# Positive and negative numbers

In mathematics, the real numbers are divided into positive and negative numbers without the zero ( ). A number which is greater than zero, such as  3 , is called positive; if it is less than zero, such as  −3 , it is called negative. Positive numbers (more precisely: number constants) have a plus sign (+) and negative numbers have a minus sign (-) as a sign . The plus sign is usually omitted when noting the number. The number zero is neither positive nor negative. ${\ displaystyle \ mathbb {R} \ backslash \ {0 \}}$ The same distinction can be made with subsets of real numbers, such as rational numbers or whole numbers .

There are sets of numbers for which one cannot divide into positive numbers, negative numbers and the number zero, which can be brought into agreement with the addition and multiplication of these numbers at the same time (e.g. the set of complex numbers ). This is always the case when you cannot define a total order that is compatible with both operations. (Number) bodies with this property are called "not arrangeable".

## presentation

Positive numbers are indicated without a sign or with a plus sign, negative numbers with a minus sign . The sign is attached directly to the first digit without any space . Especially in the financial sector, negative numbers are alternatively written in brackets.

On the number line , the range of positive numbers is mirror-symmetrical to the range of negative numbers. The distance between a number a and 0 is equal to the distance between 0 and number −a and is called the amount of the number.

## properties

### sign

The sign indicates a number as positive or negative. The sign function returns an integer value depending on the sign, −1 for negative numbers, 0 for the 0 and +1 for positive numbers.

### Opposite number

The number −a is called the opposite of a ; a is the opposite of −a . But if a is a negative number, then −a is a positive number. The opposite number to a has a different sign than a , with one exception: 0 is its own opposite number.

### amount

The amount of a number is equal to the distance between the number and the number 0 . The amount of a number a is equal to the amount of its opposite number −a .

### Natural numbers

The term “positive” or “negative number” can be transferred to the whole numbers (embedded in the real numbers). The natural numbers are the positive (or if so defined, the nonnegative) whole numbers. The notation (only positive whole numbers) or (non-negative whole numbers) has become established here. Non-negative only means that the zero is also considered in this set. ${\ displaystyle \ mathbb {N}}$ ${\ displaystyle \ mathbb {N} _ {0}}$ ### Multiplication group

If we combine the positive real (or rational) numbers to the set P and the negative real (or rational) numbers to the set N , then the union of the sets P and N , i.e. the set of all non-zero numbers, is an Abelian group with respect to the Multiplication .

Since z. For example, if the integer 2 (in terms of multiplication) does not have an inverse integer (1/2 is not an integer), this does not apply to the integers.

### "Minus times minus equals plus"

If you multiply two negative or two positive numbers, you always get a positive number. If you multiply a positive number by a negative number, the result is always negative.

## Sign error

Many calculation errors are based on a mix-up of the sign.

A well-known example is the Hochrhein Bridge, where the correction value of 27 cm was used with the wrong sign when calculating the bridge piers. Switzerland and Germany have a height system that differs by 27 cm at the common border.

## Practical use

In some areas, special terms have been established that avoid the use of negative numbers. So one speaks z. B. of “debt” instead of “negative credit” or “braking” instead of “negative acceleration”. On the other hand, scales with positive and negative numbers have become established in some places where negative numbers are not even necessary, such as for temperature measurement ( Celsius and Fahrenheit scales instead of the Kelvin scale ).

## Individual evidence

1. Ebbinghaus et al .: Numbers . 3. Edition. Springer-Verlag, Berlin / Heidelberg / New York / London / Paris / Tokyo / Hong Kong / Barcelona / Budapest 1992, ISBN 3-540-55654-0 , 3.2.4 The body cannot be arranged ( n.d. ).${\ displaystyle \ mathbb {C}}$ 2. Katrin Terpitz: Babylon is everywhere. Projects in companies. In: handelsblatt.com. September 30, 2007, accessed July 10, 2013 .