Operational Integers
Operations with Integers
Introduction to Integers
- Integers are whole numbers including positive numbers, negative numbers and zero.
- The set of integers is denoted by "Z", which comes from the German word "Zahlen" meaning "to count".
Addition and subtraction of integers
- When adding two positives or two negatives, just add the numbers as normal and keep the sign. For example, 6 + 4 = 10 and -6 - 4 = -10.
- When adding a positive and a negative, treat it like a subtraction of the absolute values (ignoring the signs). Keep the sign of the larger number. For example, 7 + (-3) = 4.
- When subtracting an integer, we can think of it as adding the negative. So 7 - 3 = 7 + (-3) = 4.
Multiplication and division of integers
- If the signs of the integers are the same (either both positive or both negative), the result is positive. For instance, 4 x 5 = 20 and -4 x -5 = 20.
- If the signs are different (one positive, one negative), the result is negative. For instance, -4 x 5 = -20 or 4 x -5 = -20.
- The same rules apply for division. For example, -20 ÷ 5 = -4 and 20 ÷ -5 = -4.
Ordering integers
- Integers can be ordered on a number line. Higher positive integers are to the right and lower (or more negative) integers are to the left.
- The integer 0 is the pivot point between positive and negative integers.
Properties of Integer Operations
- The commutative property states that the order of addition or multiplication doesn't affect the result. For example, 2 + 3 = 3 + 2 and 2 x 3 = 3 x 2.
- The associative property states that the grouping of integers doesn't change the result in addition or multiplication. For example, (2 + 3) + 4 = 2 + (3 + 4) and (2 x 3) x 4 = 2 x (3 x 4).
- Distributivity of multiplication over addition or subtraction: For example, 2 x (3 + 4) = (2 x 3) + (2 x 4).
- Zero is the identity element for addition: any number plus zero equals the original number. For example, 5 + 0 = 5.
- One is the identity element for multiplication: any number times one equals the original number. For example, 5 x 1 = 5.
- Every integer has an additive inverse: for any positive integer, there is a negative integer that, when added to the positive integer, equals zero. For example, 3 + (-3) = 0.
- All integers except zero have a multiplicative inverse, which is its reciprocal. This is not another integer, except in the case of 1 and -1. For example, the multiplicative inverse of 2 is 0.5.