1. Place value past the point
You've already met the place-value rule for whole numbers: each position is worth ten times the one to its right. The decimal system simply continues that rule past the units column.
To the right of the decimal point, each place is worth one tenth the one before it:
So $307.425$ means $3 \cdot 100 + 0 \cdot 10 + 7 \cdot 1 + 4 \cdot \tfrac{1}{10} + 2 \cdot \tfrac{1}{100} + 5 \cdot \tfrac{1}{1000}$. The decimal point is just a marker for where the ones place is; nothing about the underlying place-value system actually changes when you cross it.
$3.5$, $3.50$, $3.5000$ are all the same number — the trailing zeros add nothing. But trailing zeros do communicate something: they're often used to signal precision. "The bridge is $3.50$ km long" suggests a measurement to the nearest centimetre; "the bridge is $3.5$ km" suggests to the nearest hundred metres.
2. From fractions to decimals
Every fraction can be expressed as a decimal: just perform the division. $\tfrac{3}{4}$ means "$3$ divided by $4$," and long division gives $3 \div 4 = 0.75$.
Some fractions land on a clean finite decimal — these are terminating:
$$ \tfrac{1}{4} = 0.25, \qquad \tfrac{3}{8} = 0.375, \qquad \tfrac{7}{50} = 0.14. $$Others produce a decimal that never ends but settles into a repeating cycle — these are repeating:
$$ \tfrac{1}{3} = 0.333\ldots = 0.\overline{3}, \qquad \tfrac{1}{7} = 0.\overline{142857}, \qquad \tfrac{5}{12} = 0.41\overline{6}. $$The bar over the digits marks the part that repeats. Notice $\tfrac{1}{7}$'s cycle is six digits long — there's no upper limit to how long a repeating block can be, but it must always be shorter than the denominator.
3. Terminating versus repeating
The pattern of which fractions terminate is a small but lovely piece of number theory.
A fraction in lowest terms terminates as a decimal if and only if its denominator has no prime factors other than $2$ and $5$.
Why? Because a terminating decimal is just $\tfrac{\text{integer}}{10^n}$, and $10^n = 2^n \cdot 5^n$ — its only prime factors are $2$ and $5$. So a fraction $\tfrac{a}{b}$ in lowest terms can be rewritten with denominator $10^n$ only if $b$ already divides some $10^n$, which happens iff $b$ is built entirely from $2$s and $5$s.
Quick check:
- $\tfrac{1}{4}$: denominator $4 = 2^2$. Terminates ($= 0.25$).
- $\tfrac{1}{6}$: denominator $6 = 2 \cdot 3$. The $3$ ruins it: repeats ($= 0.1\overline{6}$).
- $\tfrac{17}{200}$: denominator $200 = 2^3 \cdot 5^2$. Terminates ($= 0.085$).
- $\tfrac{1}{7}$: denominator $7$. Repeats ($= 0.\overline{142857}$).
For repeating decimals, there's a deeper fact: the length of the repeating block always divides $\varphi(b)$ (Euler's totient of the denominator). For $\tfrac{1}{7}$, $\varphi(7) = 6$, and the cycle is exactly $6$ digits long. That's number theory you'll meet again later.
Yes, really. The repeating decimal $0.999\ldots$ equals $1$ exactly, not just "approximately." One quick argument: $\tfrac{1}{3} = 0.333\ldots$, so $3 \cdot \tfrac{1}{3} = 0.999\ldots$, but $3 \cdot \tfrac{1}{3} = 1$. So $0.999\ldots = 1$. This isn't a fluke; it's a consequence of how infinite decimals are defined as limits. Some numbers have two decimal representations.
4. From decimals to fractions
Going the other direction, two cases.
Terminating decimals
Read the decimal as "$\tfrac{\text{the digits}}{\text{the power of ten}}$" and reduce. $0.625$ has three digits after the point, so the denominator is $10^3 = 1000$:
$$ 0.625 = \tfrac{625}{1000} = \tfrac{5}{8}. $$(The reduction used $\gcd(625, 1000) = 125$.)
Repeating decimals
There's a clean algebra trick. Let $x$ be the repeating decimal, then multiply by a power of $10$ that lines the repeating part up with itself, and subtract.
Example: convert $0.\overline{27}$ to a fraction. Let $x = 0.272727\ldots$. The cycle is two digits long, so multiply by $100$: $100x = 27.272727\ldots$. Subtract:
$$ 100x - x = 27.272727\ldots - 0.272727\ldots = 27. $$So $99x = 27$, hence $x = \tfrac{27}{99} = \tfrac{3}{11}$. The decimal-cancellation trick works because the infinite tails of $100x$ and $x$ are identical, so they vanish when you subtract.
5. Adding and subtracting decimals
Same as for whole numbers, with one rule: line up the decimal points. Stacked so that ones sit under ones, tenths under tenths, and so on, every column is a single-digit operation as before, with the same carrying or borrowing.
$$ \begin{array}{r} 12.347 \\ +\;\;\,5.092 \\ \hline 17.439 \end{array} $$If one number has fewer decimal places, pad it with trailing zeros — they don't change the value, but they make column alignment obvious:
$$ 3.5 - 1.024 = 3.500 - 1.024 = 2.476. $$Lining up by visual edge instead of by decimal point is a classic source of errors. Always align by the point.
