# Category: 数字表达式

## Commutative Property of Addition

Commutative property of addition:

$$$\color{purple}{a+b=b+a}$$$

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:

$$$\color{purple}{a+\left(b+c\right)=\left(a+b\right)+c}$$$

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:

$$$\color{purple}{a+0=a}$$$

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:

$$$\color{purple}{a+\left(-a\right)=\left(-a\right)+a=0}$$$

$$$-a$$$ is called the additive inverse of $$${a}$$$.

Inverse property is true for any real number $$${a}$$$.

## Commutative Property of Multiplication

Commutative property of multiplication:

$$$\color{purple}{a\times b=b\times a}$$$

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 $$${3}\times{2}={6}$$$.

## Associative Property of Multiplication

Associative property of multiplication:

$$$\color{purple}{a\times\left(b\times c\right)=\left(a\times b\right)\times c}$$$

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:

$$$\color{purple}{a\times\left(b+c\right)=a\times b+a\times c}$$$

Intuitively, we understand, that it is correct.

Indeed, multiplication is just a shorthand for addition.

## Identity Property of Multiplication

Identity property of multiplication:

$$$\color{purple}{a\times 1=a}$$$

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:

$$$\color{purple}{a\times\frac{1}{a}=\frac{1}{a}\times a=1}$$$

$$$\frac{1}{a}$$$ is called the multiplicative inverse of $$$a$$$.

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

## 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.

Operations, that rank equally, can be performed from left to right.