Introduction to Subtraction
Subtraction is the operation of finding how much remains when one quantity is taken from another. It is the inverse of addition.
Subtraction has been recorded for almost as long as addition. Ancient Egyptian scribes used hieroglyphic tally marks to represent "taking away" quantities in trade and taxation records over 3,000 years ago, and the minus sign (−) itself first appeared in print alongside the plus sign in Johannes Widmann's 1489 German arithmetic textbook. Long before either symbol existed, merchants tracked debts and remaining stock simply by removing physical tokens or scratching out tally marks — subtraction, at its core, is the mathematics of "what's left."
Three Meanings of Subtraction
| Situation | What it means | Example |
|---|---|---|
| Take away | Remove an amount from a total | 12 − 5 = 7 |
| Find the difference | How far apart are two numbers? | 12 − 5 = 7 |
| Find the missing part | Total and one part known | 12 − ? = 5, so ? = 7 |
Visualising Subtraction: Take-Away and Distance
The two main meanings of subtraction look different visually, and it helps to see both. “Take away” shows one bar shrinking as a piece is removed. “Difference” shows two separate bars lined up side by side, and asks how much longer one is than the other. Both pictures give the same answer for 12 − 5, but they represent different real-world questions.
Take-away: start with a bar of length 12. Remove (take away) a piece of length 5 from the end. What's left is the answer.
Difference (comparison): instead of removing anything, line up two separate bars — one of length 12, one of length 5 — starting from the same point. The extra length that sticks out is the difference between them.
Both pictures show 12 − 5 = 7, but notice the difference in what's being asked: the take-away picture asks “what's left after removing 5?”, while the comparison picture asks “how much bigger is 12 than 5?” — the same subtraction answers both questions. This comparison model is also exactly how “counting up” works: the distance you count from the smaller bar's end to the larger bar's end is the difference (explored further on the Number Line Subtraction page).
Key Properties
- Subtraction is not commutative: 8 − 3 ≠ 3 − 8.
- Subtracting zero gives the original: 7 − 0 = 7.
- Subtracting a number from itself gives zero: 9 − 9 = 0.
Checking with Addition
The inverse of subtraction is addition. Always verify: difference + subtrahend = minuend.
Worked Examples
9 − 4 = 5. Check: 5 + 4 = 9 ✓
73 − 28 = 45. Check: 45 + 28 = 73 ✓
1000 − 347 = 653. Check: 653 + 347 = 1000 ✓
Real-Life Applications
- Calculating change from a purchase.
- Finding how many days remain until an event.
- Calculating profit (revenue − cost).
Key Takeaways
- Subtraction finds what remains, the difference, or the missing part.
- It is not commutative — order matters.
- Always check with addition: difference + subtrahend = minuend.
Practice: Subtract and Check
A subtraction question appears below. Work it out, then we'll check it with addition. If you get it wrong (or leave it blank), we'll show the check together.