Understanding Algebra

James W. Brennan
 

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Order of Operations

When we encounter an expression such as 3 + 15 ÷ 3 + 5 × 22+3, it makes quite a difference how we choose which operations to perform first. We need a set of rules that would guide anyone to one unique value for this kind of expression. Some of these rules are simply based on convention, while others are forced on us by mathematical logic. In the chapter on the Properties of Real Numbers, you will see how the distributive law is consistent with these rules. The universally agreed-upon order in which to evaluate a mathematical expression is as follows:

1. Parentheses from Inside Out

By “parentheses” we mean anything that acts as a grouping symbol, including anything inside symbols such as [ ], { }, |  | and . Any expression in the numerator or denominator of a fraction or in an exponent are also considered grouped and should be simplified before carrying out further operations.

 

·        If there are nested parentheses (parentheses inside parentheses), you work from the innermost parentheses outward.

2. Exponents

Also other special functions such as log, sin, cos, etc.

3. Multiplication and Division, left to right

The left-to-right order does not matter if only multiplication is involved, but it matters for division.

4. Addition and Subtraction, left to right

The left-to-right order does not matter if only addition is involved, but it matters for subtraction.

Example:  Going back to our original example, 3 + 15 ÷ 3 + 5 × 22+3

Given:

3 + 15 ÷ 3 + 5 × 22+3

The exponent is an implied grouping, so the 2 + 3 must be evaluated first:

= 3 + 15 ÷ 3 + 5 × 25

Now the exponent is carried out:

= 3 + 15 ÷ 3 + 5 × 32

Now the multiplication and division, left to right, using 15 ¸ 3 = 5 and 5 ´ 32 = 160:

= 3 + 5 + 160

Now the addition, left to right:

= 168

  Calculator Note: Most modern calculators “know” the order of operations, and you can enter expressions pretty much as they are written. Some older calculators will carry out each operation as soon as its key is pushed, which can result in the operations being carried out in the wrong order. Try some examples if you are not sure how your calculator behaves.

For example, if you enter

3 + 4 × 5 =

The correct answer should be 23, because the multiplication should be performed before the addition, giving 3 + 20. But if your calculator carries out the “3 + 4” before getting to the “ × 5”, it will show a result of 35 because it will see it as 7 × 5.

  Calculator  Note: Use the parenthesis keys to force grouping. If you are evaluating an expression such as

the denominator needs to be simplified before doing the division. If you enter it into your calculator as 4 ÷ 3 + 5, it will evaluate the “ 4 ÷ 3 “ first, and then add 5 to the result, giving the incorrect answer of 6.3333. To make it perform the addition first, use parentheses:

4 ÷ (3 + 5) = 0.5

 In our example problem above, the “2 + 3” in the exponent is an implied grouping, and you would need to use parentheses. To enter that expression in your calculator, you would enter it as

3 + 15 ÷ 3 + 5 × 2 ^ (2 + 3) =

(on some calculators the exponent button is labeled “ ^ ”, while on others it is labeled “ yx ”)


 

      

 


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James W. Brennan