The story of Maths | Part-1 Getting familiar with Numbers

Natural Numbers

1, 2, 3, 4, 5….

Counting Numbers are termed as Natural Numbers by Mathematician.

Zero(0) is not part of the Natural Number as we cannot point to some object and say it’s counting is Zero (0).

There is NO LARGEST Number, there is no smallest number.

Whole Numbers

The Natural Numbers along with Zero form the collection of whole numbers.

Closure Property

A set has closure under an operation if the performance of that operation on numbers of the set always produces a member of the same set; in this case, we also say that the set is closed under the operation.

Sum of any two whole number is a whole number i.e. the collection of whole numbers is closed under addition. — Closure Property for the addition of whole numbers.

• Whole Numbers are closed under addition and also under multiplication.

Commutative

Involving the condition that a group of quantities connected by operators gives the same result whatever the order of the quantities involved.

You can add two whole numbers in any order.

Addition and multiplication is commutative for whole numbers.

e.g.: 1 + 2+ 3 = 6; 3 +2 + 1 = 6} Order of addition doesn’t matter.

Associative

Associative property states that you can add or multiply regardless of how number are grouped. (Parenthesis)

(2 + 4) + 6 = 12;   2 + (4 + 6) = 12} Associative Property} Grouping doesn’t affect the addition.

 

When you add Zero to any whole number, it will return the same number, Zero is called an identity for an addition of whole numbers or additive identity for whole numbers.

1 is the identity for multiplication of whole numbers or multiplicative identity for whole numbers.

Playing with Numbers

Factors and Multiples:

1. A number is a multiple of each of its factors.
2. 1 is a factor of every number.
3. Every number is a factor of itself.
4. Every factor of a number is an exact divisor of that number.
5. Every factor is less than or equal to the given number.
6. The number of factors of a given number is finite.
7. Every multiple of a number is greater than or equal to that number.
8. The number of multiples of a given number is infinite.
9. Every number is a multiple of itself.

 

Prime and Composite Numbers.

The number other than 1 whose only factors are 1 and the number itself is called Prime Number.

Numbers having more than two factors are called Composite Numbers.

1. 1 is neither a prime nor a composite number.
2. Sieve of Eratosthenes to find out the prime number from 1 to 100.

Even and odd numbers:

Numbers which are multiples of 2 are called even number and numbers which are not multiple of 2 are called odd numbers.

# 2 is the smallest prime number which is even.

# Every prime number except 2 is Odd.

Divisibility

Divisibility rules:

2: Numbers which carry 2, 4, 6, 8, 0 at One’s place are divisible by 2.

3: If the sum of the digit is a multiple of 3 then the number is divisible by 3.

6: If a number is divisible by 2 & 3 both then it is divisible by 6.

4: A number with 3 or more digits is divisible by 4 if the number formed by its least two digits (i.e. one’s and ten’s) is divisible by 4.

8: A number with 4 or more digits is divisible by 8 if the number formed by the last three digits is divisible by 8.

9: If the sum of the digits of a number is divisible by 9, then the number is divisible by 9.

11: Find the difference between the sum of the digits at odd places (from the right) and the sum of the digits at the even places (from the right) of the number. If the difference is either zero (0) or divisible by 11, then the number is divisible by 11.

 

Common factors and common Multiples

• Two number having only 1 as a common factor is called co-prime numbers. (e.g. 4 & 15)
• If a number is divisible by another number then it is divisible by each of the factors of their number.
• If a number is divisible by two co-prime numbers then it is divisible by their product also.
• If two given numbers are divisible by a number, then their sum is also divisible by that number.
• If two given numbers are divisible by a number then their difference is also divisible by that number.

 

Highest Common Factor/ Greatest Common Divisor (GCD)

The highest common factor (HCF) of two or more given numbers is the highest (or greatest) of their common factors.

LCM (Lowest Common Multiple)

Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples.

2|20 25 30

2|10 25 15

3|5 25 5

5|1 5 1

5|1 1 1

LCM= 2x2x3x5x5= 300

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Yash

Bibliography:

Maths textbooks and Wiki