Number circle

The number circle is a term from machine data processing . It is particularly helpful with the finite number systems as required in arithmetic units . Only a limited number of positions are available in these . This must be carefully taken into account when calculating in order to avoid errors caused by exceeding a limit. Mathematically speaking, the number circle treats a remainder class .

basis

With the notation of numbers common in everyday life and science in Europe , a number is represented in a place value system using digits, signs and separators (such as commas, spaces). A digit stands for a number from a limited supply. An arrangement of several digits creates an unlimited supply of numbers. Every place in this arrangement that a digit occupies or is intended to occupy is a place. Each digit is to be evaluated differently depending on the position ("ones digit", "tens digit", ...).

The place value system contains digits, which are usually assigned the values of the natural numbers from to . If the ones place, which has the place value 1, is also numbered, then the -th place has the place value . ${\ displaystyle b}$ ${\ displaystyle 0}$ ${\ displaystyle b-1}$ ${\ displaystyle i = 0}$ ${\ displaystyle i}$ ${\ displaystyle b ^ {i}}$

Among the many important systems are preferably the familiar decimal system with common and in digital technology, the dual system with or Sedezimalsystem with that summarizes four locations of the dual system to a point. ${\ displaystyle b = 10}$ ${\ displaystyle b = 2}$ ${\ displaystyle b = 16}$

Unlimited numbers

The place value system develops in the following way: Through the calculation step "Addition of a one", the digit with the next higher value is written in the ones place instead of the existing digit. If the digit with the highest value is already there, a zero is written in its place and the calculation step is carried out in the next most significant place. If there is no higher digit, a zero is placed in front of the sequence of digits and the calculation step is carried out. The system is unlimited for higher-value positions to the left. Any number of zeros can be written in front of a number, which does not change its value and is usually omitted.

In the case of negative numbers, a sign is placed to the left of the sequence of digits. This then stands for the amount of the number.

Number line with the unlimited arrangement of signed (real) numbers on both sides

The infinity of the number supply is illustrated by the infinity of a number line .

Limited number of numbers

If there is limited space for digits, such as in the arithmetic unit of a microprocessor, the use of an unlimited number system is not possible. For numbers from a limited range of numbers, all digits are written, including those with leading zeros. Corresponding to the application in digital computers, the description here only deals with binary numbers. With two digits the numbers 000… 000 to 111… 111 are possible with the number of digits determined by the design of the calculator.

Number circle with eight-digit binary numbers and different decimal number assignments. A horizontal line marks the limits of the number range.

With locations can be natural numbers from to represent. The calculation step "Addition of a one" is carried out in the same way as with an unlimited number of numbers, except for the binary number 111 ... 111, where there is no carry over to a higher digit, so that the addition leads back to the number 000 ... 000. This is illustrated by a circle of numbers . ${\ displaystyle n}$ ${\ displaystyle 2 ^ {n}}$ ${\ displaystyle 0}$ ${\ displaystyle 2 ^ {n} -1}$

To represent negative numbers from the number range to , the two's complement is common in the dual system , see also Integer (data type) . This means that processing in the arithmetic unit remains simple, because the calculation step “addition of a one” remains unchanged in its execution. However, the two's complement for a negative number is not structured as in a place value system. ${\ displaystyle -2 ^ {n-1}}$ ${\ displaystyle 2 ^ {n-1} -1}$

Furthermore, an offset (zero point offset) is used to represent signed numbers. The smallest (most negative) number then includes the (conventionally smallest) binary number 000 ... 000, the largest number belongs 111 ... 111, and the zero immediately above the middle in a symmetrical number range belongs to the binary number directly above the middle, to 100 ... 000. The calculation step “addition of a one” also runs here as in the place value system, but none of the numbers is structured as in a place value system. ${\ displaystyle 2 ^ {n}}$

If a limit of the number range is exceeded during an addition or subtraction, an incorrect result occurs, which is identified by the calculator by its status bits .

Individual evidence

↑ ^a ^b DIN 1333 , numbers , 1992, chap. 10.1
↑ DIN 1333, chap. 8th
↑ Erich Leonhardt: Fundamentals of digital technology. Carl Hanser, 1984, page 147 ff
↑ Erwin Samal, Wilhelm Becker: Floor plan of the practical control technology. Oldenbourg, 2004, p. 507
↑ Axel Böttcher, Franz Kneißl: Computer Science for Engineers: Fundamentals and Programming in C. Oldenbourg, 2012, page 29 ff

[D1333-1] DIN 1333 , numbers , 1992, chap. 10.1

[2] DIN 1333, chap. 8th

[3] Erich Leonhardt: Fundamentals of digital technology. Carl Hanser, 1984, page 147 ff

[4] Erwin Samal, Wilhelm Becker: Floor plan of the practical control technology. Oldenbourg, 2004, p. 507

[5] Axel Böttcher, Franz Kneißl: Computer Science for Engineers: Fundamentals and Programming in C. Oldenbourg, 2012, page 29 ff