Ziffernüberträge

Ziffernüberträge, also known as carries, are a fundamental concept in elementary arithmetic, particularly in the process of addition and subtraction. When performing column addition, a carry occurs when the sum of the digits in a particular place value column exceeds the base of the number system, which is typically 10 in the decimal system. For example, in adding 7 and 8, the sum is 15. The '5' is placed in the units column, and the '1' is carried over to the tens column. This carry represents one group of ten.

Similarly, in subtraction, a borrow, which is the inverse of a carry, is sometimes necessary. When a digit in the minuend is smaller than the corresponding digit in the subtrahend, a borrow is taken from the next higher place value column. This effectively increases the digit in the current column by the base (10) and decreases the digit in the next higher column by one. For instance, subtracting 5 from 12 requires borrowing from the tens place. The '2' becomes '12', and the '1' in the tens place becomes '0'. The result is 7.

Ziffernüberträge are crucial for accurately representing and manipulating numbers in positional notation systems. They are the mechanism by which the value of digits in higher place values is adjusted to account for the overflow or deficit from lower place values, ensuring that the final sum or difference is correct. This principle applies across various number bases, though the specific value of the carry or borrow changes accordingly.