# How can rational numbers be negative?

## Expect negative numbers and minus brackets

### Calculating with rational numbers

In the following we consider the basic arithmetic operations of addition, subtraction, **Multiplication and division in the field of rational numbers**. We pay particular attention to the negative numbers.

### Addition and subtraction

The **subtraction** of a rational number is the same as adding its opposite number. So the following applies:

$-3 - 8 = -3 + (-8) = -11$

$-4 - 1 = -4 + (-1) = (-5)$

Since, consequently, every subtraction of a rational number can be regarded as an addition, we only need the following **Addition rules** Note: If the summands have the same sign, the sum also has this sign:

$5 + 8 = 13$

The summands are positive, so the result is positive. We consider yet another example:

$-7 - 10 = (-17)$

Here the summands are negative, so the result is negative. If the summands have different signs, we calculate the difference between their amounts. The result has the sign of the number with the highest amount. We consider the following example:

$-3 + 12 = 9$

As the difference between their amounts, we get $ 12 - 3 = $ 9. The answer to the question is positive because the amount of $ + $ 12 is greater than that of $ -3 $. Another example follows for a better understanding:

$8 - 17 = -9$

The difference is $ 17 - 8 = $ 9. The result is negative, however, since the amount of $ -17 $ is greater than that of $ 8 $.

### Commutative law

This applies to the addition **Commutative law**. This means that we can swap the summands without changing the result. The following applies: $ a + b = b + a $. We consider a few examples below:

$6 + 9 = 9 + 6 = 15$

$-6 + 9 = 9 + (-6) = 9 - 6 = 3$

With a subtraction, the application of the commutative law does not work, as the following example shows:

$ 9 - 6 \ neq 6 - 9 $

Namely, it's $ 3 \ neq -3 $. However, the addition rule can be used if we make a sum from the difference as follows:

$9 -6=9 + (-6) = -6 + 9 = 3$

### Associative law

To add or subtract sums, we apply the rule of brackets, i.e. the associative law. By that we mean that **Dissolving parentheses**.

If there is a plus in front of the brackets, we add each individual summand without changing the sign:

$ \ begin {array} {lll} -13 + (-3 + 5 - 8) & = & -13 + (-3) + (+5) + (-8) \ & = & -13 - 3 + 5 - 8 \ & = & -19 \ end {array} $

**Note:** There is a plus in front of the bracket **(Plus bracket)**, then the bracket may be omitted. All numbers in brackets keep their signs.

There is a minus in front of the bracket **(Minus bracket)**, the corresponding opposite number is added for each addend:

$ \ begin {array} {lll} -13 - (-3 + 5 - 8) & = & -13 + (+3) + (-5) + (+8) \ & = & -13 + 3 - 5 + 8 \ & = & -7 \ end {array} $

**Note:** In the case of minus brackets, the brackets can be omitted if all numbers within the brackets have the opposite sign. So we form the opposite numbers.

### Multiplication and division

The multiplication of rational numbers is an abbreviation for the multiple addition of the same summands:

$ 3 \ times 4 = 4 + 4 + 4 = 12 $

Multiplication by zero always results in zero:

$ 5 \ cdot 0 = 0 $

How do we multiply negative numbers? For example, multiplying by negative numbers can be thought of as taking on $ 5 in debt three times. Mathematically, we can express this idea as follows:

$ 3 \ cdot (-5) = -5 + (-5) + (-5) = -15 $

As we have already stated above, the multiplication can be written as the addition of equal summands. We now look at two series of numbers in which we can recognize the calculation rules for the multiplication of rational numbers:

It is noticeable that the results are always positive if both factors have the same sign. It is for example:

$ \ begin {array} {lll} 2 \ cdot4 & = & 8 \ -2 \ cdot (-4) & = & 8 \ end {array} $

If the factors have different signs, the results are negative. We can determine this connection with the following examples:

$ \ begin {array} {lll} -2 \ cdot 4 & = & -8 \ 2 \ cdot (-4) & = & -8 \ end {array} $

**Note:** If rational numbers are multiplied with the same sign, the result is always positive. So it is:

$ (+) \ cdot (+) = (+) \ (-) \ cdot (-) = (+) $

If, on the other hand, rational numbers are multiplied with different signs, the result is always negative:

$ (+) \ cdot (-) = (-) \ (-) \ cdot (+) = (-) $

In the **division** with negative numbers we imagine that $ 15 € debt is paid back in three equal installments. Since we see in division the opposite arithmetic operation to multiplication, the following applies:

$-15 :3 = -5$

Namely, it is $ -5 \ cdot 3 = -15 $. If we want to divide two negative numbers, we can also find the result via multiplication:

$-20 : (-4) = 5$

For the multiplication, $ (- 4) \ cdot 5 = (-20) $ applies. Division by zero is not defined.

**Note:** If rational numbers with the same sign are divided, the result is always positive. The following applies:

$(+):(+) = (+) \\ (-):(-) = (+)$

If rational numbers with different signs are divided, the result is always negative:

$(+):(-) = (-) \\ (-):(+) = (-)$

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