Foundations — Addition and subtraction — carrying, borrowing, word problems

Before you can carry or borrow, you need to own every symbol the parent note throws at you. This page builds each one from nothing — plain words first, then a picture, then the reason the topic can't live without it. Read top to bottom: each idea is the floor the next one stands on.

1. Digits — the ten symbols

Plain words. A digit is one of the ten marks we write numbers with: $0,1,2,3,4,5,6,7,8,9$. That's the whole alphabet of numbers — there is no single symbol for "ten".

The picture. Think of a single slot that can show any face from $0$ to $9$, like one wheel on a car's odometer. When the wheel is on $9$ and you add one more, it can't show "ten" — it clicks back to $0$ and nudges the wheel to its left.

Why the topic needs it. Because there is no symbol for ten, the moment a column reaches ten it is forced to reset and push something leftward. That forced push is exactly what the parent note calls carrying. Digits being capped at $9$ is not a rule someone invented — it is the reason carrying exists at all.

2. Place value — a digit's column decides its worth

Plain words. The same digit means different amounts depending on which column it sits in. The $4$ in $472$ is worth four hundreds, not four. Place value is this rule: worth $=$ digit $\times$ (the column's size).

The picture. Three labelled envelopes in a row. The rightmost holds single coins (ones), the middle holds coins each worth ten (tens), the left holds coins each worth a hundred (hundreds). Drop a $4$-coin into the hundreds envelope and it is worth $400$; the coin didn't change, the envelope did.

Why the topic needs it. Every column is a power of ten — a repeated multiplication of ten by itself:

So a number expands like this: $472=4\times 10^2+7\times 10^1+2\times 10^0=400+70+2.$

We use powers of ten and not some other pattern because we write in base ten — ten fingers, ten digits, ten of anything trades up. See Place Value and Base-Ten System for the full story; here we only need the trade rule below.

3. Reading the operation signs

Plain words.

• $+$ means combine two amounts into one bigger amount. The result is the sum.
• $-$ means take away the second amount from the first. The result is the difference.
• $=$ means the two sides are the same amount, just written differently.

The picture. For $+$: two coin piles pushed into one. For $-$: one pile with some coins physically removed. The $=$ sign is a perfectly balanced beam — whatever weight sits left must equal whatever sits right.

Why the topic needs it. These three signs are the verbs of arithmetic. The parent note leans on one deep fact linking them:

Word problems (see Word Problem Translation) are really the job of turning English verbs — "gave away", "bought more" — into these signs.

4. The two special numbers being subtracted

Plain words. In a subtraction, the number on top (the one you start with) is the minuend. The number below it (the one you remove) is the subtrahend.

The picture. A subtraction written vertically: minuend on the roof, subtrahend under it, a line, and the difference on the floor.

$\begin{array}{r}\text{minuend}\\ -\text{subtrahend}\\ \text{difference}\end{array}$

Why the topic needs it. These names pin down direction. Subtraction is not "big minus small" — the minuend must stay on top even when its digit is smaller than the one below. That single fact is what makes borrowing necessary; without the vocabulary you can't state the rule cleanly.

5. Carry and borrow — repackaging between columns

Plain words.

• A carry is a single $1$ pushed up to the next-bigger column when a column's total reaches ten or more.
• A borrow is a single $1$ (worth ten) pulled down from the next-bigger column when the top digit is too small to subtract.

The picture. Carrying: a full ones-envelope (ten coins) is bundled and dropped whole into the tens-envelope as one coin. Borrowing: one coin from the tens-envelope is smashed into ten coins and poured into the ones-envelope.

Why the topic needs it — this IS the topic. Because digits stop at $9$ (idea 1) and each column is worth ten of the one below (idea 2), any overflow or shortfall must be traded across columns. Carry and borrow are the two directions of that trade. Nothing is created or destroyed — the same amount is just re-boxed.

6. The tools that describe the trade exactly

The parent note uses three mathematical tools to make "repackaging" precise. Here is each, from zero.

The remainder split $a=10q+r$. When you add two digits you might get, say, $13$. We want to know two things: which single digit to write ($3$) and what to carry ($1$). Splitting $13=10\times 1+3$ answers both at once — this is the Division Algorithm with divisor $10$. The $q$ (the carry) and the $r$ (the written digit) are guaranteed unique.

We choose these two tools — floor and mod — because they are exactly the two halves of the $10q+r$ split: floor grabs $q$, mod grabs $r$. No other pair of tools gives you both pieces in one clean sweep.

Why is the carry never bigger than $1$? The most a column can total is two digits plus one incoming carry: $9+9+1=19$. Since $19<20$, the floor $\lfloor 19/10\rfloor =1$. So in base ten the carry is always $0$ or $1$ — never more. That bound is why the whole method stays simple.

7. When numbers go below zero

Plain words. Borrowing across a chain of zeros feels like going below nothing. $0-1$ in a column looks impossible — but it just means that column owes, so it reaches further left. The full theory of amounts below zero is Negative Numbers and Integers; here we only meet its shadow, which resolves the moment the borrow lands.

The picture. An empty ones-envelope told to pay one coin: it can't, so it forces the tens-envelope to break a coin, which forces the hundreds-envelope in turn. A middle $0$ borrows and immediately lends, so it settles at $9$.

Why the topic needs it. The classic "borrow across zeros" mistake happens exactly because a beginner refuses to let a column go temporarily short. Knowing that "below zero" is fine for one step — because the borrow instantly repays it — is what makes zero-chains work.

Prerequisite map

Equipment checklist

Test yourself — read the left side, answer, then reveal.

Connections

• Place Value and Base-Ten System — the full base-ten story these columns come from.
• Division Algorithm — $a=10q+r$ is the carry split itself.
• Negative Numbers and Integers — where "below zero while borrowing" leads.
• Multiplication as Repeated Addition — the same carries return there.
• Estimation and Rounding — sanity-check before you compute.
• Word Problem Translation — turning English verbs into $+$ and $-$.
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