Basic Operations

Ah, the familiar four basic operations:

But there is more to the story!

Addition and Subtraction

Addition says how many steps to take.

Subtraction is actually addition, we just change the sign of the second number before we add.

Example: 7 − 5 = 7 + (−5) = 2

Play with it here (sliders):

images/number-line-add.js?sub=n

We only need an efficient way to add. Then to subtract we simply change the sign of the second number, then add.

Multiplication says how many adds to do.

Division is actually multiplication, we just do the reciprocal (1value) of the second number before we multiply.

Example: 12 ÷ 4 = 12 × 14 = 3

This idea is very useful!

It helps explain fractions: 34 is multiplication of 3 and 14

In some areas of mathematics (such as Matrix Algebra) we cannot divide, but we can do an inverse, so we multiply by the inverse and the job is done.

So the four basic operators are just two?

Much simpler!

We just need to be happy with the concept of inverse:

They are also both better behaved:

Addition is commutative: 3 + 5 = 5 + 3. But subtraction is not: 3 − 5 ≠ 5 − 3

Multiplication is commutative: 3 × 5 = 5 × 3. But division is not: 3/5 ≠ 5/3

Subtraction and division are both still important ... we just see them now from a higher level.

We can go one step further:

Exponents say how many multiplies to do.

The inverse of exponent is logarithm.

Now order of operations becomes simpler. PEMDAS becomes PEMA:

Knowing the concept of inverses lets us simplify the basic operations to: