Subtraction

Subtraction is an arithmetic operation used to find what remains when one number is taken away from another. Its symbol is "−"; for example, 7 − 3 = 4 is read as "seven minus three equals four". It is one of the four basic operations: addition, subtraction, multiplication, and division.

Subtraction is the operation that reverses addition. If 3 + 4 = 7, then 7 − 4 = 3 and 7 − 3 = 4. For this reason, subtraction can be verified by addition: when the result is added to the number that was subtracted, it should return the original number.

Parts of subtraction

The number from which another is subtracted is called the minuend. The number being subtracted is called the subtrahend. The result is called the difference or remainder. When written vertically, numbers are arranged by their place value (ones, tens, hundreds) and the operation begins from the right.

Properties

Subtraction is not commutative: 7 − 3 and 3 − 7 are not equal. It is also not associative: (9 − 4) − 2 and 9 − (4 − 2) give different results. For this reason, when performing multiple subtractions in sequence, the order and grouping of the terms matter.

Negative numbers

When working only with natural numbers (1, 2, 3, ...), subtraction is not always possible: 3 − 7 has no answer. To enable the operation to always work, mathematics uses negative numbers, making 3 − 7 = −4 possible. The system of integers is one in which subtraction is completely closed.

The borrowing method

When the digit in the lower row is larger than the digit in the upper row in the same place value, a method called borrowing or regrouping is used: one unit is taken from the next higher place value and its ten is added to the current place. For example, in 52 − 27: two and seven cannot be subtracted directly, so we borrow from the tens place, making 12 − 7 = 5, then 4 − 2 = 2, giving a result of 25. Other methods such as complement subtraction are also used, particularly in computers.

Applications

Subtraction operates in daily life: calculating change from money, comparing heights, measuring time intervals between two dates, and computing remaining stock. In higher mathematics, the concept of subtraction has extended to vectors, matrices, and functions, where the same basic principles of reversing addition apply.