Fractional part of a real number: questions

algebra-precalculusfractional-part

I was reading this question Evaluate the following integral $ \int_1^{\infty} \frac{\lbrace x\rbrace-\frac{1}2}{x} dx$ and I have seen that the user have used the mantissa of a real number or the fractional part.

I know that the mantissa of $x\in\Bbb R$ is defined by $\text{mant}(x)=x-\lfloor x\rfloor$ and $0\leq \text{mant}(x)<1$. I add the different symbologies.

$$\text{mant}(x)=\begin{cases}
\{x\}& \\
\text{frac}(x) &\\
M(x)
\end{cases}
$$

  1. the mantissa of a real number is connected to the logarithms and in what way?
  2. If I have $x=-0.43$, $\text{mant}(x)= -0.43-\lfloor (-0.43)\rfloor=\color{red}{ -0.43-(-1)\cdot \lfloor(+0.43)\rfloor}$? i.e. can I bring out the minus sign from if $x\in\Bbb R$, $\lfloor (-x)\rfloor \iff -\lfloor x\rfloor$? Is there a specific property?
  3. What is the relevance of this mantissa function?

Best Answer

A short answer, there is much more to say about this topic.

$2$: You can bring a minus outside of a floor, but when you do this, the floor becomes a ceiling. In other words, $$ \lfloor -0.43\rfloor = -1 $$ since the floor function means to round down. On the other hand, $$ \lfloor -0.43\rfloor = -\lceil 0.43\rceil=-1. $$ Note that the fractional part of a negative number is sometimes defined in other ways (such as rounding towards zero).

$1$ and $3$: One big application of the mantissa is in how numbers are stored on a computer. There are various ways to do this (and tricks which I am omitting). But, a number is stored as a mantissa $m$ and an exponent $e$. From this pair, the corresponding number is $m2^e$ (on a real computer the representation is slightly different, but I'm just trying to go for the idea). This also shows where logarithms come into play, the exponent $e$ is a (rounded) logarithm of the desired number.

Example (using base $10$, instead of base $2$, which a computer would use): The number $1,578$ could be written as $$ 0.1578\cdot 10^4. $$ The $0.1578$ would be the mantissa and the $4$ is the exponent. Note that $4\leq\log_{10}1578<5$, a connection to logarithms.

Aside: On a real computer, the representation could be closer to $$ 1.578\cdot 10^3, $$ where only the mantissa $0.578$ would be stored (on a computer, there is only one leading digit since binary has only numbers, $0$ and $1$.

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