Decimal
Fractions & DecimalsA decimal is a number that uses a decimal point to show values that are parts of a whole, based on powers of ten.
Definition
A decimal is a number that has a dot (called a decimal point) in it. Numbers after the decimal point are smaller than $1$ - they are parts of a whole. The number system we use is based on tens, so decimals are just fractions written in base-ten shorthand.
Example
$0.5$ is the same as $1/2$ (five tenths). $0.25$ is the same as $1/4$ (twenty-five hundredths). $\$3.75$ means $3$ whole dollars and $75$ hundredths of a dollar.
Key Insight
Decimals are fractions in disguise. Instead of writing $3/10$, you can write $0.3$. Decimals are easier to add on a calculator and easier to compare - you just line up the decimal points.
Definition
A decimal is a base-ten positional representation of a real number. Each digit after the decimal point represents a power of one-tenth: the first digit is tenths ($10^{-1}$), the second is hundredths ($10^{-2}$), the third is thousandths ($10^{-3}$), and so on. Decimals are either terminating, repeating, or non-repeating/non-terminating.
Example
$3.147 = 3 + 1/10 + 4/100 + 7/1000 = 3 + 0.1 + 0.04 + 0.007$. The decimal $0.333\ldots = 1/3$ (repeating), while $3.14159\ldots = \pi$ (non-terminating, non-repeating - irrational).
Key Insight
A decimal is rational if and only if it terminates or repeats. This elegant fact connects the arithmetic of fractions to the base-ten representation system, and it provides a practical test for rationality.
Definition
A decimal expansion is the representation of a real number $x$ as $\sum_{n=-k}^{\infty} a_n \cdot 10^{-n}$, where each $a_n$ is in $\{0,1,\ldots,9\}$. For rational $p/q$ in lowest terms, the decimal terminates iff $q = 2^a \cdot 5^b$, and otherwise repeats with period equal to the multiplicative order of $10$ modulo $q/\gcd(q, 10^\infty)$. Non-repeating infinite decimals are irrational.
Example
$1/7 = 0.142857142857\ldots$ has period $6$, since $\text{ord}_7(10) = 6$ ($10^6 = 1000000 = 142857 \times 7 + 1$ is not right; correct: $10^6 - 1 = 999999 = 142857 \times 7$). The repeating block $142857$ is a cyclic number: its multiples are cyclic permutations of itself.
Key Insight
The decimal expansion is a group homomorphism from $\mathbb{Q}$ to the additive group of formal power series in $1/10$. The kernel consists of numbers with two decimal representations (e.g., $0.999\ldots = 1.000\ldots$). This non-uniqueness arises from the non-injectivity of the representation at dyadic-type points.