Introduction to Integers
Integers expand our number system beyond just natural numbers. They include all the natural numbers (1, 2, 3...), their negatives (-1, -2, -3...), and zero (0).
Definition and Notation
The set of integers is denoted by the letter Z (from the German word "Zahlen," meaning numbers).
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Integers do NOT include fractions or decimals.
Types of Integers
- Positive Integers: {1, 2, 3, ...} (Same as Natural Numbers)
- Negative Integers: {..., -3, -2, -1}
- Zero: 0 is strictly an integer. It is neither positive nor negative.
Number Line Representation
Integers can be represented on a number line.
- Zero is at the center.
- Positive integers are to the right of zero.
- Negative integers are to the left of zero.
As you move to the right, numbers increase. As you move to the left, numbers decrease.
Example: -5 < -2 < 0 < 3
Properties of Addition and Subtraction
- Closure: Adding or subtracting two integers always results in an integer. (e.g., 2 - 5 = -3, which is an integer). Note: Subtraction IS closed for integers (unlike natural numbers).
- Commutative (Addition): a + b = b + a (e.g., -2 + 5 = 3 and 5 + (-2) = 3). However, subtraction is NOT commutative (5 - 2 ≠ 2 - 5).
- Associative (Addition): (a + b) + c = a + (b + c).
- Additive Identity: a + 0 = a.
- Additive Inverse: For every integer 'a', there exists '-a' such that a + (-a) = 0.
Multiplication Rules
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Negative × Positive = Negative