Subtraction

Subtraction is the basic operation of “taking away” one quantity from another. In this chapter, we focus on how subtraction works with whole numbers, how to write and read subtraction statements, and how to perform subtraction in different situations.

What Subtraction Means

Subtraction answers questions like:

• “How many are left?”
• “How much more does one have than another?”
• “What is the difference between these two numbers?”

If you start with $a$ things and you take away $b$ of them, the result is called the difference:
$$
a - b = \text{difference}
$$

For example, $9 - 4 = 5$ means: starting with 9, removing 4, leaves 5.

Subtraction is closely related to addition, but it is not the same:

• In addition, you combine quantities.
• In subtraction, you compare or remove quantities.

Basic Subtraction Facts

Just as there are basic addition facts, there are basic subtraction facts that are useful to know by memory, especially for small numbers (0 to 10 or 0 to 20).

Some key patterns:

1. Subtracting zero does not change the number:
   $$
   a - 0 = a
   $$
   Example: - 0 = 7$.
2. Subtracting a number from itself gives zero:
   $$
   a - a = 0
   $$
   Example: - 9 = 0$.
3. Subtracting one gives the previous number (the “number before”):
   $$
   a - 1 = \text{the number just before } a
   $$
   Example: - 1 = 9$.
4. Order matters in subtraction. Swapping the numbers usually changes the result:
   $$
   9 - 4 = 5 \quad\text{but}\quad 4 - 9 \neq 5
   $$
   For now, when working only with whole numbers (non-negative integers), we usually subtract a smaller number from a larger one to avoid negative results.

Subtraction on the Number Line

A number line is a simple way to picture subtraction:

• Numbers increase to the right.
• Numbers decrease to the left.

To subtract $b$ from $a$ (that is, to compute $a - b$):

1. Start at $a$ on the number line.
2. Move $b$ steps to the left.
3. The place where you land is $a - b$.

Example: To find $8 - 3$:

• Start at $8$.
• Move 3 steps left: $8 \to 7 \to 6 \to 5$.
• You land on 5, so $8 - 3 = 5$.

This “moving left” idea matches the idea of “taking away” or “having less.”

Relationship Between Addition and Subtraction

Addition and subtraction are opposite operations. They “undo” each other.

If
$$
a + b = c
$$
then
$$
c - b = a \quad\text{and}\quad c - a = b
$$

Example:

• $6 + 3 = 9$
• So $9 - 3 = 6$ and $9 - 6 = 3$.

This means:

• You can check a subtraction by turning it into an addition.
• You can find a missing number in an addition by using subtraction.

Example (checking):

• You think $15 - 7 = 8$.
• Check: $8 + 7 = 15$? Yes, so the subtraction is correct.

Writing and Reading Subtraction

A subtraction expression has three main parts:

• The minuend: the number you start with (the first number).
• The subtrahend: the number being taken away (the second number).
• The difference: the result.

In $13 - 5 = 8$:

• Minuend: 13
• Subtrahend: 5
• Difference: 8

We read $13 - 5$ as “thirteen minus five” or “thirteen take away five.”

Other common phrases that often mean subtraction:

• “How many are left?”
• “How much more … than …?”
• “What is the difference between … and …?”
• “Decrease by”
• “Less than” (be careful with the order; this is often reversed in wording)

Example: “How much more is 12 than 7?” means $12 - 7$.

Subtracting Single-Digit Numbers

For small numbers, you can often subtract by:

• remembering basic facts, or
• counting backward (mentally using a number line).

Example: $9 - 4$.

You can:

• Think: “4 more to reach 9 is 5,” so $9 - 4 = 5$, or
• Start at 9 and count back 4 steps: $9, 8, 7, 6, 5$ → you land on 5.

Practicing all the subtractions up to 10 (or 20) helps make later work faster and easier.

Subtracting Multi-Digit Numbers Without Regrouping

Subtraction with larger whole numbers is often done with column subtraction (also called “vertical subtraction”).

Without regrouping means that, in each place (ones, tens, hundreds, etc.), the top digit is at least as big as the bottom digit.

Example:
$$
73 - 21
$$
Write it vertically, lining up the digits by place value:
$$
\begin{array}{r}
73 \\
-21 \\
\hline
\end{array}
$$

Subtract by place, starting from the right (ones):

1. Ones: $3 - 1 = 2$
2. Tens: $7 - 2 = 5$

So:
$$
73 - 21 = 52
$$

Another example:
$$
504 - 302
$$

Vertical form:
$$
\begin{array}{r}
504 \\
-302 \\
\hline
\end{array}
$$

Subtract by place:

1. Ones: $4 - 2 = 2$
2. Tens: $0 - 0 = 0$
3. Hundreds: $5 - 3 = 2$

So:
$$
504 - 302 = 202
$$

No regrouping was needed because the top digit in each column was not smaller than the bottom digit.

