# Negative numbers

## What are negative numbers?

So far we only used natural numbers;
numbers we use in counting. Natural numbers (apart from zero) are always greater than zero.
But why not extent the set of numbers to numbers less than zero?
After all, many real-life situations can be seen as things toggling to the opposite at zero, rather than starting or ending at zero.
Numbers less than zero would then be appropriate to represent these opposite values.
For instance, someone can have $100 in the bank or a debt of $100 owed to the bank.
It's both a hundred dollars, but on opposite sides of zero. These numbers less than zero are called *negative numbers*.

Negative numbers are written with a minus sign in front, e.g. −3, pronounced as "minus three" or "negative three". The opposite of negative numbers are "positive numbers", and vice versa. Positive numbers actually have a plus sign, but this sign is usually omitted. For instance, +3 = 3, sometimes pronounced as "plus three" or "positive three" to emphasize the distinction between its negative opposite. Both +3 and −3 have the same so called absolute value, but a different sign. Zero is the only number not negative and not positive. Zero has no sign.

Positive natural numbers, their negative opposites, and zero are called *integers*.
So we distinguish positive integers, negative integers and zero.
Later we will see that there are more kinds of numbers, like fractional numbers. They can also be positive or negative.

From a mathematical point of view, the definition of negative numbers provides a solution for subtracting a number from a smaller number.

5 − 7 = −2

This corresponds to the way in which negative numbers are conceived in real-life contexts. For example,
withdrawing $7 from a bank account that has only $5 in it, leaves you with an account $2 overdrawn, that is, an account balance of negative $2, or
taking the elevator from the 5th storey (5th floor) of a building, down 7 storeys, will bring you to the 2nd basement floor, i.e., storey −2 ^{(*)}.

^{(*)},
In some parts of the world, like in North America and in China, they number street level (ground floor) as the 1st floor, instead of the 0th floor.
Numbering basement floors (with negative numbers) is also more common in Europe.

### The number line

In mathematics, numbers and their mutual relation are often visualized as points on the *number line* (figure 2).
The number line is a (horizontal or vertical) straight line with a fixed distance between each two successive natural numbers.
Every natural number has a negative opposite mirrored in zero and consequently with both the same distance from zero (the absolute value).
The number line is endless or "infinite" in both directions, indicated with arrow heads.

Comparing quantities is sometimes confusing when dealing with negative numbers. Any number left from (or below) any other number on the number line is the least of the two. Negative 1000 is less than negative 1. Negative 100 is less than positive 10. Positive 2 is greater than negative 2 (but they have the same magnitude).

Negative 1000 is less than negative 1 but a dept of $1000 is a greater dept than a debt of $1. A dept of $1000 puts you in a worse financially position than a dept of $1.

### Number line arithmetic

Addition and subtraction can be visualized using the number line:

7 − 5 = 2: go 5 steps to the left (in "negative direction") from 7, this will bring you to 2.

−5 + 7 = 2: go 7 steps to the right (in "positive direction") from −5, this will again bring you to 2.

5 − 7: go 7 steps to the left (in "negative direction"), starting from 5, this will bring you to −2.

−2 − 3: go 3 steps into negative direction, starting from −2, and end at −5.

Mind chain subtraction:

5 − 7 | = | 5 − (5 + 2) | |

= | 5 − 5 − 2 | ||

= | 0 − 2 | ||

= | −2 |

−2 − 3 | = | 0 − 2 − 3 | |

= | 0 − (2 + 3) | ||

= | 0 − 5 | ||

= | −5 |

## Arithmetic with negative numbers

### Adding a negative number

The signs for the operations addition and subtraction (+ and −) are also used for the sign of a number. This is not a coincidence. Consider the subtraction 7 − 5 = 2:

7 − 5 | = | 7 − +5 |

= | 7 + −5 |

"7 minus (positive) 5" *or* "7 plus negative 5".

"Remove" 5 from 7 *or* add a "shortage" of 5 to 7;
there are two ways to look at a subtraction,
but in essence they mean the same thing! Subtracting a number is exactly the same as adding the opposite of that number!

