Real Numbers
Chapter 1 | Class 9–10 Mathematics | ThePiPath.Online
In previous classes, we have studied various types of numbers. Let us now gain more knowledge about them.
2² = 4, (−3)² = 9, (2/3)² = 4/9, (−3/4)² = 9/16, (√−5)² = −5
Therefore, 2, −3, 2/3, −3/4 etc. are real numbers, but √−5 is not a real number.
| Type | Definition | Set / Symbol | Examples |
|---|---|---|---|
| Natural Numbers | All counting numbers are called Natural Numbers. | N = {1, 2, 3, 4, 5, ...} | 1, 2, 3, 100 |
| Whole Numbers | All natural numbers together with zero form the set of Whole Numbers. | W = {0, 1, 2, 3, 4, ...} | 0, 1, 2, 50 |
| Integers | All natural numbers, zero, and their negatives together form Integers. | Z = {..., −3, −2, −1, 0, 1, 2, 3, ...} | −4, 0, 7 |
| Even Numbers | Numbers of the form 2m, where m is a natural number. | {2, 4, 6, 8, 10, ...} | 2, 4, 6, 8 |
| Odd Numbers | Numbers of the form (2m − 1), where m is a natural number. | {1, 3, 5, 7, 9, ...} | 1, 3, 7, 9 |
| Prime Numbers | Numbers divisible only by 1 and themselves. | {2, 3, 5, 7, 11, 13, ...} | 2, 3, 5, 7, 11 |
| Composite Numbers | Numbers divisible by at least one number other than 1 and themselves. | {4, 6, 8, 9, 10, 12, ...} | 4, 6, 9, 15 |
Draw a straight line XY which can be extended infinitely in both directions. Mark a point O on it representing zero (0). Taking a fixed unit distance, mark points on both sides of zero.
Points at distances 1, 2, 3, 4 units to the right of O represent integers 1, 2, 3, 4 respectively.
Points at distances 1, 2, 3, 4 units to the left of O represent integers −1, −2, −3, −4 respectively.
Since line XY can be extended infinitely on both sides, every integer can be represented by a unique point on it. This straight line is called the Number Line.
Since 0 can be written as 0/1, zero is also a rational number.
Every natural number is also a rational number because:
Similarly, every integer is a rational number:
Therefore, all integers are rational numbers.
2/5 = 4/10 = 6/15 = 8/20 = 20/50 ...
All such numbers are called Equivalent Rational Numbers.
We can write 3/5 as follows:
Therefore, five rational numbers equivalent to 3/5 are:
- (i) Every whole number is a natural number.
- (ii) Every integer is a rational number.
- (iii) Every rational number is an integer.
(i) FALSE — because 0 is a whole number but not a natural number.
(ii) TRUE — because every integer n can be written as n/1, which is of the form p/q. Hence every integer is a rational number.
(iii) FALSE — because the rational number 4/7 is not an integer.
Draw a straight line XY with a point O representing 0. Mark a point A to the right of O such that OA = 1 unit. Let B be the midpoint of OA.
Then OB = 1/2 unit. Now mark equal distances OB, 2OB, 3OB, 4OB to the right, representing 1/2, 1, 3/2, 2 etc. Similarly mark to the left, representing −1/2, −1, −3/2, −2 etc.
Draw a straight line XY.
Mark O as 0. To the right: OP = 1, OQ = 2, OR = 3 units. Divide QR into 5 equal parts. QA = 2/5 of a unit, so point A represents 2⅖ on the number line.
To the left: OS = 1, OT = 2, OU = 3, OV = 4 units. Divide UV into 5 equal parts. BU = 1/5 of a unit, so point B represents −3⅕ on the number line.
There are many methods to find rational numbers between two given rational numbers. The primary method is as follows:
Let p and q be two rational numbers. Add them and divide by 2. The result (p + q) / 2 is a rational number that lies between p and q. By continuing this process, infinitely many rational numbers can be found between any two rational numbers.
First rational number
Second rational number (between 2 and 5/2)
Third rational number (between 5/2 and 3)
Social Plugin