1 Order of Operations & Integers
Learning Outcomes
By the end of this chapter, learners will be able to:
- perform operations in the correct arithmetic order of precedence,
- understand the operation symbols,
- practice skills using acronym BEDMAS,
- review basic operations with integers and exponents.
Introduction
Try It – Solve the Question
You have just won [latex]\$50,000[/latex] in a contest.
Congratulations! But before you can claim it, you are required to answer a mathematical skill-testing question, and no calculators are permitted. After you hand over your winning ticket to the redemption agent, she hands you your time-limited skill-testing question:
[latex]2\times5+30\div5[/latex].
What does this question equal?
Solution
If you figured out the solution is [latex]16[/latex], you are on the right path.
If you thought it was something else, this is a great time to review order of operations.
The Symbols
While some mathematical operations such as addition use a singular symbol (+), there are other operations, like multiplication, for which multiple representations are acceptable. With the advent of computers, even more new symbols have crept into mathematical symbology. The table below lists the various mathematical operations and the corresponding mathematical symbols you can use for them.
| Mathematical Operation | Symbol or Appearance | Comments |
| Brackets | [latex]()[/latex] or [latex][\; ][/latex] or [latex]\{\}[/latex] | In order, these are known as round, square, and curly brackets. |
| Exponents | [latex]2^3[/latex] or [latex]2\hat{}3[/latex] | In [latex]2^3[/latex], [latex]3[/latex] is the exponent; an exponent is always written as a superscript. On a computer, the exponent is recognized with the ^ symbol; [latex]2^3[/latex] is represented by 2^3. |
| Multiplication | [latex]\times[/latex] or [latex]\ast[/latex] or [latex]\cdot[/latex] or [latex]2(2)[/latex] or [latex](2)(2)[/latex] | In order, these are known as times, star, or bullet. The last two involve implied multiplication, which is when two terms are written next to each other joined only by brackets. |
| Division | [latex]/[/latex] or [latex]\div[/latex] or [latex]\frac42[/latex] | In order, these are known as the slash, divisor, and divisor line. Note in the last example that the division is represented by the horizontal line. |
| Addition | [latex]+[/latex] | There are no other symbols. |
| Subtraction | [latex]−[/latex] | There are no other symbols. |
You may wonder if the different types of brackets mean different things. Although mathematical fields like calculus use specialized interpretations for the different brackets, use of all the brackets helps the reader visually pair up the brackets. Consider the following two examples:
Example 1: [latex]3 × (4 / (6 - (2 + 2)) + 2)[/latex]
Example 2: [latex]3 × \{4 / [6 - (2 + 2)] + 2\}[/latex]
Notice that in the second example you can pair up the brackets much more easily, but changing the shape of the brackets did not change the mathematical expression. This is important to understand when using a calculator, which usually has only round brackets. Since the shape of the bracket has no mathematical impact, solving example 1 or example 2 would involve the repeated usage of the round brackets.
BEDMAS
In the section opener, your skill-testing question was [latex]2\times5+30\div5[/latex]. Do you just solve this expression from left to right, or should you start somewhere else? To prevent any confusion over how to resolve these mathematical operations, there is an agreed-upon sequence of mathematical steps commonly referred to as BEDMAS.
How to use BEDMAS
BEDMAS is an acronym for Brackets, Exponents, Division, Multiplication, Addition, and Subtraction.
Step 1: Brackets must be resolved first. As brackets can be nested inside of each other, you must resolve the innermost set of brackets first before proceeding outwards to the next set of brackets. When resolving a set of brackets, you must perform the mathematical operations within the brackets by following the remaining steps in this model (EDMAS). If there is more than one set of brackets but the sets are not nested, work from left to right and top to bottom.
Step 2: If the expression has any exponents, you must resolve these next. Remember that an exponent indicates how many times you need to multiply the base against itself. For example, [latex]2^3=2\times2\times2[/latex].
Step 3: The order of appearance for multiplication and division does not matter. However, you must resolve these operations in order from left to right as they appear in the expression.
Step 4: The last operations to be completed are addition and subtraction. The order of appearance doesn’t matter; however, you must complete the operations working left to right through the expression.
