Subtraction

Subtraction (which is signified by the minus sign, −) is one of the four arithmetic operations along with addition, multiplication and division. It is an operation that represents taking away or removing objects from a group. For example, if you have 5 peaches and take away 2, you are left with 3 peaches. This is written as 5 − 2 = 3. The result of subtraction is called the difference.

Subtraction is closely related to addition. In fact, it is the opposite of addition. If you subtract a number from another, adding the same number back will give you the original number. For example, if c = a − b, then c + b = a.

Subtraction has some important patterns. Changing the order of the numbers being subtracted changes the sign of the answer. Also, the order in which you subtract more than two numbers matters. These patterns help us understand how numbers work and can be studied in more advanced math.

Notation and terminology

Subtraction is usually written using the minus sign “−” between the numbers. The answer is shown with an equals sign. For example, 2 − 1 = 1 (pronounced as “two minus one equals one”) and 4 − 6 = −2 (pronounced as “four minus six equals negative two”).

In some cases, like in accounting, subtraction can be understood without a symbol. A column of two numbers, with the lower number in red, usually means that the lower number should be subtracted, and the difference is written below.

The number being taken away is called the subtrahend, and the number it is taken from is the minuend. The answer is called the difference. This comes from Latin words meaning “to take away” and “to reduce.”

Of integers and real numbers

Integers

Imagine a straight line where each point has a number. If you start at a point labeled 3 and move 2 steps to the left, you reach the point labeled 1. This is what subtraction means: 3 − 2 = 1.

If you start at 3 and move 4 steps to the left, you reach a point labeled −1. This shows that subtraction can give us numbers that are less than zero, called negative numbers: 3 − 4 = −1.

Natural numbers

When we only use whole numbers starting from zero (like 0, 1, 2, 3), we can’t always subtract bigger numbers from smaller ones. For example, we can’t subtract 26 from 11 and still get a whole number. One way to handle this is to say it’s not possible. Another way is to allow answers that are negative numbers, so 11 − 26 would be −15.

Real numbers

With all numbers, including fractions and decimals, subtraction works by adding the opposite, or “additive inverse,” of the number being subtracted. For example, 3 − π (where π is about 3.14159) is the same as 3 + (−π).

Properties

Subtraction has some interesting properties. It is anti-commutative, which means if you reverse the order of the numbers, the result becomes the opposite of the original result. For example, if you have two numbers, a and b, then a − b is the same as −(b − a).

Subtraction is also non-associative. This means that when you subtract three numbers in a row, the way you group them changes the answer. For example, a − b − c can mean either (a − b) − c or a − (b − c), and these can give different results. To avoid confusion, we use an order of operations to decide how to calculate it.

In integers, subtracting one from a number gives you the largest number that is smaller than the original number. This is called the predecessor of the original number. For any integer a, the number (a − 1) is its predecessor.

Units of measurement

When you subtract numbers that have units like kilograms or pounds, they need to be the same unit. Usually, the answer will have the same unit as the numbers you started with.

Percentages

We can talk about changes in percentages in two ways: percentage change and percentage point change. Percentage change shows how much things have changed compared to the original amount, in percentage terms. Percentage point change is just the difference between the two percentages.

For example, imagine 30% of toys made in a factory are broken. After six months, only 20% of toys are broken. The percentage change is ⁠20% − 30%/30%⁠ = ⁠−1/3⁠ = ⁠−33+1/3⁠%, while the percentage point change is −10 percentage points.

In computing

The method of complements is a way to subtract one number from another by only adding positive numbers. This method was used in old mechanical calculators and is still used in modern computers.

To subtract a binary number y from another number x, you add the ones' complement of y to x and then add one to the sum. The leading digit "1" of the result is dropped. This method works well with binary numbers because flipping each bit (changing "0" to "1" and "1" to "0") makes it easy to get the ones' complement, and adding 1 gives the two's complement. For example:

01100100 (x, equals decimal 100)

• 00010110 (y, equals decimal 22)

becomes the sum:

01100100 (x)

• 11101001 (ones' complement of y)

•    1  (to get the two's complement)
  

—————————— 101001110

Dropping the initial "1" gives the answer: 01001110 (equals decimal 78)

The teaching of subtraction in schools

Different countries teach subtraction in their schools in various ways. In the United States, students usually learn a method called "borrowing" or "regrouping." This method helps them subtract larger numbers by adjusting digits. Teachers often use special marks to help students remember the steps.

In some European countries, schools use a different method called the Austrian or additions method. This method doesn’t involve borrowing. Instead, it focuses on adding numbers in a certain way to find the difference. Each country may have its own small marks or symbols to help students remember how to do this.

Both methods help students break down subtraction into smaller, easier steps, working from the smallest digit to the largest.

Subtraction by hand

There are several ways to do subtraction by hand. One way is the Austrian method, where you find the difference between numbers step by step. Another way is the American method, where you start from the right and "borrow" from the next digit if needed.

You can also use the partial differences method, where you subtract parts of the numbers and add or subtract those results. Other methods include counting up from the smaller number to the larger one, or breaking the subtraction into smaller, easier steps.