Order of Operations
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Other tools you may find usefulOperator Priority Rules in Mathematics and Logic
Sequencing is a set of strict mathematical rules that determine the order in which arithmetic operations (addition, subtraction, multiplication, division, exponentiation) should be performed in complex numerical expressions. These rules have been developed over the centuries to avoid ambiguities in mathematical notations - thanks to them, every person in the world (and every calculator and computer) will receive exactly the same result from a given numerical notation. Ignoring these rules is the most common cause of errors in school assignments and the source of famous online disputes on discussion forums.
Our website explains the logical structure of performing mathematical operations step by step, making it easier to understand more difficult examples and helping to eliminate errors in everyday and exam calculations.
Golden rules for the order of operations
To solve any mathematical expression correctly, the following hierarchy of priorities must be followed:
- Operations in brackets:We always perform them first. If the brackets are nested, we start calculations from the deepest bracket.
- Exponentiation and square root:We perform before multiplication and division.
- Multiplication and division:Have equal priority. We perform them in the order they are written -from left to right.
- Addition and subtraction:They have the lowest, equal priority. We perform them at the very end, also in the orderfrom left to right.
Table of priorities of mathematical operations
The table below arranges the order of operations from the most important to the least important:
| Order (Degree) | Type of mathematical operation | Sample notation | Solution diagram and result |
|---|---|---|---|
| 1. | Operations in brackets( ) |
5 × (2 + 3) | First 2 + 3 = 5, then 5 × 5 = 25 |
| 2. | Exponentiationx²and square root√x |
3 + 2³ | First 2³ = 8, then 3 + 8 = 11 |
| 3. | Multiplying×and dividing/(left to right) |
12 / 3 × 2 | First 12 / 3 = 4, then 4 × 2 = 8 |
| 4. | Adding+and subtracting-(left to right) |
10 - 4 + 2 | First 10 - 4 = 6, then 6 + 2 = 8 |
Analysis of the famous internet puzzle:6 / 2(1 + 2)
This equation regularly goes viral in social media, causing arguments where people argue about two outcomes:9or1. What is the correct solution according to the official rules of algebra?
- We start with the operation in brackets:
1 + 2 = 3. The expression takes the form:6 / 2 × 3(the multiplication sign is hidden between 2 and the brackets). - Division and multiplication have the same priority, so we perform them from left to right.
- First from the left is division:
6 / 2 = 3. - Then we perform the multiplication:
3 × 3 = 9. - The correct result is 9.The result 1 is produced by incorrectly assuming that the multiplication associated with the bracket has higher priority than the division on the left.
Frequently asked questions (FAQ)
What do we do first: multiplication or division?
Multiplication and division have equal priority. We perform them in the order in which they appear in the notation, from left to right.
What is the full order of operations?
The order is: 1. Operations in brackets, 2. Exponentiation and radicals, 3. Multiplication and division, 4. Addition and subtraction.
Do parentheses always change the order of operations?
Yes, the brackets are there to force the calculations to take precedence. The operation inside the bracket must be performed before all other operations outside it.
What does the "left to right" rule mean?
This means that if the expression contains operations with the same priority (e.g. only addition and subtraction or only multiplication and division), we perform them in sequence as we read the text - starting from the left.
How do computer calculators deal with the order of operations?
Modern scientific calculators and programming languages use a standard operator hierarchy algorithm (often based on the so-called reverse Polish notation or grammatical parsing trees), thanks to which they always correctly interpret the sequence of operations.