Here’s a breakdown of key mathematical terms and concepts:
1. Prime Numbers
- Definition: Numbers with only two divisors: 1 and themselves.
- Examples: 2, 3, 5, 7, 11, 13.
- 2 is the only even prime number.
- 1 is NOT a prime number (it has only one divisor).
- Fun Facts:
- Infinite primes exist (proven by Euclid).
- Useful in computing and simplifying fractions.
2. Squares and Square Roots
- Squares: A number multiplied by itself, written as ( x^2 ).
- Example: ( 5^2 = 25 ).
- Applications: Area calculations (e.g., 5m × 5m = 25m²).
- Square Roots: A number that gives the original value when squared, denoted as ( \sqrt{x} ).
- Example: ( \sqrt{25} = 5 ).
- Not all numbers have whole square roots (e.g., ( \sqrt{13} = 3.60555 )).
3. Exponents, Indices, and Powers
- Definition: Repeated multiplication of a number.
- ( 2^3 = 2 * 2 * 2 = 8 ).
- Also called powers or orders.
- Standard Form (Scientific Notation): Expresses very large or small numbers as ( a * 10^n ).
- Example: ( 2,000,000 = 2 * 10^6 ).
- Negative powers represent fractions (e.g., ( 1 * 10^{-3} = 0.001 )).
4. Factors and Multiples
- Factors: Numbers that divide another number completely.
- Example: Factors of 30 are 2, 3, 5, 6.
- Multiples: Numbers obtained by multiplying a given number by others.
- Example: 30 is a multiple of 15, 6, and 5.
5. Infinite (Irrational) Numbers?
- Definition: Numbers that never end or repeat.
- Example: Pi (( \pi )) = 3.14159…
- Recurring Numbers: A form of infinite numbers with repeating decimals.
- Example: ( \frac{1}{3} = 0.333. ).
- Finite Numbers: Have a fixed number of digits.
- Example: 1.5, 3, 0.625.
6. Real, Imaginary, and Complex Numbers
- Real Numbers: Numbers that exist on the number line (positive, negative, integers, decimals).
- Imaginary Numbers: Used for the square root of negative numbers (( i = \sqrt{-1} )).
- Practical use: Electrical engineering.
- Complex Numbers: Combine real and imaginary numbers.
- Example: ( 3 + 4i ).
Why These Concepts Matter
While some concepts may seem abstract, they have real-world applications:
- Prime numbers: Cryptography.
- Squares: Area and volume calculations.
- Imaginary numbers: Electronics and wave theory.
- Standard Form: Simplifies scientific data.
With these basics, you'll better understand the "language" of mathematics and its practical relevance!?