Number Systems
A number system is a way of expressing numbers using a consistent set of symbols and rules. Understanding number systems is essential in mathematics, computer science, and everyday life.
1. Natural Numbers (â„•)
Natural numbers are the counting numbers starting from 1:
- Example:
1, 2, 3, 4, 5, ...
2. Whole Numbers (W)
Whole numbers are natural numbers plus zero:
- Example:
0, 1, 2, 3, 4, ...
3. Integers (ℤ)
Integers include all whole numbers and their negatives:
- Example:
..., -3, -2, -1, 0, 1, 2, 3, ...
4. Rational Numbers (â„š)
Rational numbers can be written as a fraction p/q where p and q are integers and q ≠0.
Their decimal expansions either terminate or repeat.
- Example:
1/2 = 0.5,2/3 = 0.666…
5. Irrational Numbers
Irrational numbers cannot be expressed as fractions. Their decimal expansions are non-repeating and non-terminating.
- Example:
π ≈ 3.14159…,√2 ≈ 1.414…
6. Real Numbers (â„)
Real numbers include both rational and irrational numbers. They represent any number on the number line.
7. Imaginary and Complex Numbers (â„‚)
Imaginary numbers are multiples of i, where i² = -1. Complex numbers combine real and imaginary parts: a + bi.
- Example:
3 + 4i,-2 - 5i
8. Summary Table
| Number System | Symbols | Example |
|---|---|---|
| Natural | ℕ | 1, 2, 3… |
| Whole | W | 0, 1, 2… |
| Integers | ℤ | …-2, -1, 0, 1, 2… |
| Rational | â„š | 1/2, -3/4, 0.75 |
| Irrational | — | √2, π |
| Real | â„ | All numbers on the number line |
| Complex | â„‚ | 3 + 4i, -2 - 5i |
Number systems help us categorize and understand the properties of numbers. Mastery of number systems is essential before moving on to algebra, functions, and more advanced math topics.