A number is a mathematical object which is used to count and measure.
Let's recall the classification of numbers that we learned in the previous grades.
Natural Numbers:
- The numbers that are used for counting which starts from 1.
- Natural number is denoted by \(N\). \(N = \{1, 2, 3, 4, \dots\}\).
Whole numbers:
- Whole numbers are a set of numbers including the set of natural numbers (1 to infinity) and the integer '0'.
- Whole number is denoted by \(W\). \(W = \{0, 1, 2, 3, 4, \dots\}\).
Integer:
- An integer is a whole number (not a fractional number) that can be positive, negative, or zero.
- Integer is denoted as \(Z\). \(Z = \{\dots -3, -2, -1, 0, 1, 2, 3, \dots\}\)
Rational Numbers:
- A number is called a rational number, if it can be written in the form \(p/q\), where \(p\) and \(q\) are integers and \(q \neq 0\).
- Rational is denoted by \(Q\). \(Q = \{\dots -25, \dots, -1/2, \dots, 0, \dots, 47/31, \dots, 100/5, \dots\}\)
Important!
- Every natural number is a whole number, integer and rational.
- Every whole number is an integer and rational.
- Every integer is a rational number.
Method to find the rational number between any two rational numbers.
Method 1: [Average method]
Find the midpoint of any two rational numbers by calculating \(\frac{a+b}{2}\). Repeat the process with the new values to find more.
Example: Between \(6\) and \(7\), the first number is \(\frac{13}{2}\).
Method 2: [Same denominator method]
To find \(n\) numbers, use \(n+1\) as a common denominator.
Multiply the rational numbers, \(a\) and \(b\) by \(\frac{n+1}{n+1}\).
Rational number \(Q\) does not have a unique representation in the form of \(p/q\). These are called an equivalent rational number.
Irrational Numbers
The Pythagoreans in Greece was the first to discover the numbers which were not rational, around \(400 BC\). These numbers are called an irrational number.
An irrational number is a number that cannot be expressed as a fraction \(p/q\), where for any integers \(p\) and \(q\).
Irrational numbers have decimal expansions that neither terminate nor become periodic.
Irrational numbers have decimal expansions that neither terminate nor become periodic.
Important!
As there are infinitely many rationals, so there are infinitely many irrational numbers too.
Example:
\(\sqrt{2}= 1.4142135623730950488…\)
\(\sqrt{3}= 1.7320508075688772935…\)
\(\pi= 3.1415926535897932384626433...\)
Real Numbers
The real numbers are numbers that include both rational and irrational. In short, the real number is the union of rational and irrational numbers. Therefore, the real number is either rational or irrational.
The real number is denoted by \(R\).
Important!
Each real number can be represented by a unique point on the number line. Conversely, each point on the number line represents a unique real number. Every real number is either rational or irrational.
Analysis of the decimal expansions of different types of numbers:
|
Numbers
|
Nature of the decimal expansion
|
Type of number
|
| \(1\div 125 = 0.008\) | Terminating expansion. | Rational |
| \(152\div 333 = 0.456456456... =\overline{0.456}\) | Recurring and non-terminating decimals. | Rational |
| \(\pi=3.14159 26535 89793 23846\) | Non -repeating and non- terminating decimals. | Irrational |
The decimal expansion of a rational number is will terminating or non-terminating and recurring. Conversely, the decimal expansion of a number is terminating, or non-terminating recurring is a rational number.
Example:
1. Prove that \(0 .2363636... = 0.2\overline{36}\) is a rational number. That is, show that \(0.2\overline{36}\) can be expressed in \(p/q\), where \(p\) and \(q\) are integers with \(q \neq (0\).
Solution:
Let us take the provided number as \(x\).
That is \(x = 0.2363636...\)
Note the number \(x\) - two of digits\(36\) repeats here.
Here we have to make multiplies of \(x\) in such a way that the repeated decimals will be the same.
\(10x = 2.363636...\)
Multiply \(x\) by \(1000\).
\(1000x = 236.363636...\)
Subtract \(10x\) from \(1000x\),
\(1000x - 10x = 236.363636... - 2.363636...\)
\(990x = 234\)
\(x = 234/990\)
Therefore, the fractional form of the rational number \(0.2\overline{36}\) is \(\frac{234}{990}\).
Sometimes it is easy to identify the provided number is irrational or not by checking the following properties.
Properties of irrational numbers:
- Addition, subtraction, multiplication and division of two irrational number is may or may not be irrational.
- Addition of rational and irrational number is always irrational.
- Subtraction of rational and irrational number is always irrational.
- Multiplication of a non-zeror rational and irrational is always irrational.
- Division of a non-zero rational and irrational is always irrational.