Home > Negative and non-negative numbers

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

1 Negative 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.

2 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.

3 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):

We then have (except for x=0):

where |x| is the absolute value of x and H(x) is the Heaviside step functionThe Heaviside step function named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative inputs and one elsewhere: : The function is used in the mathematics of signal processing to represent a signal that switches on at.

4 Arithmetic involving signed numbers

4.1 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)

Subtracting a positive number from a smaller positive number yields a negative result:

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

Subtracting a positive number from any negative number yields a negative result:

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

Subtracting a negative is equivalent to adding the corresponding positive:

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

Also:

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