Negative And Non-negative Numbers 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 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):

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:

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

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

4.2 Multiplication

MultiplicationArithmetic In its simplest form, multiplication is a quick way of adding identical numbers. The result of multiplying numbers is called a product''. The numbers being multiplied are called coefficients or factors and individually as the multiplicand and m 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 lawIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example: : 4 · (2 + 3) (4 · 2) + (4 · 3) In the left-hand side of the above equatio to work:

The left hand side of this equation equals 0 × (−4) = 0. The right hand side is a sum of −12 + (−3) × (−4); for the two to be equal, we need (−3) × (−4) = 12.

4.3 Division

DivisionThis article is about the arithmetic operation. For other uses, see Division (disambiguation). In mathematics, especially elementary arithmetic, division is an arithmetic operation which is the reverse operation of multiplication and sometimes can be inte is similar to multiplication. If both the dividend and the divisor have different signs, the result is negative:

If both numbers are of the same sign, the result is positive (even if they are both negative):

5 Formal construction of negative and non-negative 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:

We define an equivalence relation ~ upon these pairs with the following rule:

if and only if

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N²/~, 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