Decimal Fractions

Summary: Decimal fractions extend place value to the right of the point, and Euler uses them as the practical notation for tabular logarithms. (source: chapter-1.1.23)

Sources: chapter-1.1.23

Last updated: 2026-04-24

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Place Value

Digits to the left of the point increase by powers of $10$, while digits to the right decrease by powers of $10$. Thus in $36.54892$, the digit $5$ after the point means five tenths, the next digit means four hundredths, and so on. (source: chapter-1.1.23)

Use in Logarithms

Euler represents logarithms such as $\log 2$ and $\log 3$ by decimal approximations: $\log 2=0.3010300,\log 3=\mathrm{0.4771213.}$ (source: chapter-1.1.23)

Characteristic

The integer part of a logarithm is called the characteristic. Euler explains that it is one less than the number of digits of the corresponding number when that number exceeds $1$. (source: chapter-1.1.23)

Numbers that differ only by powers of $10$ have the same decimal part in their logarithms and differ only in characteristic. (source: chapter-1.1.23)

Related pages

• ch1.1.23-expressing-logarithms
• logarithms
• powers-and-exponents
• roots
• fractions