Foundations — Decimals — place value, reading and writing

Before you can read, write, or compare a decimal, you must own every symbol the parent note leans on. Below is every piece of notation, in build-order: each one uses only the ones above it.

1. A digit — the atom

Why the topic needs it: every decimal is built entirely out of digits placed into boxes. Without the idea "a digit is a count," none of the place talk makes sense.

2. The box — a place, and its "worth"

Here is the whole secret of our number system: the same digit means different amounts depending on which box it sits in.

Look at the figure. The digit $3$ appears three times, but:

• in the left box it means $3\times 100=300$,
• in the middle box it means $3\times 10=30$,
• in the right box it means $3\times 1=3$.

Why the topic needs it: the parent note's central formula — "each digit's value = digit × the place's worth" — is exactly this idea. See Place Value in Whole Numbers for the whole-number version this extends.

3. Base-10 — why every box is 10× its right neighbour

Watch the pattern grow to the left by multiplying by 10 each step: $\underset{\text{ones}}{\underbrace{1}}\stackrel{\times 10}{\to }\underset{\text{tens}}{\underbrace{10}}\stackrel{\times 10}{\to }\underset{\text{hundreds}}{\underbrace{100}}\stackrel{\times 10}{\to }\underset{\text{thousands}}{\underbrace{1000}}$

Why the topic needs it: decimals are not a new rule. They are this one rule ($÷10$ each step right) refusing to stop at the ones place.

4. The symbols $\times$ and $÷$ — grow and shrink

Why the topic needs it: the whole engine is "$\times 10$ going left, $÷10$ going right." You cannot follow the place pattern without these two.

5. The fraction bar $\frac{a}{b}$ — "worth less than one"

When we divide $1$ by $10$, the answer is smaller than one whole. We need a way to write "a piece of one."

The figure shows one bar cut into $10$ equal slices. One slice is $\frac{1}{10}$ — that is what "one step right of ones" is worth.

Why the topic needs it: every place to the right of the point is one of these fractions. Tenths $=\frac{1}{10}$, hundredths $=\frac{1}{100}$. Study Fractions and this section together.

6. The decimal point "$.$" — the fence

7. Decimal digits are named after their place

Now combine everything: boxes to the right of the fence, each $÷10$ of the one before, each worth a power-of-ten fraction.

Why the topic needs it: these names are the vocabulary for reading ("point two five six") and writing ("seven and five hundredths"). Comparing decimals also uses these places from the left — see Comparing and Ordering Decimals — and Rounding Decimals needs their names to know which place to round to.

8. The "$=$" sign and why $0.3=0.30$

This is why a trailing zero changes nothing: $0.3=\frac{3}{10}=\frac{30}{100}=0.30$ Cutting each of $3$ tenths into $10$ gives $30$ hundredths — the same shaded amount, just more (smaller) pieces. But a zero in the middle ($0.03$) moves the $3$ into a smaller box, so it is a different value. Never confuse "same value" with "same digits."

Prerequisite map

Equipment checklist

Test yourself — say the answer before revealing.

Connections

• Place Value in Whole Numbers — the box-and-worth idea, before the fence
• Base-10 Number System — the master ÷10 / ×10 rule
• Fractions — what every decimal place actually is
• Comparing and Ordering Decimals — uses these places from the left
• Rounding Decimals — needs the place names
• Parent: Decimals — where all this equipment is used