# Negative and non-negative numbers

A **negative number** is a number that is less than zero, such as -3. A **positive number** is a number that is greater than zero, such as 3. Zero itself is neither negative nor positive. The **non-negative** numbers are the positive numbers together with zero.
Note that some numbers are neither negative nor non-negative, for example the imaginary unit *i*.

## Negative numbers

These include negative integers, negative rational numbers, negative real numbers, negative hyperreal numbers, and negative surreal numbers.

Negative integers can be regarded as an extension of the natural numbers, such that the equation *x* − *y* = *z* has a meaningful solution for all values of *x* and *y*. The other sets of numbers are then derived as progressively more elaborate extensions and generalizations from the integers.

Negative numbers are useful to describe values on a scale that goes below zero, such as temperature, and also in bookkeeping where they can be used to represent debts. In bookkeeping, debts are often represented by red numbers, or a number in parentheses.

## Positive numbers

In the context of complex numbers positive implies real, but for clarity one may say "positive real number".

## Non-negative numbers

A number is nonnegative if and only if it is greater than or equal to zero, i.e. positive or zero. Thus the *nonnegative integers* are all the integers from zero on upwards, and the *nonnegative reals* are all the real numbers from zero on upwards.

A *real* matrix *A* is called **nonnegative** if every entry of *A* is nonnegative.

A *real* matrix *A* is called **totally nonnegative** by matrix theorists or totally positive by computer scientists if the determinant of every square submatrix of *A* is nonnegative.

## Sign function

It is possible to define a function sgn(*x*) on the real numbers which is 1 for positive numbers, -1 for negative numbers and 0 for zero (sometimes called the signum function):

*x*=0):

*x*| is the absolute value of

*x*and

*H(x)*is the Heaviside step function.

This can in a sense be extended to complex numbers, by writing then as *z* = *r e ^{iθ}*, with

*r*>0 and real, and looking at

*e*, i.e. . Again 0 needs to be treated as a special case.

^{iθ}## Arithmetic involving signed numbers

### Addition and subtraction

For purposes of addition and subtraction, one can think of negative numbers as debts.

Adding a negative number is the same as subtracting the corresponding positive number:

- (if you have $5 and acquire a debt of $3, then you have a net worth of $2)

- (if you have $4 and spend $6 then you have a debt of $2).

- (if you have a debt of $3 and spend another $6, you have a debt of $9).

- (if you have a net worth of $5 and you get rid of a debt of $2, then your new net worth is $7).

- (if you have a debt of $8 and get rid of a debt of $3, then you still have a debt of $5).

### Multiplication

Multiplication of a negative number by a positive number yields a negative result: (-2) × 3 = -6. The reason is that this multiplication can be understood as repeated addition: (-2) × 3 = (-2) + (-2) + (-2) = -6. Alternatively: if you have a debt of $2, and then your debt is tripled, you end up with a debt of $6.

Multiplication of two negative numbers yields a positive result: (-3) × (-4) = 12. This situation cannot be understood as repeated addition, and the analogy to debts doesn't help either. The ultimate reason for this rule is that we want the distributive law to work:

### Division

Division is similar to multiplication. If both the dividend and the divisor have different signs, the result is negative:

## Formal construction of negative and nonegative integers

In a similar manner to rational numbers, we can extend the natural numbers **N** to the integers **Z** by defining integers as an ordered pair of natural numbers (*a*, *b*). We can extend addition and multiplication to these pairs with the following rules:

- if and only if

**Z**to be the quotient set

**N**

^{2}/~, i.e. we identify two pairs (

*a*,

*b*) and (

*c*,

*d*) if they are equivalent in the above sense.

We can also define a total order on **Z** by writing

- if and only if

*additive zero*of the form (

*a*,

*a*), an

*additive inverse*of (

*a*,

*b*) of the form (

*b*,

*a*), a multiplicative unit of the form (

*a*+1,

*a*), and a definition of subtraction

## Computing

See negative and non-negative in binary.