# Operations with integers

## The sum of integers

As the integers have $+$ or $-$ before, when we operate with them we will write these numbers in brackets in order to not confuse them with $+$ and $-$ while adding up and subtracting.

For example, if we want to add $+5$ and $-6$ we will write: $$(+5)+(-6)$$ and if we want to subtract them, we will write: $$(+5)-(-6)$$

The absolute value of an integer is simply the number without any sign (without plus or less. It is written between two vertical bars, as follows:

$|+16|=16$ (that is, the absolute value of $+16$ is $16$) $|-8|=8$ (that is, the absolute value of $-8$ is $8$)

1. If both numbers have the same sign: (that is, both are positive, or both are negative). First, the absolute value of each number is calculated. Later the absolute values are added up and finally the sign that they had is written.

For example, to do the operation: $(+9)+(+5)=$ we must follow:

• Both have the same sign (+).
• We calculate the absolute values: $|+9|=9, |+5|=5$.
• We add up the absolute values: $9+5=14$.
• We put the sign they had before: $+14$. So: $(+9)+(+5)=14$

2) If both numbers have different signs: (that is, one is positive and the other negative). First, calculate the absolute values of every number. Later both absolute values are subtracted. Finally, put the sign for the number which has the highest absolute value.

For example, to do: $(-7)+(+2)=$ we must follow:

• They have different signs: $-7$ is negative and $+2$ is positive
• We calculate the absolute values: $|-7|=7$,$|+2|=2$.
• We do the subtraction: $7-2=5$
• As the absolute value of $-7$ is greater than the absolute value of $2$ (because $7$ is greater than $2$), we put the negative sign: $-5$ And so, the result is: $(-7)+(+2)=-5$

## Subtraction of integers

The opposite of an integer is the same number but with an opposite sign. That is, if you have a plus, then you write a minus, and vice versa: if you have a minus, you write a plus.

For example, the opposite of $+4$ is $-4$, and the opposite of $-5$ is $+5$. In the case of the zero, which is the only integer without a $+$ or $-$, we will say that its opposite is itself: the opposite of $0$ is $0$.

If you subtract two numbers, the first is called the minuend digit and the second is called the subtrahend digit.

For example, in the subtraction: $(-19)-(+4)$ the minuend is $-19$, and the subtrahend $+4$.

To subtract two integers we must follow these steps:

1. Identify the minuend and subtrahend
2. Calculate the opposite of the subtrahend
3. Add up the minuend and the opposite of the subtrahend
4. We now have the result of the subtraction.

## Multiplication and division of integers

Multiplications of integers are done as follows:

1. First we do the multiplication or the division, regardless of the signs (that is, as if natural numbers).
2. To know the sign of the result, we must look the following table:
 **Sign** $+$ $-$ $+$ + - $-$ - +

If both numbers have the same sign, the result will be positive. If both have different signs, then the result will be negative.

## Power of integers

A power is an expression of the following type: $(+7)^3$ The bottom number is called the base and the small number above it (to the top right of the bottom number) is called the exponent. In the example above, the exponent is $3$ and the base is $7$, and it is said that $7$ is raised to the power $3$. The powers are used to write, in a shorter way, the multiplication of a number by itself. If we multiply $3$ four times, we will write: $$(+3)\cdot(+3)\cdot(+3)\cdot(+3)=(+3)^4$$

The exponent (the top number) tells how many times you multiply the base (the bottom number).

When we multiply two powers with the same base, the exponents are added up. For instance:

$$(-6)^3\cdot(-6)^2=(-6)^{3+2}=(-6)^5$$

When we divide two powers with the same base, the exponents are subtracted.

For example: $$(+4)^7\cdot(+4)^5=(+4)^{7-5}=(+4)^2$$

But it is very important to understand that this can only be done when the base of the powers is the same.

When a power is raised to another power, we must multiply exponents:

$$\big((-6)^2\big)^3=(-6)^{2\cdot3}=(-6)^6$$

Furthermore, if any number is raised to $0$, the result is always $1$:

$(+13)^0=1$ $(-9)^0=1$

On the other hand, a power can be raised to a negative exponent. In this case, the result is 1 divided by the power with the exponent positive. For example:

$$(+14)^{-6}=\dfrac{1}{(+14)^{+6}}$$