Tenths
Fractions & DecimalsTenths is the decimal place value immediately after the decimal point, representing one part out of ten equal parts.
Formula
1 \text{ tenth} = \frac{1}{10} = 0.1
Definition
Tenths is the first place after the decimal point. One tenth ($0.1$) means $1$ out of $10$ equal pieces. If you break $1$ whole into $10$ equal parts, each part is one tenth.
Example
$0.3$ means $3$ tenths, or $3/10$. If you cut a candy bar into $10$ equal pieces and take $3$, you have $3$ tenths of the candy bar.
Key Insight
Tenths connect the world of fractions to the world of decimals. The fraction $1/10$ and the decimal $0.1$ are two ways to write the exact same amount.
Definition
The tenths place is the first position to the right of the decimal point, representing the digit multiplied by $10^{-1} = 0.1$. A number like $4.7$ has $4$ ones and $7$ tenths. Tenths are the decimal equivalent of fractions with denominator $10$.
Example
$3.6 = 3 + 6/10 = 3 + 3/5$. To add $2.4 + 1.9$: align decimal points, add tenths ($4+9=13$ tenths, carry $1$), giving $4.3$.
Key Insight
The metric system is built on tenths: a millimeter is a tenth of a centimeter, a centimeter is a tenth of a decimeter. Understanding tenths is understanding why metric units convert so easily - they are all decimal multiples.
Definition
The tenths digit of a real number $x$ is $\lfloor 10x \rfloor \bmod 10$, where $\lfloor \cdot \rfloor$ denotes the greatest-integer function. In the decimal expansion $x = \sum a_n \cdot 10^n$, it is the coefficient $a_{-1}$. The tenths place is the leading coefficient in the fractional part of $x$ when expressed in base $10$.
Example
$\lfloor 10\pi \rfloor \bmod 10 = \lfloor 31.4159\ldots \rfloor \bmod 10 = 31 \bmod 10 = 1$. So the tenths digit of $\pi$ is $1$, confirmed by $\pi = 3.14159\ldots$
Key Insight
Extracting the $n$th decimal digit of a transcendental number like $\pi$ is a deep problem. The BBP (Bailey-Borwein-Plouffe) formula allows computing the $n$th hexadecimal digit of $\pi$ without computing all preceding digits - a remarkable result with no known base-$10$ analogue.