**Numbers**

- We use numbers mainly for counting.
- For instance if we see 7 balls or 7 pencil then we need to know that they are same number of things and for this we use number 7.
- So, 7 represent what is common to these 2 objects that there are 7 balls and 7 pencil.
- So, 7 is an abstract concept in that sense and it refers to a quantity.
- Therefore when we see a numbers of things we can count them.

**0( Zero) As A Whole Number Or Natural Number**

- The most important number of all which of the Indian origin is 0.
- So, it is important to have a way to represent something when there is nothing to count because without the 0 we can not use our place numbering system that we use to manipulate numbers.
- Numbers starting with zero are called natural numbers.
- As some books state that natural numbers start with 1,2,……… so, we use N with double line across it to represent the set of the natural numbers.
- To emphasize that we use in the natural number so we will put subscript o below N ( Nₒ)
- So we will write either N or Nₒ , but when we are talking about natural numbers it always iclude in natural numbers

**What can we do with natural numbers?**

- We can add them, substract them, multiply them, divide them.

**Do we always get natural number when we apply any of the above mention operation? the answer is No. **We always dont get natural number when we apply above mention operations. Hence it is explained below

**Subtraction**

- This operations fails in this test as because if we substract a large number form smaller number for eg 5-6 then we will get -1 which is not a natural number.
- So, we need to expand the scope of are numbers to allow above mention to happen. This is how we get negative numbers.
- Set which is the natural numbers extended with a negative numbers is what we called integers and we use Z with bar to denote them.
- So, natural number and integers are infinite set
- Thus subtraction takes away natural umbers ad give us what we called them integers.

**Mutiplication**

- When we say 7 times 4 what we actually wants to say is that take 7 objects and make their 4 copies.
- This is how we multiply when we take a number m and multiply it by n.
- So what we taking m plus m plus m plus n times this is what we called repeated addition
- We often use time sign the x sign for multiplication but this often cumbersome when we write out equations.
- So sometimes we replace this times sign by a dot and sometimes we dont write nothing at all.

But we dont use this in numbers as because imagine that if i write 7 4 like this then you do not know whether it is a number or it is 7 times 4. - Therefore we normally write a dot explicitly between them like 7 times 4 but when we have a names like m or n standing for numbers , then if we write mn then we assume m times n.
- Now we have integers , an integers have signs, they are positive and negative numbers.

So, we have to remember that when we multiply numbers with signs, the resulting number will also have that sign. - The sign rile state that if we have a negative sign and then we multiply it with the positive sign sign then the resulting number will also have negative sign but if multiply two negative sign together then the resulting number will have the positive sign.
- In short if we have even number of minus sign the we will get a positive number but if we have a odd number of negative sign then we will get the negative number.
- Instead of doing m plus m we can takes m times m and this is called square.
- And if we multiply m times then what we called is cube.
- So, to emphasize multiplication is repeated addition and exponentiation is repeated multiplication.

**Division**

- Division is actually repeated subtraction.
- You keep subtracting by the number you are trying to divide and finally if you hit zero then you divide it actually.
- The quotient the number of times you can actually divide without getting unto a fractional part and the is after you have a little bit over which you can not subtract one more is the reminder.
- The notation for the reminder is that modulus ( 1 mod5 = 4)
- Modulus is another word for reminder and it is written as mod.
- So, 19 mod5 is 4, and it is same as the reminder when 19 divided by 5 is 4.

**What is factor?**

- A factor is a number which divides a bigger number evenly without any reminder.
- So, a divides b, if b mod a is 0 and we write this with a vertical bar a|b.
- a|b, So, on the left is smaller number, and on the right is the bigger number.
- So, a divides b what this is supposed to say the other way of thinking about it is that b is some multiple of a.
- So, b is a divides b then a times some k is equal b ( axb = b)
- So, We have some multiple the some number of times that a goes into b.
- Therefore b is a multiple of a.
- The symbol that we use for not being a divisor is just to put a stroke across that vertical line.
- So, a divides b is the same as saying that a is a factor of b and it is easy to see that factors must come in pairs because is a divides b then a goes into b 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.
- For every number n, 1 times n is n, the pair for 1 is always the number itself.
- If it is the same as the number itself and this happens when the number actually happens to be a pefect squre that is, it is some number multiplied by itself.
- For instance consider 36; so, 36 is 6 times 6.
- So, if you look at the factors of 36 and groups them in pairs then we have 1 and 36.
- If we have something which is not a square you will have even number of factors , you will have 2 plus 2 plus 2 if something is a square you will have an odd number of factors.
- When you finally come to the number of which it is a square that number will come only once in the list of factors.

**Prime Numbers**

- A prime number is one which has no factors other than 1 and itself.
- So, 1 is a factor always and 1 times n is 1.
- Therefore we write p for a prime number.
- So, a prime number has only two factors 1 and p.
- It is important that it must have two factors, two separate factors.
- One technically is not a prime number because it has only one factor in itself because one times 1 is 1 and so, only factor that 1 has is 1.
- Therefor the smallest prime number is 2 because it has two factors 1 and itself 2 and no other factors.
- After 2 no even numbers can be primes because they are all multiples of 2 and 2 divides them.
- Sieve of Eratoshthenes to generate prime numbers which is whenever you discover a prime, you knock off all the numbers which are multiple of it.
- Every number can not only be factorized but it can actually factorize uniquely into the prime numbers that form it.
- Integer which can be decomposed into a product of primes in a unique way.

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