Characteristic and Mantissa

Summary: §112–§113 of Chapter 6: the integer–fractional split of a base-10 logarithm. The integer part — the characteristic — equals one less than the digit count of the number. The fractional part — the mantissa — depends only on the digit pattern of the number, not on where the decimal point sits. Two numbers with the same mantissa share digits and differ only by a power of 10.

Sources: chapter6 (§112–§113)

Last updated: 2026-04-26

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The split

For $N>0$, write the common-logarithm

${\log }_{10}N=c+m,c\in \mathbb{Z},m\in [0,1).$

• $c$ is the characteristic.
• $m$ is the mantissa.

This decomposition is unique (source: chapter6, §112).

The characteristic counts digits

For $N\ge 1$ with $k$ digits in its integer part, $10^{k-1}\le N<10^k$, so $\log N\in [k-1,k)$ and hence $c=k-1$.

The characteristic of a number is one less than the number of digits which express the number.

(source: chapter6, §112)

Examples:

$N$	Digits	Characteristic
$1$	1	0
$78509$	5	4
$1,000,000$	7	6

Conversely, from a tabulated $\log N=7.5804631$, one reads off characteristic $7$, hence $N$ has 8 integer digits — without computing $N$ itself (source: chapter6, §112).

For $0<N<1$ the characteristic is negative.

The mantissa is invariant under $\times 10^k$

Multiplying $N$ by $10^k$ adds $k$ to $\log N$ — i.e. it changes the characteristic by $k$ and leaves the mantissa unchanged. So the mantissa encodes only the digit string of $N$, independent of decimal placement (source: chapter6, §113):

$\log N$	$N$
$4.9130187$	$81850$
$6.9130187$	$8,185,000$
$3.9130187$	$8185$
$0.9130187$	$8.185$

Same mantissa $0.9130187\Rightarrow$ same digit string $8185$, with the decimal point shifted according to the characteristic. From $\log N=2.7603429$, the mantissa $0.7603429$ gives the digit string $5758945$, and the characteristic 2 says the integer part has 3 digits, giving $N=575.8945$ (source: chapter6, §113).

Negative characteristics by convention

A logarithm like $\log 0.5758945=-1+0.7603429$ has integer part $-1$ and fractional part $0.7603429$. To preserve the digit-pattern reading of the mantissa, tables write this as

$\log 0.5758945=9.7603429-10$

— i.e. with characteristic $9$ “diminished by 10.” Likewise $-2$ is written as $8-10$, $-3$ as $7-10$, etc. (source: chapter6, §113). This convention keeps the mantissa in $[0,1)$ and the table lookup unchanged.

Why this is useful

The split is what makes a printed log table small enough to be portable. The mantissa table lists, for every digit pattern from 1 to (typically) 100,000, the seven-place mantissa. The characteristic is computed by inspection of the number itself. Together:

• For multiplication: add logs, separately combining characteristics and mantissas; carry from the mantissa to the characteristic on overflow.
• For finding a number from its log: read the mantissa table backward to recover the digit string; the characteristic positions the decimal point.

§113’s closing example: $2^{16777216}$

To find the digit count of the 25th term of the progression $2,4,16,256,\dots$ (each term the square of its predecessor):

1. The terms are $2,2^2,2^4,2^8,\dots ,2^{2^{n-1}}$. The 25th term is $2^{2^{24}}=2^{16777216}$.
2. $\log 2=0.301029995663981195$.
3. ${\log }_{10}(2^{16777216})=16777216\cdot 0.301029995663981195=5050445.25973367$.
4. Characteristic 5050445 ⇒ the number has 5,050,446 digits.
5. Mantissa $0.25973367$ ⇒ the leading digits are $181858\dots$; pushing to more decimal places of $\log 2$, Euler reports the eleven leading digits as $18185852986$ (source: chapter6, §113).

The actual 5,050,446-digit number is uncomputable in any direct sense at the time — but its digit count and leading digits drop out of one multiplication and a table lookup. This is the kind of computation Chapter 6’s machinery makes routine.

Related pages

• common-logarithm
• logarithm
• change-of-base
• geometric-mean-method-for-logarithms
• chapter-6-on-exponentials-and-logarithms