# Interesting negative decimal number notation

I was studying logarithms, and had to solve the problem:

If $\log 8 = 0.90$, find $\log 0.125$.

I found out the answer to be $-0.90$. That was easy. But my text book has given the answer as:

$$-0.90 = \bar{1}.10$$

Now what that bar above $1$ is for ? My understanding is that it is used to signify a repeating digit. But how can $-0.90 = \bar{1}.10$ ?

Can someone please explain this to me.

Thank you!

• Which text book is this? Looks like very strange notation to me...
– 5xum
Mar 20 '14 at 9:59
• @5xum This Mar 20 '14 at 10:01

This was a very common notation when decimal logarithms were involved. The starting point is that if $$M/N=10^k$$ for two positive numbers $M$ and $N$, then their decimal logarithms have the same mantissa (fractional part) and we can write $$\log_{10} M= k + \log_{10} N$$ In particular, any number $M>0$ can be uniquely written as $$M=10^k N$$ with integer $k$ and $1\le N<10$. So the logarithmic tables were compiled, say, for numbers from $1$ to $1000$, but only showed the mantissa; when the logarithm of, say, $10.4$ was needed, one looked at $104$ finding the mantissa to be $01703$ and so could conclude $\log_{10}10.4=1.01703$. If one knows the decimal logarithm of $2$ $$\log_{10}2=0.30103$$ (equality is meant as “approximate”), then $\log_{10}0.02=-2+0.30103$, which was written as $$\log_{10}0.02=\overline{2}.30103$$ to ease lookup in logarithmic tables, because, as already said, only mantissas were shown.

In your case, $0.125=1/8$, so $$\log_{10}0.125=-\log_{10}8 = -3\log_{10}2=-0.90309=-1+0.09691=\overline{1}.09691$$ (use the “complement to $9$” rule for writing the mantissa).

• where you say the equality is "approximate", you can actually use the $\LaTeX$ symbol. May I edit?(+1)
– Guy
Mar 20 '14 at 10:12
• @Sabyasachi I preferred the notation that was standard when doing those computations. Mar 20 '14 at 10:13
• Okay. Cheers.${}$
– Guy
Mar 20 '14 at 10:13
• @egreg Wonderful answer! Thank you! But I was wondering if at the last, it should be $-0.90309$ instead of $0.90309$ ?? Mar 20 '14 at 10:16
• @GaurangTandon just to be clear for others who may be reading this $0.9691=\log_{10} 1.25$ so the expression given by egreg is the logarithm of $1.25 \times 10^{-1}=0.125$. The notation was used because it allowed easier computation in the days before computers and calculators (and also made sense in the era of slide rules). Mar 20 '14 at 10:29