Counting & Place Value

1. What counting really is

Counting feels too obvious to need a definition. You point at things and say "one, two, three" and that's that. But there's a precise idea sitting underneath the routine, and it matters:

Two ideas hide inside that definition. The first is that counting is a one-to-one pairing. If you count five apples, you're really saying "I can match those apples to the numbers 1, 2, 3, 4, 5 — one apple per number, no leftovers on either side." Two collections have the same count exactly when their objects can be paired up with nothing left over. This is the seed of an idea that grows into the modern definition of "how many" — but at this level, you can just feel it.

The second is that the numbers themselves never run out. Whatever number you stop at, you can always say "and one more." Mathematicians call this the successor property: every natural number has a next.

2. Infinite numbers, finite symbols

Here is the puzzle anyone who has ever invented a number system had to solve. There are infinitely many counting numbers, but any usable writing system has only a finite stock of symbols. How do you write infinitely many things using finitely many marks?

You can see various ancient attempts as failed or partial answers to this question:

• Tally marks. One mark per thing. Honest, but unusable past about thirty.
• Egyptian numerals. A separate hieroglyph for $1$, $10$, $100$, $1{,}000$, and so on. Better, but you need a new symbol every power of ten and arithmetic is awkward.
• Roman numerals. Combine letters with positional rules (IV = 4, VI = 6). Compact for small numbers, but multiplication of MCDXXXII by CXLIX is a nightmare — and there's no easy way to write a billion.

The system that conquered the world — sometimes called Hindu–Arabic numerals, developed in India and transmitted through the medieval Arab world to Europe — solved the puzzle with one beautifully economical idea: let the position of a digit, not just the digit itself, carry meaning. The same ten symbols $0$ through $9$ can mean different amounts depending on where they sit.

That single move makes arithmetic mechanical, makes very large numbers writable, and is the foundation of every later number system you'll meet.

3. Base-10 positional notation

Read the number $3{,}427$. Each digit occupies a place, and each place is worth ten times the place to its right:

That's the whole idea. The digit $3$ in $3{,}427$ isn't worth $3$ — it's worth $3{,}000$, because it sits three places left of the ones. The digit $4$ is worth $400$. Read column-by-column, the number decomposes into a sum of multiples of powers of ten.

In symbols, any whole number with digits $d_n d_{n-1} \ldots d_1 d_0$ (read left to right) equals

So $3{,}427 = 3 \cdot 10^3 + 4 \cdot 10^2 + 2 \cdot 10 + 7$. The system has a name worth knowing:

4. Zero, the unsung hero

Notice what happens when one of the place columns is empty. The number $407$ has a $4$ in the hundreds place, nothing in the tens place, and a $7$ in the ones. If you just wrote "$4\,7$" with a gap, that wouldn't survive being copied. You need a symbol that means this place is occupied by nothing. That symbol is $0$.

Zero is doing two different jobs here, and conflating them is the source of a lot of confusion:

• As a number, zero means "none of them" — the count of an empty collection.
• As a placeholder, zero means "this column has no contribution, but it exists." It's what distinguishes $47$ from $407$ from $4{,}007$.

The placeholder use is what most ancient number systems lacked, and it's why their arithmetic was so hard. Roman numerals never needed a "zero" because each symbol stood for a fixed amount — but that's also why you can't easily write a hundred million in Roman numerals.

5. Reading and writing large numbers

Once numbers get past four digits or so, reading them digit-by-digit becomes painful. The convention that fixes this is to group digits into blocks of three from the right, and give each block a name.

In writing, the boundary between blocks is marked with a thousands separator. In the US and UK that separator is a comma: $1{,}200{,}540$. In much of continental Europe it's a period or a space: $1{.}200{.}540$ or $1\,200\,540$. The international SI standard recommends a thin space, partly because both the comma and the period are used as the decimal separator somewhere in the world — and mixing them is exactly the kind of thing that crashes spaceships.

To read $1{,}200{,}540$ aloud, identify the blocks from the right:

• $540$ — the ones-thousands-millions block (just "five hundred forty")
• $200$ — the thousands block ("two hundred thousand")
• $1$ — the millions block ("one million")

Read them in left-to-right order with each block followed by its name: "one million, two hundred thousand, five hundred forty."

7. Other bases — same idea, different alphabet

Nothing about positional notation requires the number ten. The same trick works with any whole number $b \geq 2$ as the base (or radix): use $b$ symbols, and let each place be worth $b$ times the place to its right.

Two non-decimal bases come up enough to be worth knowing:

• Binary (base $2$) uses only the symbols $0$ and $1$. Every computer at the hardware level works in binary, because a transistor is happiest representing one of two states (on / off).
• Hexadecimal (base $16$) uses $0$–$9$ and then $A$, $B$, $C$, $D$, $E$, $F$ for ten through fifteen. It's the language of memory addresses, color codes (#D8763A is hex), and any place humans need to read binary without going cross-eyed.

Here's the same number, thirteen, written in different bases:

It's the same quantity in every row — what changes is the writing system. This is one of the most liberating ideas in arithmetic: the number is not the same thing as the way you write it. Practitioners who only ever saw base $10$ tend to confuse the two; once you've seen the same number in three or four bases the distinction becomes obvious.

8. Comparing and ordering

Once you have place value, comparing two whole numbers is mechanical. Walk through the columns from left to right:

1. If one number has more digits than the other (after stripping leading zeros), it's bigger. Done.
2. Otherwise, compare the leftmost digits. If they differ, whichever is larger wins.
3. If they're equal, move one column right and repeat.

For example, compare $30{,}471$ and $30{,}417$. Both have five digits. Left to right: $3=3$, $0=0$, $4=4$, then $7 > 1$ — so $30{,}471 > 30{,}417$. The remaining column doesn't matter, because the difference at the tens place ($70$ vs. $10$) is already larger than anything the ones column could undo.

The notation for ordering whole numbers is standard:

• $a < b$ — "$a$ is less than $b$"
• $a > b$ — "$a$ is greater than $b$"
• $a \leq b$ — "$a$ is less than or equal to $b$"
• $a \geq b$ — "$a$ is greater than or equal to $b$"
• $a = b$ — "$a$ equals $b$"

The strict inequalities point at the smaller number, like a mouth open toward what it wants to eat — a memory trick worth keeping if you ever flip them by accident.

9. Common pitfalls

10. Worked examples

Try each one yourself before opening the solution.