This document discusses different types of number systems. It begins by introducing natural numbers, which are counting numbers formed by repeated addition of 1. Whole numbers include all natural numbers and 0. Integers extend whole numbers infinitely in both the positive and negative directions. Rational numbers are numbers that can be written as fractions p/q where p and q are integers. Irrational numbers have non-repeating decimal expansions and cannot be written as fractions. Real numbers include all rational and irrational numbers and are represented on the number line. Methods for finding rational numbers between two given numbers and representing different types of numbers on the number line are also described.

1. MATHS
PROJECT

2. NUMBER SYSTEMS

3. INTRODUCTION
NATURAL NUMBERS
WHOLE NUMBERS
INTEGERS
RATIONAL NUMBERS
 FINDING RATIONAL NUMBERS
IRRATIONAL NUMBERS
 RATIONALISING DENOMINATORS
 REAL NUMBERS
 REAL NUMBERS AND THEIR DECIMAL
EXPANSION
NUMBER LINES

4. INTRODUCTION :
 It's the way we categorize numbers. There
may be an infinite amount of them, but they all
fall nicely in several ranges.
The first group is part of the second, which is
part of the third, which is part of the fourth,
and the pattern continues.

5. NATURAL NUMBERS :
Natural numbers (also called
counting numbers) can be
formed by repeated addition of
the number 1.
1, 2, 3, 4, 5, 6, 7... and so on

6. WHOLE NUMBERS :
The group of whole
numbers is another name
for the natural numbers but
always includes 0:
0, 1, 2, 3, 4, 5...

7. INTEGERS :
Integers include all whole numbers
but also extend infinitely into the
negative numbers. Except for zero
(which is neither positive nor
negative), all integers are assumed to
be positive if they do not have a sign
marking them negative.
...-4, -3, -2, -1, 0, 1, 2, 3, 4, 5....

8. RATIONAL NUMBERS :
Rational numbers are any number that can be written
in the form of p/q , where p and q are any integer and
q does not equal zero.
This includes fractions and whole numbers (The
whole number 32 can be represented as 32/1). Many
decimals are rational numbers, too, even non-terminating
repeating ones such as 0.333.... and
0.412412412.... 0.333... can be expressed as 3/9
0.412412412... can be expressed as 412/999.

9. Finding rational numbers between two numbers:
There are two methods to find rational
numbers. They are:
 mid-value method
 denominator method

10. finding rational numbers by mid-value method
• Find out a rational number lying halfway between 2/7 and 3/4.
Solution:
Required number = 1/2 (2/7 + 3/4)
= 1/2 ((8 + 21)/28)
= {1/2 × 29/28)
= 29/56
Hence, 29/56 is a rational number lying halfway between 2/7 and
3/4.

11. finding rational numbers by denominator mehod :
Find 7 rational numbers between ¾ and 7/4 .
Solution:
Multiply numerator and denominator by 7+1=8.
¾*8/8=24/32
7/4*8/8=56/32
The 7 rational numbers are :
25/32,26/32,27/32,28/32,29/32,30/32,31/32.

12. IRRATIONAL NUMBERS :
This group is completely exclusive from all the
aforementioned groups. It is its own group.
Irrational numbers are any number which can
not be written in the form of p/q. The square
root of any number other than the square of an
integer (0, 1, 4, 9, 16 ...) is irrational. Irrational
numbers have non-repeating decimal
expansions. Any number which is a repeating
decimal is rational.

14. Real nmbers:
 The collections of rational and
irrational numbers are known as real
numbers.
 The real numbers are “all the
numbers” on the number line.

15. REAL NUMBERS AND THEIR DECIMAL
EXPANSION:

16. Let x=0.333…..(1)
Multiply both sides by 10.
10x=3.333….(2)
Subtract (1) from (2),
10x-x=3.333-0.333
9x=3
x=3/9

17. Number lines :
Representation of
natural numbers
 whole numbers
 integers
 rational numbers
 irrational numbers
on number lines.

18. Natural numbers on number line

19. Whole numbers on number line

20. Integers on number line

21. Rational numbers on number line

22. Irrational numbers on number line
1. First of all draw the number line.
2. Mark point A at "0" and B at "1".
This means AB = 1 Unit.
3. Now, at B, draw BX perpendicular
to AB.
4. Cut off BC = 1 Unit.
5. Join AC.
6. By Pythagoras theorem in right
triangle ABC, we get AC = √2 Units.
7. Now, with radius AC and centre A,
mark a point on the number line.
Let the marked point is M. M
represents √2 on the number line.

23. Thank you