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Skill 13 of 14
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Addition & the four basic operations

You have 3 apples. Your friend gives you 6 apples more. How many apples do you have in total?

9, of course.

3 and 6 is 9’ or ‘if you have 3 and add 6, you have 9’ or the familiar ‘3 plus 6 is 9’. Let’s use the $+$ symbol to mean ‘and’ or ‘add to’ or simply ‘plus’ (the same symbol we use as a ‘positive sign’):

$$3\,\text{apples}+6 \,\text{apples} =9 \,\text{apples} $$

  • Addition ($\;+\;$) adds some more to increase the amount, while
  • subtraction ($\;-\;$) takes some away again. They are each other’s counterparts. Likewise,
  • multiplication ($\;\cdot\;$) adds more portions by multiplying the existing amount, whereas
  • division ($\;/\;$) splits this amount into portions again. They are each other’s counterparts.

These four basic math operations are the root of all math. They are the basic part of the mathematical elementary branch called arithmetic1 that deals with numbers.[2] Everything else in math is just combinations or rewritings of these. We might use different symbols now and then:

Arithmetic operations
Addition$+$Gives a sum 3
Subtraction$-$Gives a difference
Multiplication$\cdot$ or $\times$ or $*$ or noneGives a product
Division$/$ or $:$ or $\div$Gives a fraction or quotient 2

All the parts involved in these operations have gotten fancy names. See an overview in Resource: The four basic operations.

In the supermarket you take 3 clusters of 5 bananas and 4 bags of 10 apples. How many pieces of fruit are you bringing home? That would be 15 bananas and 40 apples, so 55 pieces in total:

$$\underbrace{3\cdot5}_\text{First this…}+ \underbrace{4\cdot10}_\text{and this…} = \underbrace{15+40}_\text{…then this}=55 $$

There is clearly a priority, an order of operations. Multiplications first; addition next. You cannot do, for instance, addition first and then the multiplications:

$$\xcancel{3\cdot \underbrace{5+4}_\text{Not first}\cdot 10=3\cdot9\cdot10=270}$$

Let’s set up a full prioritised list:

Order or operations (PEMDAS)[10]
First do...Parentheses
Then do...Exponents (powers)
Then do...Multiplications
as well as...Divisions
And finally do...Additions
as well as...Subtractions

It’s easy to remember the order by remembering ‘PEMDAS’, the first letter of each term.4


References:

  1. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com
  2. Encyclopædia Britannica’ (encyclopedia/dictionary), www.britannica.com