# Category: Number Expressions

## Commutative Property of Addition

Commutative property of addition:

` huge color(purple)(a+b=b+a) `

What does it mean?

It means, that order of numbers doesn't matter.

Indeed, suppose your first friend (let's call it Tom) gave you 3 apples, then your another friend (let's call it Jim) gave your 4 apples. You got total of 7 apples.

## Associative Property of Addition

Associative property of addition:

`huge color(purple)(a+(b+c)=(a+b)+c)`

Intuitively, we understand, that it is correct.

Indeed, suppose your friend Ann has 3 apples, Bob has 5 apples and Cliff has 4 apples.

## Identity Property of Addition

Identity property of addition:

` huge color(purple)(a+0=a) `

What does it mean?

It means, that addition of zero to the number will not change the number.

Indeed, suppose your first friend (let's call it Tom) gave you 3 apples, but your second friend (let's call it Jim) gave you nothing. So, you are left with 3 apples.

## Inverse Property of Addition

Inverse property of addition:

`huge color(purple)(a+(-a)=(-a)+a=0) `

`-a` is called the additive inverse of `a`.

Inverse property is true for any real number `a`.

Notice, that we wrote, that `a+(-a)=(-a)+a`. This is true, according to the commutative property of addition.

## Commutative Property of Multiplication

Commutative property of multiplication:

` huge color(purple)(a times b=b times a) `

What does it mean?

It means, that order of numbers doesn't matter.

Indeed, as can be seen from illustration, we can count there are 3 circles in a row, and there are 2 rows, so total number of squares is `3times2=6`.

## Associative Property of Multiplication

Associative property of multiplication:

`huge color(purple)(a times (b times c)=(a times b) times c)`

As with commutative property order is not important.

Indeed, you can make sure on a couple of examples, that it is correct.

## Distributive Property of Multiplication

Distributive property of multiplication:

`huge color(purple)(a times (b+c)=a times b+a times c)`

Intuitively, we understand, that it is correct.

Indeed, multiplication is just a shorthand for addition.

## Identity Property of Multiplication

Identity property of multiplication:

` huge color(purple)(a times 1=a) `

What does it mean?

It means, that multiplication of the number with one will result in original number.

Recall, that multiplication is shorthand for addition.

## Inverse Property of Multiplication

Inverse property of multiplication:

`huge color(purple)(a times 1/a=1/a times a=1) `

`1/a` is called the multiplicative inverse of `a`.

Inverse property is true for any real number `a`.

Notice, that we wrote, that `a times 1/a=1/a times a`. This is true, according to the commutative property of multiplication.

## Order of Operations (PEMDAS)

Order of Operations (PEMDAS):

- Parentheses
- Exponent
- Multiplication
- Division
- Addition
- Subtraction

Note, that multiplication and division rank equally. Also addition and subtraction rank equally.