Decimal Point
Fractions & DecimalsThe decimal point is the dot in a decimal number that separates the whole number part from the fractional part.
Definition
The decimal point is the small dot in a number like $3.14$ or $0.5$. Everything to the left of the dot is a whole number; everything to the right of the dot is less than one whole.
Example
In $12.75$, the decimal point separates $12$ (twelve whole things) from $.75$ (seventy-five hundredths, or $3/4$). The dot is the dividing line between "wholes" and "parts."
Key Insight
Moving the decimal point one place to the right multiplies by $10$; moving it one place left divides by $10$. The decimal point is not just a dot - it is the key to the entire positional value system.
Definition
The decimal point separates the integer part of a number from its fractional part in base-ten notation. Each position to the right of the point represents a successively smaller power of $10$: $10^{-1}$, $10^{-2}$, $10^{-3}$, etc. In scientific notation, the decimal point is always placed after the first nonzero digit.
Example
$2{,}450$ has an implied decimal point after the last zero: $2450. = 2450.0$. Moving the decimal left: $245.0$, $24.50$, $2.450$, $0.2450$ - each shift left divides by $10$. Scientific notation: $2.45 \times 10^3$.
Key Insight
Different countries use different decimal separators: the United States uses a period (.), while many European countries use a comma (,). Mathematical software and international standards must account for this ambiguity - a seemingly minor notational choice with real practical consequences.
Definition
The decimal point marks the boundary between non-negative and negative exponents of the base in a positional numeral system. In base $b$, the value of the digit $d$ at position $n$ (counting from the decimal point, positive to the left) is $d \cdot b^n$. The choice of base $10$ is conventional; binary (base $2$), hexadecimal (base $16$), and other bases use analogous "radix points."
Example
In base $2$, $1011.01 = 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 + 0 \cdot 2^{-1} + 1 \cdot 2^{-2} = 8+2+1+0.25 = 11.25$ in base $10$. The binary "radix point" plays the exact role of the decimal point in base $10$.
Key Insight
Floating-point representation in computers is essentially scientific notation in base $2$. The IEEE 754 standard specifies a radix point and a fixed number of binary digits, which is why computers cannot represent $0.1$ exactly in binary - causing the floating-point rounding errors familiar to every programmer.