you have a little bit left over which you cannot subtract one more time is the remainder

is 4 so 19 divided by 5 the quotient is 3 and the remainder is 4 now very often we will need to use this remainder and

there is a notation for remainder so this is this notation called modulus so modulus is another word for remainder

and it's written as mod so 19 mod 5 is the same as the remainder when 19 is divided by 5 so instead of saying the

remainder of 19 divided by 5 is 4 we will often say 19 mod 5 is 4 so with this notation we can now define what is

a factor so a factor is a number which divides a bigger number evenly without any remainder so a divides B if B mod a

is 0 remember what this mean is means is that if B is divided by a there is no remainder and we write this with this

vertical bar so on the left is the smaller number on the right is a bigger number so a divides B this is what this

is supposed to say and the other way of thinking about it is that B is some multiple of a so if a divides B then a

times some K is equal to B so we have some multiple some number of times that a goes into B

so therefore B is a multiple of it so here are some examples we have already seen that 4 divides 20 because 4 times 5

is 20 7 divides 63 because 7 times 9 is 63 32 divides thousand 24 because 32 times 32 is 1,024 and so on now the

symbol that we use for not being a divisor is just to put a stroke across a vertical line so 4 does not divide 19

because there is no way to multiply anything by 4 and get 19 similarly 9 does not divide hundred

evenly because we get 9 11th and 99 and then we go 208 so we say formally that a is a factor of B if a divides B so a

divides B is the same as saying that a is a factor of B and it's easy to see that factors must come in pairs because

if a divides B then a goes in to be some K times so K also divides B right so K times a is equal to B so both K is a

factor and a is a factor so for instance if you take a number 12 then 1 is a factor because 1 divides everything and

in fact for every number n1 times n is n so the pair for 1 is always the number itself now in this case 12 is divisible

by 2 and 2 goes in 6 times so the pair 2 6 formed two factors 6 times 2 is 12 2 times 6 is 12 and similarly 3 times 4

now of course there is an important site condition which is that sometimes the pair is the same as the number itself

and this happens when the number actually happens to be a perfect square that is it is some number multiplied by

itself so for instance consider 36 so 36 is 6 times 6 so if we look at the factors of 36 and group them in pairs

then we have 1 and 36 we have 2 and 18 we have 3 and 12 we have 4 and 9 and finally we have the factor 6 but 6 is

multiplied by 6 so 6 doesn't produce a new factor as its pair it's just itself so another way of thinking about it is

that if you have something which is not a square you will have an even number of factors you will have 2 plus 2 plus 2

plus 2 if something is a square you'll have an odd number of factors we'll have 2 plus 2 plus 2 and finally

when you come to the number of which it is a square that number will come only once in the list of factors so once we

talk about factors we come to a very interesting class of numbers which are the prime numbers so prime number is 1

which has no factors other than 1 and itself so 1 is a factor always and 1 times n is 1 so 4 we usually write P for

a prime number so a prime number has only two factors 1 and P now it's important that it must have two factors

two separate factors so one technically is not a prime because it has only one factor one itself because one times one

is one and so the only factor that one has is 1 so the smallest prime actually is 2

because it has two factors 1 and itself - and no other factors 3 is also a prime because it has only two factors 1 and 3

2 doesn't go into 3 and so on so we are all familiar with the smaller prime numbers so 2 is the first prime number 3

is the next prime number then 5 then 7 notice that after 2 no even numbers can be primes because they are all multiples

of 2 and so 2 divides them now we come to 9 and 9 is not a prime number because it's a multiple of 3 but 11 is a prime

number and so on so there is actually one clever way which is called the sieve of eratosthenes to generate prime

numbers which is whenever you discover a prime you knock off all the numbers which are multiples of it so we can do

this for instance to get all the prime numbers from 1 200 so what we do is we first lay out a grid like this right we

know that 1 is not a prime so the first prime that we have as a candidate is 2 right so this is how the sieve of

eratosthenes works you lay out the numbers in a grid and now we can try and mark off all the prime numbers which are

up to hundred so we know that 1 is not a prime so we leave one off the grid and we start with 2 so 2 is our first prime

number and what the sieve of eratosthenes is you knock off all multiples of 2 so you knock off all the

even numbers and of course now you can do it in one shot so you can knock off this whole column this whole call them

so all these numbers are not prime so now once you have you have a target so we are looking only up to 100 so up

to 100 we have knocked off all the powers of two or all the multiples of two so now we look at the first number

which has not been marked off and we notice that 3 is a prime because 3 is not yet marked off so now we start

marking multiples of 3 some of them are already marked off because they are multiples of 2 so 6 is already gone but

9 is also gone 12 is already gone but 15 is also gone and so on so we can mark off all the other multiples of 2 which

are not multiples of 3 and so on right so we get this kind of a picture and now having done this assuming we have done

it all the way then we will come and find that 5 is a prime right so this is the process by which if you hunt and

count all the primes up to a certain number n you can write out all the numbers up to N and starting at the left

you can take the first unmarked number call it a prime and mark all its multiples to the right as non primes and

the next unmarked number will be the next prime and so on now this is not necessarily an efficient way to do the

prime numbers but this is a good way to generate them without missing out any one of the important facts that we use

all the time is that every number cannot only be factorized as we have seen into a number of different pairs of factors

it can actually factorize uniquely into the prime numbers that form it so for instance if we look at 12 we said that

12 was 2 times 6 it was also 4 times 3 it was 1 times 12 and so on but fundamentally it has 3 prime factors 2 2

again and 3 so depending on how we combine them for instance we can get 4 times 3 or we can get 2 times 6 and so

on but 2 times 2 times 3 is the absolute unique way of writing 12 as a product of prime numbers and using our

exponentiation notation we can condense this and put the two twos together and say it is 2 to the power 2 or 2 squared

times 3 similarly if we take a number like 126 then it is 2 times 3 6 times 3 18 times 7 so the prime factors are

precisely 2 3 twice and 7 and we can write this as 2 into 3 squared into 7 so this is very important because we use

it implicitly along a lot and we will see later how we use this so this is called the prime factorization right so

every integer can be decomposed into a product of primes in a unique way so to summarize we started with the natural

numbers which we use for counting which are the numbers 0 1 2 3 4 and so on then we extended these numbers with the

negative numbers and this gave us the set of integers so the integers include all the natural numbers as well as the

negative numbers 0 1 2 3 and so on minus 1 minus 2 minus 3 and so on we saw some basic arithmetic operations

on these the usual addition subtraction multiplication division and exponentiation we also looked at what

happens when we divide integers and we don't want to look at fractions then we talked about the quotient which is the

integer number of times that the dividend goes into the number and the remainder is also written as a mod B so

using this notation of a mod B we can talk about divisibility which we write with a vertical bar so a divides B if a

mod B is 0 so the factors of a number are those numbers which divide it and a prime number has exactly two factors 1

and itself and we can always decompose any integer uniquely into the list of factors prime factors that multiply out

to form that number