6. Multiplying decimals
Forget the decimal points temporarily and multiply as integers. Then count the total number of decimal places in the two original numbers, and put the decimal point that many places from the right of the result.
Example: $1.2 \times 0.35$.
- Multiply as integers: $12 \times 35 = 420$.
- Count decimal places: $1.2$ has $1$, and $0.35$ has $2$. Total: $3$.
- Put the point three places from the right of $420$: $0.420$, which is $0.42$.
Why does this work? Because $1.2 = \tfrac{12}{10}$ and $0.35 = \tfrac{35}{100}$. Multiplying:
$$ \frac{12}{10} \cdot \frac{35}{100} = \frac{12 \cdot 35}{10 \cdot 100} = \frac{420}{1000} = 0.420. $$The "count decimal places" trick is just the bookkeeping for how many factors of $10$ end up in the denominator.
7. Dividing decimals
Dividing a decimal by a whole number is straightforward: do the long division as usual, and place the decimal point in the quotient directly above the one in the dividend.
$$ 25.6 \div 4 = 6.4. $$Dividing by a decimal is two steps. First, shift the decimal point of the divisor right until it becomes a whole number; shift the dividend's point by the same number of places. Then divide.
Example: $1.44 \div 0.12$. Shift both points two places to the right: this becomes $144 \div 12 = 12$.
That move is legal because shifting both numerator and denominator by the same factor of $10$ is the equivalent-fractions rule:
$$ \frac{1.44}{0.12} = \frac{1.44 \cdot 100}{0.12 \cdot 100} = \frac{144}{12} = 12. $$So "shift both points the same way" is just "scale numerator and denominator by the same power of $10$."
$1.44 \div 0.12$ — round to $1.5 \div 0.1 = 15$. The exact answer ($12$) is close. If you slipped a decimal point and got $0.12$ instead, the estimate would tell you something's wrong by two orders of magnitude.
8. Common pitfalls
$12.5 + 3.45$ is not computed by stacking flush-right; you'd be adding tenths to hundredths. Align the decimal points: $12.50 + 3.45 = 15.95$. This is the most common slip when working without graph paper.
If $1.2 \times 0.35 = 0.42$ but you forget to count places, you might write $42$ or $4.2$ or $0.042$. The estimate $1 \times 0.4 = 0.4$ tells you the answer should be near $0.4$, and that's enough to catch a misplaced point.
When dividing $25.6 \div 4$, the decimal point in the answer goes directly above the one in the dividend. Skipping this step means you end up with $64$ instead of $6.4$ — a ten-fold error, easy to miss.
$\pi$ is an irrational number — its decimal expansion never repeats. $3.14$ is convenient, $3.14159$ is closer, but no decimal will ever be exact. Don't write $\pi = 3.14$; write $\pi \approx 3.14$ or use the symbol $\pi$ until the very end. This becomes important the moment errors start to accumulate.
9. Worked examples
Example 1 · Convert $\tfrac{7}{8}$ to a decimal
Long-divide $7 \div 8$:
- $8$ into $7$: $0$ times. Write $0.$ and put the point.
- Bring down: $7$ becomes $70$ (tenths). $8$ into $70$ is $8$ ($8 \cdot 8 = 64$). Remainder $6$.
- $8$ into $60$: $7$ ($7 \cdot 8 = 56$). Remainder $4$.
- $8$ into $40$: $5$ ($5 \cdot 8 = 40$). Remainder $0$. Stop.
Sanity check: $8 = 2^3$, which has only $2$ as a prime factor. By the rule, $\tfrac{7}{8}$ terminates ✓.
Example 2 · Convert $0.\overline{6}$ to a fraction
Let $x = 0.\overline{6} = 0.666\ldots$. The cycle length is $1$, so multiply by $10$:
$$ 10x = 6.666\ldots $$Subtract the original $x$:
$$ 10x - x = 6.666\ldots - 0.666\ldots = 6 \quad\Longrightarrow\quad 9x = 6 \quad\Longrightarrow\quad x = \tfrac{6}{9} = \tfrac{2}{3}. $$Answer: $\boxed{\tfrac{2}{3}}$. (Confirming $\tfrac{2}{3} = 0.666\ldots$ ✓.)
Example 3 · $12.347 + 5.92$
Pad the second number with a trailing zero so both have three decimal places: $5.92 \to 5.920$. Stack and add, aligning decimal points:
$$ \begin{array}{r} 12.347 \\ +\;\;\,5.920 \\ \hline 18.267 \end{array} $$Answer: $\boxed{18.267}$. Estimate check: $12 + 6 = 18$ ✓.
Example 4 · $1.2 \times 0.35$
Ignore the points and multiply as integers: $12 \times 35 = 420$.
Count decimal places: $1.2$ has $1$, $0.35$ has $2$, total $3$.
Place the point three from the right of $420$: $0.420 = 0.42$.
$$ 1.2 \times 0.35 = \boxed{0.42}. $$Estimate: $1 \times 0.4 = 0.4$, so $0.42$ is in the right neighbourhood ✓.
Example 5 · $1.44 \div 0.12$
Shift the divisor's point right by $2$ places to make it whole; shift the dividend's point by the same amount:
$$ \frac{1.44}{0.12} = \frac{144}{12} = 12. $$Answer: $\boxed{12}$. Estimate: $1.5 \div 0.1 = 15$ — same order of magnitude ✓.