NUMBER SYSTEM

MATHS

PROJECT

NUMBER SYSTEMS

CONTENTS :INTRODUCTIONNATURAL NUMBERSWHOLE NUMBERSINTEGERSRATIONAL NUMBERS FINDING RATIONAL NUMBERSIRRATIONAL NUMBERS RATIONALISING DENOMINATORS REAL NUMBERS REAL NUMBERS AND THEIR DECIMAL EXPANSIONNUMBER LINES

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.

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

WHOLE NUMBERS :

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

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....

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.

Finding rational numbers between two numbers:

There are two methods to find rational numbers. They are:

mid-value method

denominator method

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.

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.

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.

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.

REAL NUMBERS AND THEIR DECIMAL EXPANSION:

Represent 0.333….in the form of p/q.

Let x=0.333…..(1)Multiply both sides by 10.10x=3.333….(2)Subtract (1) from (2),10x-x=3.333-0.3339x=3x=3/9

Number lines :

Representation of natural numbers whole numbers integers rational numbers irrational numbers

on number lines.

Natural numbers on number line

Whole numbers on number line

Integers on number line

Rational numbers on number line

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.

Thank you