Hundredths
Fractions & DecimalsHundredths is the second decimal place value, representing one part out of one hundred equal parts.
Formula
1 \text{ hundredth} = \frac{1}{100} = 0.01
Definition
Hundredths is the second place after the decimal point. One hundredth ($0.01$) means $1$ out of $100$ equal pieces. Money uses hundredths: one cent is one hundredth of a dollar.
Example
$0.47$ means $47$ hundredths, or $47/100$. Forty-seven cents is $47$ hundredths of a dollar ($\$0.47$).
Key Insight
Hundredths and percents are the same thing! $47$ hundredths ($0.47$) is the same as $47$ percent ($47\%$). The "per cent" literally means "per hundred."
Definition
The hundredths place is the second position to the right of the decimal point, representing the digit multiplied by $10^{-2} = 0.01$. In financial calculations, amounts are typically rounded to the nearest hundredth (the nearest cent). A percent is defined as a ratio per $100$, making hundredths the natural decimal representation of percentages.
Example
$0.63 = 6/10 + 3/100 = 60/100 + 3/100 = 63/100 = 63\%$. To round $3.478$ to the nearest hundredth: look at the thousandths digit ($8 \ge 5$), so round up: $3.48$.
Key Insight
The connection between hundredths and percent is not a coincidence - percent means "per hundred," and the second decimal place is literally the "per-hundred" position. This makes converting between decimals and percents as simple as shifting the decimal point two places.
Definition
The hundredths digit of $x$ is $\lfloor 100x \rfloor \bmod 10$. Rounding to the nearest hundredth applies the rounding function $\text{round}(100x)/100$. In floating-point arithmetic, numbers are not stored with infinite precision, so "hundredths" in a computer are binary approximations - $\$0.01$ in binary is $0.00000010100011110101110\ldots$ (repeating), causing accumulation of rounding errors in financial software.
Example
In IEEE 754 double precision, $0.1 + 0.2 = 0.30000000000000004$, not $0.3$. This is because $0.1$ and $0.2$ have non-terminating binary expansions. Financial software uses decimal arithmetic libraries or integer arithmetic (storing cents as integers) to avoid this.
Key Insight
The imprecision of binary floating-point at the hundredths place is a practical reason why financial systems avoid floating-point. The "Patriot missile bug" (1991) was caused by floating-point time accumulation error at the millisecond level - a real-world consequence of decimal/binary place-value mismatch.