Regrouping (Borrowing) in Subtraction

Sometimes the digit in the minuend is smaller than the digit in the subtrahend in a certain place. Then we cannot subtract that column directly. We use regrouping (often called borrowing).

The basic idea:

• You “take” 1 from the next higher place (where possible).
• That 1 represents 10 of the smaller place.
• Add 10 to the current place, then subtract.

Example 1: Subtracting with Regrouping in the Ones

Compute $52 - 38$.

Write it vertically:
$$
\begin{array}{r}
52 \\
-38 \\
\hline
\end{array}
$$

Start from the ones place:

1. Ones: We want $2 - 8$, but $2$ is smaller than $8$.
• Regroup: take $1$ ten from the tens digit (5 tens become 4 tens).
• Convert that 1 ten to 10 ones: $2$ ones become $12$ ones.
• Now compute $12 - 8 = 4$.
2. Tens: After regrouping, the tens digit on top is $4$ (not $5$ anymore).
• Compute $4 - 3 = 1$.

So:
$$
52 - 38 = 14
$$

Example 2: Regrouping Across a Zero

Compute $402 - 178$.

Vertical form:
$$
\begin{array}{r}
402 \\
-178 \\
\hline
\end{array}
$$

Step by step:

1. Ones: $2 - 8$ is not possible directly.
• Regroup from the tens place. But the tens digit is $0$, so we first need to regroup from the hundreds.
2. Regroup hundreds to tens:
• Take 1 from the hundreds: $4$ hundreds become $3$ hundreds.
• That 1 hundred becomes 10 tens: tens digit becomes $10$.

Now we have $3$ hundreds, $10$ tens, and still $2$ ones.

3. Regroup tens to ones:
• Take 1 ten from the tens: $10$ tens become $9$ tens.
• That 1 ten becomes 10 ones: $2$ ones become $12$ ones.

Now we have $3$ hundreds, $9$ tens, and $12$ ones.

4. Now subtract:
• Ones: $12 - 8 = 4$
• Tens: $9 - 7 = 2$
• Hundreds: $3 - 1 = 2$

So:
$$
402 - 178 = 224
$$

Regrouping across zeros often takes more than one step, but the idea is the same: move value from a higher place to a lower place.

Estimating Subtraction

You can use estimation to check if an answer is reasonable.

Basic method:

• Round each number to a nearby “easy” number (like to the nearest 10 or 100).
• Subtract the rounded numbers.

Example: Estimate $487 - 192$.

• Round $487$ to $490$ (or maybe $500$).
• Round $192$ to $190$ (or maybe $200$).

Using nearer tens:
$$
490 - 190 = 300
$$

So the exact answer should be around $300$. (In fact, $487 - 192 = 295$.)

Estimation does not give the exact answer, but it helps spot answers that are clearly too big or too small.

Checking Your Subtraction

There are a few simple ways to check a subtraction:

1. Use Addition

If you computed:
$$
a - b = c,
$$
then you can check by adding:
$$
c + b
$$
to see if you get $a$.

Example:

• You think $83 - 27 = 56$.
• Check: $56 + 27 = 83$? Yes, so the subtraction is correct.

2. Reverse the Order

Subtract in the opposite direction to compare results:

• If $a - b = c$,
• Then $a - c$ should equal $b$.

Example:

• You think $70 - 45 = 25$.
• Check: $70 - 25 = 45$? Yes.

Using at least one of these checks is a good habit, especially for longer problems.

Subtraction in Everyday Situations

Subtraction appears naturally in many real-world questions:

• Money: “You have \$20 and spend \$7. How much is left?” → $20 - 7$.
• Time: “The movie ends at 8:30 and started at 7:15. How long did it last?” → subtract start time from end time using appropriate methods.
• Distance: “You walked 5 km and your friend walked 3 km. How much farther did you walk?” → $5 - 3$.
• Comparisons: “There are 18 students in class A and 13 in class B. How many more are in A?” → $18 - 13$.

Often, the words “left,” “more than,” and “difference” signal a subtraction situation.

Limits with Whole-Number Subtraction

In this chapter, we are working only with whole numbers (0, 1, 2, 3, …). In this setting:

• Subtraction like $9 - 12$ is “not allowed” if we require the answer to be a whole number.
• We usually arrange problems so that the minuend is at least as large as the subtrahend.

Later, when integers are introduced, subtraction that gives negative results (like $9 - 12 = -3$) will make sense. For now, focus on whole-number cases where the result is non-negative.

Summary

• Subtraction represents “taking away” and “finding the difference.”
• $a - b$ is read “$a$ minus $b$,” where $a$ is the minuend, $b$ is the subtrahend, and the result is the difference.
• Subtraction is not commutative: $a - b$ is usually not equal to $b - a$.
• You can picture subtraction as moving left on the number line.
• Column subtraction subtracts by place value, often needing regrouping (borrowing).
• Subtraction and addition are inverse operations; use addition to check your subtraction.
• Estimation helps you judge whether a subtraction answer is reasonable.