# Calculating with real numbers

When adding real numbers with the same sign the sum will have the same sign as the numbers added.

$3+2=5$

$-7+(-2)=-9$

When adding real numbers with different signs you subtract the lesser absolute value from greater one. The sum will then have the same sign as the number with the greater absolute value.

$-7+2=-5$

There are a couple of properties of addition:

The additive commutative property tells us that the order in which you add the numbers does not change the sum.

$a+b=b+a$

And the additive associative property tells us that it also that the order in which we group three or more numbers does not affect the sum.

$\left ( a+b \right )+c=a+\left ( b+c \right )$

The additive identity property tells us that the sum of a number and 0 is always the number.

$a+0=a$

And the additive inverse property tells us that if you add a number with its opposite you will always get 0

$a+\left ( -a \right )=0$

## Subtraction

You know by now that

$10-2=10+\left ( -2 \right )=8$

This means that subtracting 2 from 10 is the same as adding negative 2 to 10. This is called the subtraction rule.

$a-b=a+\left ( -b \right )$

## Multiplication

If you remember from pre-algebra the product of two real numbers with the same sign is always positive

$3\cdot 2=6$

$\left ( -5 \right )\cdot \left ( -3 \right )=15$

This also holds true if you multiply more than two numbers. If you multiply with an odd number of negative numbers the product will be negative

$2\cdot \left ( -3 \right )\cdot \left ( -1 \right )\cdot 5\cdot \left ( -2 \right )=-60$

Whereas if you multiply with an even number of negative numbers the product will be positive

$3\cdot \left ( -4 \right )\cdot \left ( -2 \right )=24$

As with addition there are a couple of properties of multiplication as well

The multiplicative commutative property tells us the same things as its additive counterpart. The product is not affected by order in which you multiply the numbers

$a\cdot b=b\cdot a$

The same goes for the multiplicative associative property. The product is not affected by how you group three or more numbers.

$a\cdot \left ( b\cdot c \right )=\left (a\cdot b \right ) \cdot c$

The multiplicative identity property tells us that if we multiply a number with 1 the product is always the number

$a\cdot 1=a$

The multiplicative identity of 0 tells us that the product is always 0 when you multiply a number with 0

$a\cdot 0=0$

The last property of multiplication is the multiplicative property of -1 and this property tells us that a product of a number and -1 is the opposite of the number

$a\cdot \left ( -1 \right )=\left ( -a \right )$

## Division

Reciprocals are numbers that when multiplied have the product 1.

$\frac{5}{7}\cdot \frac{7}{5}=1$

These are also called multiplicative inverses. Zero does not have a multiplicative inverse since everything multiplied with 0 is 0 as we could see above.

The inverse property of multiplication can be written as:

$a\cdot \frac{1}{a}=\frac{1}{a}\cdot a=1,\: a\neq 0$

It can also be written as

$\frac{a}{b}\cdot \frac{b}{a}=1\: where \: a,\, b\neq 0$

The division rule tells us that to divide a number a with a number b is the same as multiplying a with the multiplicative inverse of b

$\frac{a}{b}=a\cdot \frac{1}{b},\: b\neq 0$

Since we can express a division as a multiplication the sign rules of multiplication holds true for division as well.

The quotient of two numbers with the same sign is positive

$\frac{8}{2}=4$

Whereas the quotient of two numbers with different signs is negative

$\frac{-8}{2}=-4$

The quotient of 0 and any nonzero real number is always 0

$\frac{0}{1000}=0$

Since 0 does not have a multiplicative inverse you cannot divide a number by 0.

## Video lesson

Simplify the expression

$\frac{20x + 5 + 10 - 5x}{5}$