#### 1. Addition
**Addition** combines two numbers to get their sum.
- **Formula:** `a + b`
- **Example:** If `a = 3` and `b = 5`, then `a + b = 3 + 5 = 8`.
**Properties of Addition:**
- **Commutative Property:** `a + b = b + a`
- **Associative Property:** `(a + b) + c = a + (b + c)`
- **Identity Element:** `a + 0 = a` (0 is the additive identity)
#### 2. Subtraction
**Subtraction** finds the difference between two numbers.
- **Formula:** `a - b`
- **Example:** If `a = 7` and `b = 4`, then `a - b = 7 - 4 = 3`.
**Properties of Subtraction:**
- **Not Commutative:** `a - b ≠ b - a`
- **Not Associative:** `(a - b) - c ≠ a - (b - c)`
#### 3. Multiplication
**Multiplication** scales one number by another.
- **Formula:** `a * b`
- **Example:** If `a = 6` and `b = 4`, then `a * b = 6 * 4 = 24`.
**Properties of Multiplication:**
- **Commutative Property:** `a * b = b * a`
- **Associative Property:** `(a * b) * c = a * (b * c)`
- **Distributive Property:** `a * (b + c) = a * b + a * c`
- **Identity Element:** `a * 1 = a` (1 is the multiplicative identity)
#### 4. Division
**Division** splits one number by another.
- **Formula:** `a / b`
- **Example:** If `a = 8` and `b = 2`, then `a / b = 8 / 2 = 4`.
**Properties of Division:**
- **Not Commutative:** `a / b ≠ b / a`
- **Not Associative:** `(a / b) / c ≠ a / (b / c)`
- **Identity Element:** `a / 1 = a` (1 is the multiplicative identity for division)
#### 5. Exponentiation
**Exponentiation** raises a number to the power of another number.
- **Formula:** `a^b` (a raised to the power b)
- **Example:** If `a = 2` and `b = 3`, then `a^b = 2^3 = 8`.
**Properties of Exponentiation:**
- **Power of a Product:** `(a * b)^c = a^c * b^c`
- **Power of a Quotient:** `(a / b)^c = a^c / b^c`
- **Power of a Power:** `(a^b)^c = a^(b * c)`
#### 6. Roots
**Roots** find the number that, when multiplied by itself a certain number of times, gives the original number.
- **Formula:** `√a` (square root of a), `∛a` (cube root of a)
- **Example:** The square root of 9 is `√9 = 3` because `3 * 3 = 9`.
**Properties of Roots:**
- **Root of a Product:** `√(a * b) = √a * √b`
- **Root of a Quotient:** `√(a / b) = √a / √b`
- **Root of a Power:** `(a^b)^(1/c) = a^(b / c)`
#### 7. Absolute Value
**Absolute Value** measures the distance of a number from zero on the number line, always as a positive value.
- **Formula:** `|a|`
- **Example:** The absolute value of -7 is `|-7| = 7`.
**Properties of Absolute Value:**
- **Non-Negative:** `|a| ≥ 0`
- **Identity Property:** `|0| = 0`
- **Triangle Inequality:** `|a + b| ≤ |a| + |b|`
#### Summary
Understanding these operations helps in performing arithmetic and solving problems involving real numbers. Here's a recap:
1. **Addition:** Combines numbers to get a sum.
2. **Subtraction:** Finds the difference between numbers.
3. **Multiplication:** Scales numbers by multiplying.
4. **Division:** Splits numbers into fractions.
5. **Exponentiation:** Raises numbers to a power.
6. **Roots:** Finds the original number by reversing exponentiation.
7. **Absolute Value:** Measures the distance from zero, always positive.
Mastering these basic operations provides a strong foundation for more advanced mathematical concepts and problem-solving skills.
