Introduction to Number Systems
This section outlines the classification of numbers on the number line, beginning with natural numbers and expanding to include whole numbers, integers, and rational numbers. The transition from one type to another emphasizes how each category encompasses the previous one.
- Natural Numbers (N): These are the counting numbers starting from 1, 2, 3, and so on.
- Whole Numbers (W): This set includes natural numbers along with zero (
0). Thus, W = {0, 1, 2, 3, ...}.
- Integers (Z): This set further encompasses negative numbers, represented as {..., -3, -2, -1, 0, 1, 2, 3, ...}.
- Rational Numbers (Q): Rational numbers consist of integers that can be expressed as a fraction of two integers, where the denominator is not zero.
The section also introduces the concept of irrational numbers, which cannot be expressed as a simple fraction and are identified as numbers with non-terminating, non-repeating decimal expansions. The section concludes by motivating students to explore the nature of numbers further through examples and exercises.
Example :
Find five rational numbers between 2 and 4.
Solution 1:
Recall that to find a rational number between \( r \) and \( s \), you can add \( r \) and \( s \) and divide the sum by 2.
\[ \frac{2 + 4}{2} = 3 \]
So, \( 3 \) is a number between 2 and 4. You can proceed in this manner to find four more rational numbers between these two values. These four numbers are: \( 2.5, 3.5, 2.75, 3.25 \).
Solution :
The other option is to find all the five rational numbers in one step. Since we want five numbers, we write 2 and 4 as rational numbers with a denominator of 5 + 1, i.e., \[ \frac{2}{1} = \frac{10}{5} \quad \text{and} \quad \frac{4}{1} = \frac{20}{5}. \]
So we can find that the five rational numbers between 2 and 4 are: \[ \frac{11}{5}, \frac{12}{5}, \frac{13}{5}, \frac{14}{5}, \frac{15}{5}. \]