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

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 .

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).