Tenths
Tenths 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, so if you break $1$ whole into $10$ equal parts, each part is one tenth. Formally, the tenths place is the first position to the right of the decimal point, representing a digit multiplied by $10^{-1} = 0.1$; a number like $4.7$ has $4$ ones and $7$ tenths, the decimal equivalent of a fraction with denominator $10$. The tenths digit of a real number $x$ is $\lfloor 10x \rfloor \bmod 10$, the coefficient $a_{-1}$ in the decimal expansion $x = \sum a_n \cdot 10^n$, the leading coefficient of the fractional part of $x$ in base $10$.
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 it. $3.6 = 3 + 6/10 = 3 + 3/5$, and to add $2.4 + 1.9$ you align the decimal points and add tenths ($4+9=13$ tenths, carry $1$), giving $4.3$. The tenths digit of $\pi$ can be found directly: $\lfloor 10\pi \rfloor \bmod 10 = \lfloor 31.4159\ldots \rfloor \bmod 10 = 31 \bmod 10 = 1$, confirmed by $\pi = 3.14159\ldots$
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, and the metric system is built on this idea, a millimeter is a tenth of a centimeter, a centimeter is a tenth of a decimeter, so metric units convert so easily because they are all decimal multiples. Extracting the $n$th decimal digit of a transcendental number like $\pi$ is in general 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.