Category: Zahlenausdrücke

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

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.

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

Intuitively, we understand, that it is correct.

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

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

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.

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

$-a$ 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):

1. Parentheses
2. Exponent
3. Multiplication
4. Division