`a` − +`b` = `a` + −`b`

And addition is commutative:

−5 + 7 | = | 7 + −5 | |

= | 7 − +5 | ||

= | 7 − 5 | ||

= | 2 |

−5 + 7 | = | −5 + (5 + 2) | |

= | −5 + 5 + 2 | ||

= | 0 + 2 | ||

= | 2 |

−2 − 3 | = | −2 + −3 |

= | −3 + −2 | |

= | −3 − 2 |

### Subtracting a negative number: minus negative

It is not immediately clear how to subtract a negative number on a number line: 2 − −3. If we use the rule that subtracting a number is exactly the same as adding the opposite:

2 − −3 | = | 2 + +3 |

= | 2 + 3 | |

= | 5 |

`a` − −`b` = `a` + `b`

Let's try a slightly different approach: If one thing is equal to the other and if you negate (i.e. take the opposite of) both sides of the equal sign, you should get a new true expression (and vice versa). And in case of adding or subtracting this means all numbers (but not the operation) change sign.

2 − −3 = 5, because −2 − +3 = −2 − 3 = −5.

Multiplying a number by −1 is exactly the same as negating the number. Negating both sides of the equal sign:

−a − b | = | −a − b |

−a − +b | = | −a + −b |

−1 × (−a − +b) | = | −1 × (−a + −b) |

a − −b | = | a + b |

In the fourth line the distributive rule is applied.

Negating both sides of the equals sign can be useful for addition and subtraction with negative numbers in general:

−7 + 5 = −2, because 7 + −5 = 2.

−2 − 3 = −2 + −3 = −5, because 2 + 3 = 5.

−117 − 95 = −212, because 117 + 95 = 212.

95 − 117 = −22, because 117 − 95 = 22.

−1000 + 56 = −944, because 1000 − 56 = 944.

### Multiplication with negative numbers

Three times a debt of 4 must be a debt of 12. And 4 times a debt of 1 must be a debt of 4.

3 × −4 | = | (−4) + (−4) + (−4) |

= | −4 − 4 − 4 | |

= | −12 |

4 × −1 | = | −1 × 4 |

= | −4 |

Multiplying a number by −1 is exactly the same as negating the number!

3 × −4 | = | −1 × 3 × 4 |

= | −3 × 4 |

3 × −4 | = | −1 × 3 × 4 |

= | −1 × (3 × 4) | |

= | −1 × 12 | |

= | −12 |

Thus:

a × −b | = | −a × b |

= | −1 × (a × b) |

When one of the factors is negative and the others are positive, you can simply apply the multiplication as if all factors were positive and negate the result. But what if two factors are negative? We need to find out what −1 × −1 means. Multiplying by −1 negates the number: −1 × 3 = −3. Two times multiplying by −1 should bring you back to the original again. After all, double negation or taking the opposite of the opposite returns the original number again.

−1 × −1 = 1

Appendix A shows a proof that −1 × −1 = 1.

−2 × −3 | = | −1 × −1 × 2 × 3 |

= | 2 × 3 | |

= | 6 |

−(−3) | = | −1 × (−3) |

= | −1 × −1 × 3 | |

= | 3 |

−(−`a`) = `a`

2 − −3 | = | 2 + ( −(−3) ) |

= | 2 + 3 | |

= | 5 |

Whatever number, negative or positive, multiplied by zero always yields zero.

−`a` × 0 = 0

In chain multiplication, an odd number of minuses yields a negative result, an even number of minuses a positive result.

−2 × −3 × −4 | = | (−2 × −3) × −4 |

= | 6 × −4 | |

= | −24 |

### Division with negative numbers

9 ÷ 4 = 2r1, because (2 × 4) + 1 = 9.

9 ÷ −4 = −2r1, because (−2 × −4) + 1 = 9.

By convention the remainder is defined to always be non-negative and less than the absolute value of the number that is divided by, otherwise the quotient and remainder in a division would not be unique (9 ÷ 4 = 2r1 or 9 ÷ 4 = 3r−3 or...).

−9 ÷ 4 = −3r3, because (−3 × 4) + 3 = −9.

−9 ÷ 4 ≠ −2r−1, although (−2 × 4) − 1 = −9.

−9 ÷ −4 = 3r3, because (3 × −4) + 3 = −9.

−9 ÷ −4 ≠ 2r−1, although (2 × −4) − 1 = −9.

Although this convention makes sense in some ways, it can be a little unsatisfying in others. Dividing a debt of 9 over 4 persons, leaving everyone with a debt of 2 and a remaining debt of 1 makes more sense than leaving everyone with a debt of 3 and a remaining surplus of 3. Also as a fraction we interpret it like: −9/4 = −2.25 (see later chapters).

## Rules

A summary of the rules:

`a` − `b` = `a` + −`b`

`a` − −`b` = `a` + `b`

a × −b | = | −a × b |

= | −1 × (a × b) |

−1 × −1 = 1

−(−`a`) = `a`

−`a` × 0 = 0