1. What addition really is
If you have three apples in one hand and five in the other and someone asks "how many apples?" — you've just done addition. Addition is the operation that joins two collections into one and reports the new total.
The operation that takes two numbers — called addends — and produces a single number called their sum. Written $a + b = c$, where $c$ is the sum of $a$ and $b$.
The "$+$" symbol came from a shorthand for the Latin et ("and"), and the verbal version — three plus five — still reads as "three and five." It's worth holding on to that everyday meaning: addition is the formal version of the word and.
Two intuitions cover almost every situation where addition is the right tool:
- Combining. Two separate piles become one. "I have $\$12$ and you give me $\$5$ — now I have $\$17$."
- Stepping forward. Starting at one number, moving along by another. "It's $3$ o'clock, in $4$ hours it'll be $7$." This is the picture you'll want when you eventually meet number lines and negative numbers.
2. The three properties that make addition easy
Addition obeys three rules that are so familiar you might not realize they're rules at all. Each one is a license to rearrange — and rearranging is what makes mental arithmetic possible.
| Property | What it says | In symbols |
|---|---|---|
| Commutative | Order doesn't matter. | $a + b = b + a$ |
| Associative | Grouping doesn't matter. | $(a + b) + c = a + (b + c)$ |
| Identity | Adding zero changes nothing. | $a + 0 = a$ |
None of these are obvious from the definition — they have to be checked. But they hold for all whole numbers, and they license moves you make without thinking:
- To add $9 + 7 + 1$, your brain probably reorders to $(9 + 1) + 7 = 10 + 7 = 17$. That's commutativity and associativity in tandem.
- To add $48 + 27$, you might split into $48 + 20 + 7 = 68 + 7 = 75$. That's associativity, after you've decomposed $27$ by place value.
The column algorithm you'll see in a moment is one specific way to add — but you're free to do addition in any order you like, because of these properties. People who are fast at mental arithmetic aren't following a different algorithm; they're using commutativity and associativity to make the numbers friendlier first.
3. Adding multi-digit numbers (the column method)
When the addends are bigger than a single digit, the trick is to add column by column, exploiting place value. Each column is a small problem you already know: a single-digit addition. The only complication is what happens when that small sum is $10$ or more — the column overflows, and the overflow has to go somewhere.
Let's add $487 + 256$ in full.
Walk through it column by column, right to left:
- Ones column: $7 + 6 = 13$. Thirteen is "one ten and three ones." Write the $3$ in the ones column of the answer; carry the $1$ to the top of the tens column.
- Tens column: $1 + 8 + 5 = 14$. That's "one hundred and four tens." Write the $4$ in the tens column; carry the $1$ to the hundreds.
- Hundreds column: $1 + 4 + 2 = 7$. Just write the $7$. (If the leftmost column had also overflowed, it would have created a brand-new column on the left — see the worked examples.)
So $487 + 256 = 743$. The whole algorithm is one idea repeated: each column holds at most one digit; anything over ten spills one place to the left, because that's exactly what the next place is worth.
Place value is the engine. When the ones column produces $13$, that $13$ is literally $1 \cdot 10 + 3 \cdot 1$. The $3$ stays in the ones place; the $1$ migrates to the tens place because that's where tens go. Carrying isn't a separate rule — it's place value being respected.
4. What subtraction really is
Subtraction is the operation that undoes addition. If $a + b = c$, then $c - b = a$. That formal description is correct but a bit colorless. The two living intuitions are these:
- Take away. Start with a collection of $c$ things, remove $b$ of them, how many are left? This is the intuition kids learn first.
- Distance / difference. How far apart are two numbers? $c - b$ is the gap between $b$ and $c$ on the number line. This view becomes more useful as you go: it survives unchanged when negative numbers appear, and it's the intuition that physicists and engineers actually use.
The operation that takes two numbers and reports their difference. In $c - b = a$, the number $c$ is the minuend, $b$ is the subtrahend, and $a$ is the difference. The "$-$" sign has been a minus sign in mathematics since at least the 15th century.
Subtraction looks like a small variant of addition, but it behaves very differently:
- It is not commutative: $5 - 3 = 2$, but $3 - 5 = -2$. Order matters absolutely.
- It is not associative: $(10 - 5) - 2 = 3$, but $10 - (5 - 2) = 7$. Grouping matters absolutely.
- Zero is still an identity, but only on the right: $a - 0 = a$. However $0 - a = -a$, which is a different number unless $a = 0$.
The lack of commutativity is the property to remember above all else. Most subtraction mistakes are really commutativity mistakes — you wrote the operands in the wrong order, or swapped them mid-computation.
5. Subtracting multi-digit numbers (with borrowing)
The column method for subtraction works exactly like the addition version, except that instead of carrying an overflow to the left, you sometimes have to borrow one from the left to make a column big enough to subtract from.
Try $503 - 178$. The ones column wants $3 - 8$, which won't fit in a single digit's worth of room — so we need to bring in a ten from the next column to the left. But that column has a $0$ in it, so first we have to bring a hundred from further left and break it into ten tens. The chain looks like this:
The play-by-play:
- Ones: we want $3 - 8$. Can't do it. We need ten more in the ones column.
- Tens: would like to borrow a ten from here, but the tens column is $0$. We have to borrow from the hundreds column first.
- Hundreds: take one of the five hundreds. We now have $4$ hundreds left, and the borrowed hundred becomes $10$ tens in the tens column. The tens column is now $10$.
- Tens (again): now we can hand a ten to the ones column. We have $9$ tens left, and the ones column is now $13$.
- Subtract column by column: $13 - 8 = 5$, $9 - 7 = 2$, $4 - 1 = 3$. The difference is $325$.
"Borrowing" is the conventional name, but it's a misleading one: nothing is given back. Regrouping is the term modern textbooks prefer — and it's accurate, because what you're really doing is rewriting $503$ as $4$ hundreds + $9$ tens + $13$ ones, which is exactly the same number, just decomposed in a way that makes the subtraction column-by-column doable.
A subtraction problem comes with a free check: add the difference and the subtrahend back together; you should get the minuend. $325 + 178 = 503$ ✓. Use this whenever you don't trust your borrowing.