Zorich's "Mathematical Analysis I" Problem 25

[Andrei1]

Here is a problem from Zorich's "Mathematical analysis I", pg.69.

25. A number $$x$$ is represented on a computer as $$x=\pm q^p\sum_{n=1}^{k}\frac{\alpha_n}{q^n}$$, where $$p$$ is the order of $$x$$ and $$M=\sum_{n=1}^{k}\frac{\alpha_n}{q^n}$$ is the mantissa of the number $$x$$ $$\left(\frac{1}{q}\leqslant M<1\right).$$ Now a computer works only with a certain range of numbers: for $$q=2$$ usually $$|p|\leqslant 64$$, and $$k=35.$$ Evaluate this range in the decimal system.

I suspect this text has misprints: is it correct that $$n=1$$ under $$\sum$$ and why, or it should be $$n=0$$? By order I understand the unique $$p\in\mathbb{Z}$$ such that $$q^{p}\leqslant x<q^{p+1}.$$

 

[I like Serena]

Here is a problem from Zorich's "Mathematical analysis I", pg.69.

I suspect this text has misprints: is it correct that $$n=1$$ under $$\sum$$ and why, or it should be $$n=0$$? By order I understand the unique $$p\in\mathbb{Z}$$ such that $$q^{p}\leqslant x<q^{p+1}.$$

Hi Andrei,

No misprint - it should really be $$n=1$$, since a mantissa is always less than 1, so with $q=2$ the first term is either $\frac 0 2$ or $\frac 1 2$.

The order, as used here, would be the least power of $p$ that is greater than the number. That is:
$$q^{p-1}\le |x| < q^p$$

I don't know where those bounds for $p$ and $k$ are coming from, but $|p| \le 1023$ and $k=53$ is about the most common, belonging to a 64-bit IEEE floating point number (see wiki).