Also note that on most calculators you have two ways to key in an exponent:
If the exponent is squaring the base ([latex]3^2[/latex]), press [latex]3[/latex] [latex][x^2][/latex] to obtain the solution of [latex]9[/latex].
If the exponent is anything other than a [latex]2[/latex], you could use the [latex][x^y][/latex] button. For [latex]2^5[/latex], you press [latex]2[/latex] [latex][x^y][/latex] [latex]5 =[/latex] to get the solution of [latex]32[/latex].
Things to Watch For
Negative Signs
Remember that mathematics can use both positive numbers (such as [latex]+3[/latex]) and negative numbers (such as [latex]−3[/latex]). Positive numbers do not need to have the [latex]+[/latex] sign placed in front of them since it is implied. Thus [latex]+3[/latex] is written as just [latex]3[/latex]. Negative numbers, though, must have the negative sign placed in front of them. Be careful not to confuse the terminology of a negative number with a subtraction or minus sign.
For example, [latex]4 + (−3)[/latex] is read as four plus negative three and not four plus minus three. A common method to key a negative number on a calculator is to enter the number first followed by the [latex]±[/latex] button, which switches the sign of the number.
Horizontal Divisor Line
One of the areas in which people make the most mistakes involves the hidden brackets. This problem almost always occurs when the horizontal line is used to represent division. Consider the following mathematical expression:
[latex](4 + 6)\div(2 + 3)[/latex]
If you rewrite this expression using the horizontal line to represent the divisor, it looks like this:
[latex]\begin{align*}\frac{4+6}{2+3}\end{align*}[/latex]
Notice that the brackets disappear from the expression when you write it with the horizontal divisor line because they are implied by the manner in which the expression appears. Your best approach when working with a horizontal divisor line is to reinsert the brackets around the terms on both the top and bottom. Thus, the expression looks like this:
[latex]\begin{align*}\frac{(4+6)}{(2+3)}\end{align*}[/latex]
Employing this technique will ensure that you arrive at the correct solution, especially when using calculators.
Hidden and Implied Symbols
If there are hidden or implied symbols in the expressions, your first step is to reinsert those hidden symbols in their correct locations. In the example below, note how the hidden multiplication and brackets are reinserted into the expression:
[latex]4 \displaystyle \frac{3+2^2 \times 3}{(2+8) \div 2} \text{ transforms into } 4\times \frac{3+2^2 \times 3}{(2+8) \div 2}[/latex]
Once you have reinserted the symbols, you are ready to follow the BEDMAS model.
Calculators are not programmed to be capable of recognizing implied symbols. If you key in [latex]3(4 + 2)[/latex] on your calculator, failing to input the multiplication sign between the [latex]3[/latex] and the [latex](4 + 2)[/latex], you get a solution of [latex]6[/latex].
Some calculators ignore the [latex]3[/latex] since they don’t know what mathematical operation to perform on it. To have your calculator solve the expression correctly, you must punch the equation through as [latex]3 \times (4 + 2) =[/latex]. This produces the correct answer of [latex]18[/latex].
Sample Exercises 1.1
Evaluate each of the following expressions:
- [latex]2\times5+30\div5[/latex]
- [latex]{(6+3)}^2+18\div2[/latex]
- [latex]\begin{align*}4\;\times\;\left[\frac{\{3+2^2\times3\}}{\{(2+8)\div2\}}\right]\end{align*}[/latex]
Question A
Solution
Step 1: B – Are there brackets? No.
Step 2: E – Are there exponents? No.
Step 3: D – Resolve the division and multiplication from left to right.
M – Resolve the division and multiplication from left to right.
Perform the multiplication:
[latex]2\times5=10[/latex]
Perform the division:
[latex]30 \div 5 =6[/latex]
After resolving the division and multiplication,
the expression now looks like this:
[latex]10+6[/latex]
Step 4: A – Perform the remaining addition.
S – As there is no subtraction, we state the final solution.
[latex]10+6 =16[/latex]
Question B
Solution
Step 1: B – Are there brackets? Yes, start with the brackets.
[latex](6+3)=9[/latex]
The expression now looks like this:
[latex](9)^2+18\div2[/latex]
Step 2: E – Are there exponents? Yes, resolve exponents next.
