Why do we follow the order of operation?

Answer 1

What the order of operations mainly does is to ensure that you do the "shorthand" operations first; this can be seen by examining what the operations mean (I'll demonstrate using Multiplication and Addition):

Consider that Multiplication (×) is a "shorthand" way of writing repeated Addition; for example 5 × 4 is the same as 5 + 5 + 5 + 5 or 5 added to itself 4 times; or 4 + 4 + 4 + 4 + 4 which is 4 added to itself 5 times - both give the same answer of 20.

Now consider if you have 5 × 4 + 1. The multiplication is "shorthand" for repeated addition and can be written out in full, giving:

• 5 × 4 + 1 = 5 + 5 + 5 + 5 + 1 = 20 + 1 = 21

or

• 5 × 4 + 1 = 4 + 4 + 4 + 4 + 4 + 1 = 20 + 1 = 21

Now, Addition has the Commutative property - that is if you add two numbers together the order of the numbers does not matter; ie 20 + 1 is the same as 1 + 20; so what about 1 + 20:

• 1 + 20 = 1 + 5 + 5 + 5 + 5 = 1 + 5 × 4

or

• 1 + 20 = 1 + 4 + 4 + 4 + 4 + 4 = 1 + 5 × 4

using that 5 + 5 + 5 + 5 = 4 + 4 + 4 + 4 + 4 = 20 from above.

Thus for 5 × 4 + 1 = 1 + 5 × 4 to be true the Multiplication (shorthand for repeated Addition) must be done first.

So there is an order of operation to ensure that these "shorthand" operations are done first.

To help remember the order of operation, there are several acronym mnemonics whereby the order of the letters gives the order of operations:

BODMAS = Brackets Order Divide Multiply Add Subtract

BIDMAS = Brackets Indices Divide Multiply Add Subtract

PEDMAS = Parentheses Exponents Divide Multiply Add Subtract

Brackets (or parentheses) are used to group parts of the expression together and represent a "number" - they are useful to override the normal order of operation, for example in 1 + 5 × 4 the multiply is normally done first, ie it is 1 + (5 × 4) = 1 + 20 = 21; but if the addition must be done first, then it is put in brackets: (1 + 5) × 4 = 6 × 4 = 24 - a different value. This comes up mostly in formulae; for example in converting Fahrenheit temperature to Celsius, the formula is C = (F - 32) × 5 ÷ 9; the subtraction must be done before the multiply.

Order/Indices/Exponents is a "shorthand" way of writing repeated multiplication, for exmaple 2³ = 2 × 2 × 2 = 8; thus should be done before multiplication.

If there is more than one of the same operation in a row, then, for:

• Brackets (Parentheses) do them inner to outer, eg ((1 + 7) - (3 + 2)) × 2 = (8 - 5) × 2 = 3 × 2 = 6
• Order/Indices/Exponents do them right to left (top to bottom), eg 2^3^4 = 2^(3^4) = 2^81
• For Divide, Multiply, Add, Subtract do them left to right, eg 5 - 3 - 1 = (5 - 3) - 1 = 2 - 1 = 1.

One thing of note is that the acronyms are slightly wrong in that if the order of the letters is strictly followed it can give a wrong answer; for example 1 - 2 + 3

They all say do the addition before the subtraction, so it look like this becomes: 1 - (2 + 3) = 1 - 5 = -4

However, the correct answer is (1 - 2) + 3 = 2 with the "subtraction" done first.

This is because subtraction, being the inverse of addition, can be considered as "shorthand" for the addition of a negative number: 1 - 2 = 1 + -2; thus:

• 1 - 2 + 3 = 1 + -2 + 3 = (1 + -2) + 3 = -1 + 3 = 2

Addition and subtraction really have the same "priority" in the order of operations and should be done in the order they are met (from left to right), or always do the subtraction (Division) first.

Similarly Division and Multiplication really have the same "priority" in the order of operations and should be done in the order met (left to right), but always doing the Division first (as the acronyms say) will get the correct answer.

Answer 2

An operation such as:1 + 2 x 3

will give you different results, depending on which operation you do first, in this case, the sum, or the multiplication. Try it out!

The order of operations commonly used is an agreement which avoids ambiguity. In this case, the answer would be 7.