The rules for integers involve understanding their basic properties and how they interact, while absolute value focuses on a number's distance from zero, regardless of its sign.
Understanding Integers
Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals.
- Positive Integers: These are numbers greater than zero (e.g., 1, 2, 3, ...).
- Negative Integers: These are numbers less than zero (e.g., -1, -2, -3, ...).
- Zero: This integer is neither positive nor negative.
Basic Integer Operations
Integers can undergo various mathematical operations, such as addition, subtraction, multiplication, and division. It's crucial to remember the rules for working with signs:
- Addition: When adding numbers with the same sign, add their absolute values and keep the same sign. For numbers with different signs, subtract the smaller absolute value from the larger one, and keep the sign of the number with the larger absolute value.
- Subtraction: Subtracting a number is the same as adding its opposite. For example, 5 - 3 is the same as 5 + (-3).
- Multiplication and Division:
- Multiplying or dividing two numbers with the same sign results in a positive number.
- Multiplying or dividing two numbers with different signs results in a negative number.
Absolute Value Explained
The absolute value of a number represents its distance from zero on the number line. The absolute value of a number is always non-negative. It is often denoted using vertical bars around the number, like |x|.
Rules of Absolute Value
- Positive Integers: The absolute value of a positive integer is the integer itself. For instance, |5| = 5.
- Negative Integers: The absolute value of a negative integer is the positive version of that integer. The reference states, "The absolute value of a negative integer will just be that integer without the negative sign." For example, |-5| = 5.
- Zero: The absolute value of zero is zero: |0| = 0.
Summary Table
| Number | Absolute Value | Explanation |
|---|---|---|
| 5 | 5 | Positive integer, remains the same. |
| -5 | 5 | Negative integer, negative sign is removed. |
| 0 | 0 | Zero remains the same. |
| 12 | 12 | Positive integer, remains the same. |
| -12 | 12 | Negative integer, negative sign is removed. |
Practical Implications
In practical terms, absolute value allows us to ignore the sign of a number when we are only interested in its magnitude. For example, when calculating distance, the displacement from a point can be negative or positive, but the actual distance traveled will always be the absolute value of the displacement.
