Thinking Mathematically The Integers: Order of Operations

Integers When zero is added to the natural numbers the result is the set of “whole numbers.” Whole numbers = {0,1,2,3,4,5,6,7, ... }. When the negative of each natural number is added to the whole numbers the result is the set of integers. Integers = { ... -5,-4,-3,-2,-1,0,1,2,3,4,5, ... }.

A Number Line If zero and one are marked on a line, then every number can be graphed on that line by marking its distance from zero in either a positive or negative direction. Such a line is called a “number line.” Plot 2 on a number line. Plot -3 on a number line. 1 2 3 -1 -2 -3

Direction on a Number Line The symbols < and > are used to indicate directions on a number line. Plot 2 and -3 on a number line. 1 2 3 -1 -2 -3 Since -3 is “to the left” of 2 the relationship between them can be written -3 < 2 or 2 > -3.

Absolute Value The absolute value of an integer a, denoted by |a|, is the distance from zero to a on the number line. Because absolute value describes a distance, it is never negative. Example: |2| = 2 Example: |-3| = 3 3 2 1 2 3 -1 -2 -3

Adding Integers Every integer can be identified by an arrow whose length is the absolute value and whose direction is the direction from zero. Two integers are added by following one arrow by the other. Example: 1 + 2 = 3 1 + 2 = 3 1 2 1 2 3 -1 -2 -3

Rules for Addition of Integers If the integers have the same sign, Add their absolute values. The sign of the sum is the same as the sign of the two numbers. If the integers have different signs, Subtract the smaller absolute value from the larger absolute value. The sign of the sum is the same as the sign of the number with the larger absolute value.

Subtracting Integers -5 -1 -2 -3 The negative of a number is called its “additive inverse.” Subtraction is simply addition of an additive inverse using the “arrow” method. Example: 2 - 5 = 2 + (-5) = -3 2 + (-5) = -3 -5 2 1 2 3 -1 -2 -3

Rules for Multiplying Integers The product of two integers with the same sign is found by multiplying their absolute values. The product is positive. The product of two integers with different signs is found by multiplying their absolute values. The product is negative. The product of 0 and any integer is 0.

Rules for Dividing Integers The quotient of two integers with the same sign is found by dividing their absolute values. The quotient is positive. The quotient of two integers with different signs is found by dividing their absolute values. The quotient is negative. Zero divided by any nonzero integer is zero. Division by 0 is undefined.

Order of Operations A list of arithmetic operations such as 2 + 5 x 4 is ambiguous without parenthesis to indicate whether the addition or multiplication should be done first. 2 + (5 x 4) = 2 + 20 = 22 (2 + 5) x 4 = 7 x 4 = 28 When using a calculator it is very important to know which operation is done first.

Order of Operations Perform all operations within grouping symbols. Evaluate all exponential expressions. Do all multiplications and divisions in the order in which they occur, working from left to right. Finally, do all additions and subtractions in the order in which they occur, working from left to right.

Thinking Mathematically The Integers: Order of Operations