transfer

The carry-over ( English carry ) is a term from mathematics and stands for the number of characters in a particular division with arithmetic operations with numbers represented by a value system are presented. In the usual calculation of numbers in the decimal system , one speaks of the transfer of tens .

In a number system with place values, numbers are represented by number signs , in which the symbolizing digits are assigned a place value after the respective place in a digit sequence in addition to their digit value. Two numbers and thus each have a number sign with consecutive digits, ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle x=x_{n}\ldots x_{i}\ldots x_{1}x_{0}\quad {\text{und}}\quad y=y_{m}\ldots y_{i}\ldots y_{1}y_{0}}$ ,

whereby a certain number of digits is available at each position, which characterizes the respective number system as a basic number or base , for example ten digits in the decimal system. If the numbers represented in this way ( -adic) are to be linked to one another by an arithmetic operation , for example the numbers and are added, one has to proceed in places. Then, at the location of a transfer , arise if the intermediate result of the linking of each item and greater than or equal , and thus a multi-digit sequence of numbers. The digits of the surplus places from are then linked with those of the places to from and . ${\displaystyle b}$ ${\displaystyle b}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle i}$ ${\displaystyle z}$ ${\displaystyle x_{i}\ }$ ${\displaystyle \ y_{i}}$ ${\displaystyle b}$ ${\displaystyle k}$ ${\displaystyle z_{k}\ldots z_{1}}$ ${\displaystyle z}$ ${\displaystyle i+1}$ ${\displaystyle i+k}$ ${\displaystyle x}$ ${\displaystyle y}$

If the range of numbers is limited, arithmetic overflows can occur with addition or subtraction .

Example: addition

Example: 786 + 457

If you add the numbers 195 and 107 in decimal representation, you get two carry-overs (shown here in red):

${\displaystyle {{\begin{matrix}\ &1_{\ }&9_{\ }&5\\+&1_{\color {Red}1}&0_{\color {Red}1}&7\end{matrix}} \over {\begin{matrix}\quad &3_{\ }&0_{\ }&2\end{matrix}}}}$

The addition results in the first calculation step to a result that can not be specified by the present digit numbers stock: . Therefore, the digit lowest position in this case will be entered at this point and transmit a digit higher position on the appropriate place, in this case, the as carry-over to the next position. The second calculation step gives . But at this point the carryover still has to be taken into account and counted. again provides a two-digit result and another as a carry over, which results in added . ${\displaystyle 5+7}$ ${\displaystyle 12}$ ${\displaystyle 2}$ ${\displaystyle 1}$ ${\displaystyle 9+0}$ ${\displaystyle 9}$ ${\displaystyle 1}$ ${\displaystyle 9}$ ${\displaystyle 9+1}$ ${\displaystyle 1}$ ${\displaystyle 1+1}$ ${\displaystyle 3}$

This method is also used to add in other number representations, such as the dual :

${\displaystyle {{\begin{matrix}\ &{\ }_{\ }&1_{\ }&0_{\ }&1\\+&{\ }_{\color {Red}1}&1_{\ }&0_{\color {Red}1}&1\end{matrix}} \over {\begin{matrix}\quad &1_{\ }&0_{\ }&1_{\ }&0_{\ }\end{matrix}}}}$

Related topics

In the information technology , the carry is a carry bit realized (carry bit).