[latex](9)^2=81[/latex]
The expression now looks like this:
[latex]81+18 \div2[/latex]
Step 3: D – Perform the division.
M – There is no multiplication.
[latex]18 \div2=9[/latex]
The expression now looks like this:
[latex]81+9[/latex]
Step 4: A – Perform the remaining addition.
S – As there is no subtraction, we state the final solution.
[latex]81+9=90[/latex]
Question C
Solution
Step 1: B – Are there brackets?
Yes, start with the innermost set of brackets and perform EDMAS.
[latex]\begin{align*}4\;\times\;\left[\frac{\{3+2^2\times3\}}{\{(2+8)\div2\}}\right]=4\;\times\;\left[\frac{\{3+2^2\times3\}}{\{10\div2\}}\right]\end{align*}[/latex]
Now solve the curly brackets, starting with the top. Perform EDMAS.
Step 2: E – Are there exponents? Yes, resolve the exponent on top next.
[latex]2^2=4[/latex]
The expression now looks like this:
[latex]\begin{align*}4\;\times\;\left[\frac{\{3+4\times3\}}{\{10\div2\}}\right]\end{align*}[/latex]
Step 3: D – Perform the division on the bottom.
M – Perform the multiplication on the top
[latex]10\div2=5[/latex]
[latex]4\times3=12[/latex]
We no longer need the curly brackets, so they can be dropped.
The expression now looks like this:
[latex]\begin{align*}4\;\times\;\left[\frac{3+12}{5}\right]\end{align*}[/latex]
Take note that it is understood that even without the curly brackets, there are hidden (or invisible) brackets on the top and the bottom. Therefore, all operations must be completed before performing the division represented by the horizontal divisor line.
Step 4: A – Perform the remaining addition on the top.
S – There is no subtraction.
[latex]3+12=15[/latex]
The expression now looks like this:
[latex]\begin{align*}4\;\times\;\left[\frac{15}{5}\right]\end{align*}[/latex]
Step 5: We can now solve the division, in the outermost brackets.
[latex]\begin{align*}4\;\times\;\left[\frac{15}{5}\right]\end{align*}[/latex]
[latex]\begin{align*}=4\times3\end{align*}[/latex]
We no longer need the square brackets so they are dropped here.
Step 6: The last step is to perform the multiplication and state the final solution.
[latex]\begin{align*}4\times3=12\end{align*}[/latex]
integers
Simplifying Negatives
If your question involves positive and negative numbers, it is sometimes confusing to know what symbol to put when simplifying or solving. Remember these two rules:
Rule #1: A pair of the same symbols is always positive.
Thus [latex]4+(+3)[/latex] and [latex]4−(−3)[/latex]. both become [latex]4+3[/latex]
Rule #2: A pair of the opposite symbols is always negative.
Thus [latex]4+(−3)[/latex] and [latex]4−(+3)[/latex] both become [latex]4–3[/latex]
One way to remember these rules is to count the total sticks involved, where a [latex]+[/latex] sign has two sticks and a [latex]-[/latex] sign has one stick. If you have an odd number of total sticks, the outcome is a negative sign. If you have an even number of total sticks, the outcome is a positive sign.
Note the following examples:
- [latex]4+(-3) = 3[/latex] total sticks is odd and therefore simplifies to negative:
[latex]4−3 = 1[/latex] - [latex](-2) \times (−2)=2[/latex] total sticks is even and therefore simplifies to positive:
[latex](−2) \times (-2) = +4[/latex]
Sample Exercises 1.2
Solve the following :
- [latex]3 + (-5) =[/latex]
- [latex]8 - (-6) + (-2) =[/latex]
- [latex](-7) - (+4) - 1 + 9 =[/latex]
- [latex]18 - 3(2 + 5) =[/latex]
- [latex]11 + 7 - (-6) + (-2)=[/latex]
Solutions
1) [latex]= -2[/latex]
2) [latex]=12[/latex]
3) [latex]= -3[/latex]
4) [latex]= -3[/latex]
5) [latex]= 22[/latex]
Chapter Credit
Adapted from 2.1: Order of Operations in Business Math: A Step-by-Step Handbook (2021B) by J. Olivier published by Lyryx Learning. CC BY-NC-SA 4.0 International License.
