# 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.

This idea is very useful!

### Example: Computer programs (or computer chips)

To handle numbers we first find a good way to "add"

And then to subtract we simply change the sign of the second number, then add.

Play with it here (sliders):

## Multiplication and Division

Multiplication says how many adds to do.

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

This idea is very useful!

## Wait, What?

So the four basic operators are just two?

Much simpler don't you think?

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

- Additive inverse (change sign)
- Multiplicative inverse (\frac{1}{value})

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.

## Exponents and Logarithms

We can go one step further:

Exponents say how many multiplies to do.

The inverse of exponent is logarithm.

## Order of Operations

Now order of operations becomes simpler. PEMDAS becomes **PEMA**:

- Parentheses (overrides the usual order)
- Exponents (how many multiplies)
- Multiply (how many adds)
- Add (how many steps)

## Summary

Knowing the concept of inverses lets us simplify to this:

**Addition**says how many steps- Inverse is changing sign/direction
**Multiplication**says how many adds- Inverse is the reciprocal (\frac{1}{value})
**Exponents**says how many multiplies- Inverse is logarithm