Numbers

Lesson no. 1: Definitions of Various Types of Numbers:

1. Natural Numbers N:

The set of numbers used for counting is Natural numbers.
1, 2, 3, 4, 5 and so on form the set of Natural numbers.
Natural numbers are denoted by N.

2. Whole Numbers W:

Natural numbers including 0 are called Whole Numbers.
So, 0, 1, 2, 3, 4, 5, and so on are all whole numbers.
Whole numbers are denoted by W.

3. Integers Z:

 

Whole Numbers 0, 1, 2, 3, 4, 5 and so on are Positive Integers and negative numbers such as -1, -2, -3, -4, -5 and so are Negative Integers.
Positive integers are denoted by Z+ and Negative Integers are denoted by Z-
Note:
1. 0 is neither a positive integer nor a negative integer.
2. Non-negative integers are Whole Numbers, i.e. positive integers including 0.

4. Rational Numbers:

Fractions in which both the numerator and denominator are integers are called Rational numbers.
1/2, 3/4, 6/7. -8/9, -6/5 are all Rational Numbers.

Note:

1. The denominator in a Rational Number should not be 0
2. Numerator and Denominator cannot have numbers that do not have perfect squares.
For example, v2/3 is not a rational number, because 2 is not a perfect square.

5. Irrational Numbers:

Numbers that cannot be written as a rational number are irrational numbers.

Examples:

1. Numbers that do not have perfect squares are Irrational Numbers.
Example: v2, v3, v5 and so on.

6. Real Numbers:

The set of numbers including both rational and irrational numbers is Real Numbers.

7. Factor:

A number that divides another number leaving remainder 0 is called a Factor.
2 is a factor of 4 as it divides 4 leaving remainder 0.
2 is not a factor of 9 as it leaves remainder 1 after dividing 9.

8. Multiple:

The definition of a Multiple is somewhat complex.
First of all, Multiples can be both Integral Multiples and Non-integral Multiples.
For example, 6 is an integral multiple of 2 and 3 as:
6 = 2 × 3.
But, 6 can also be written as:
6 = 0.5 × 12.
Here, 6 in a Non-Integral Multiple of only 0.5, but not 12.

9. Prime Number:

A positive integer greater than 1 that has only two positive factors, 1 and itself is called a Prime Number.
Examples: 3, 5, 7 and others.
The positive factors of 3 are only 1 and 3,
The positive factors of 5 are only 1 and 5,
The positive factors of 7 are only 1 and 7
Note:
1. 1 is neither a prime number nor a composite number.
2. Negative numbers such as -3, -5, -7, and so on are not prime numbers. By definition, prime numbers are greater than 1.

10. Composite Numbers:

A positive number that has positive factors besides 1 and itself is called a Composite Number.
Numbers that are not primes are composites.
Examples: 4, 9, 15 and so on.
The positive factors of 4 are 1, 4 and 2, i.e. besides 1 and 4, also 2;
The positive factors of 9 are 1, 9 and 3, i.e. besides 1 and 9, also 3;
The positive factors of 15 are 1, 15, and 3 and 5, i.e. besides 1 and 15, also 3 and 5.

11. Co-Primes:

Two positive integers that do not have any common factor other than 1 are called Co-primes or Relative Primes.
Examples: 8 and 9, 15 and 16, 17 and 19.
Note: Co-primes need not be primes themselves.
As in the above examples, 8 and 9 are both not primes, still they are primes with respect to each other as they have only 1 as their common factor.

12. Even Number:

A number divisible by 2 is called an Even number.
2, 4, 6 and so on.

13. Odd Number:

A number not divisible by 2 is called an Odd Number.
1, 3, 5, 7, and so on.

14. Prime Factors:

Factors of a number that are prime numbers are called prime factors.
The positive factors of 12 are 1, 2, 3, 4, 6 and 12. Among these factors, 2 and 3 are prime factors of 12.

15. Prime Factorization of a Number:

Expressing a positive integer as product of its prime factors is called prime factorization of the number.
36 = 4 × 9 = 22 × 32 and 100 = 4 × 25 = 22 × 52

How to find the number of numbers divisible by both 2 and 3

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