Order of operations
Order of Operations
Hello Science Bee. In this lesson, we are going to learn about the order of operations.
What is the Order of Operations?
The order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.
For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.
Thus, the expression 2 + 3 × 4 is interpreted to have the value 2 + (3 × 4) = 14, not (2 + 3) × 4 = 20.
Video: Order of Operations
Exponents
With the introduction of exponents in the 16th and 17th centuries, they were given precedence over both addition and multiplication and could be placed only as a superscript to the right of their base. Thus 3 + 52 = 28 and 3 × 52 = 75.
These conventions exist to eliminate ambiguity while allowing notation to be as brief as possible. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) (sometimes replaced by brackets [ ] or braces { } for readability) can indicate an alternate order or reinforce the default order to avoid confusion. For example, (2 + 3) × 4 = 20 forces addition to precede multiplication, and (3 + 5)2 = 64 forces addition to precede exponentiation.
List out the order of operations
The order of operations used throughout mathematics, science, technology and many computer programming languages is expressed here:
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exponents and roots
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multiplication and division
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addition and subtraction
This means that if a mathematical expression is preceded by one binary operator and followed by another, the operator higher on the list should be applied first.
The commutative and associative laws of addition and multiplication allow adding terms in any order, and multiplying factors in any order—but mixed operations must obey the standard order of operations.
It is helpful to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse). Thus 3 ÷ 4 = 3 × ¼; in other words the quotient of 3 and 4 equals the product of 3 and ¼. Also 3 − 4 = 3 + (−4); in other words the difference of 3 and 4 equals the sum of 3 and −4. Thus, 1 − 3 + 7 can be thought of as the sum of 1, −3, and 7, and add in any order: (1 − 3) + 7 = −2 + 7 = 5 and in reverse order (7 − 3) + 1 = 4 + 1 = 5, always keeping the negative sign with the 3.
The root symbol √ requires a symbol of grouping around the radicand. The usual symbol of grouping is a bar (called vinculum) over the radicand. Other functions use parentheses around the input to avoid ambiguity. The parentheses are sometimes omitted if the input is a monomial. Thus, sin 3x = sin(3x), but sin x + y = sin(x) + y, because x + y is not a monomial. Some calculators and programming languages require parentheses around function inputs, some do not.
Examples involving the order of operations
A horizontal fractional line also acts as a symbol of grouping:
For ease in reading, other grouping symbols, such as curly braces { } or square brackets [ ], are often used along with parentheses ( ). For example:
![{\displaystyle [(1+2)-3]-(4-5)=[3-3]-(-1)=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/491249623dd91c5e6b2aa952d3e43c42877e8ccf)
Mnemonics to remember the order of operations
Mnemonics are often used to help students remember the rules, involving the first letters of words representing various operations. Different mnemonics are in use in different countries.
In the United States, the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. PEMDAS is often expanded to the mnemonic "Please Excuse My Dear Aunt Sally". Note that there is no priority between multiplication and division. Same is true for addition and subtraction.
These mnemonics may be misleading when written this way. For example, using any of the above rules in the order "addition first, subtraction afterward" would incorrectly evaluate the expression.
10 − 3 + 2.
The correct value is 9 (and not 5, as if the addition would be carried out first and the result used with the subtraction afterwards).
How to use order of operations with fractions?
This video shows how to use the order of operations with fractions.
How to use order of operations with integers and exponents?
This video shows how to use the order of operations with integers and exponents.
How to use order of operations with fractions and decimals?
This video shows how to use the order of operations with fractions and decimals.
How to use order of operations with fractions and exponents?
This video shows how to use the order of operations with fractions and exponents.
How to use order of operations with mixed numbers?
This video shows how to use the order of operations with mixed numbers.
How to use order of operations with fractions and decimals?
This video shows how to use the order of operations with Fractions and Decimals.
