Section B $-$ Floating point numbers in binary

In the last section you had a reminder about Standard Index form. In this seminar, we are going to use both Standard Index form and Binary at the same time. Life doesn't get better than this.

First, how does binary handle fractions?

Columns to the right of the decimal point halve each time:

$128,64,32,16,8,4,2,1,\frac{1}{2},\frac{1}{4},\frac{1}{8}$

So ${3\mathrm{.}5}_{10}$ is the same as ${11\mathrm{.}1}_2.$

What is $4\mathrm{.}25$ is in binary?

How about ${5\mathrm{.}75}_{10}?$

This is just a side$-$issue. You already know that $\frac{1}{3}$ is the same as $0\mathrm{.}33333333$ in decimal. It requires a recurring number. In binary, this happens with a tenth. One tenth is $0\mathrm{.}1$ in decimals. That's easy. In binary it is $0\mathrm{.}000110011001100110011001100110011001100110011$dots. etc.

Now, we are ready to see how a computer stores a floating$-$point number. Admittedly, this is fairly obscure knowledge. Most computer scientists have a vague idea of how floating$-$point numbers work, but many of them either forget the details or were never taught them. Personally, I like to know this stuff and some of you will, too. But don't panic if you are finding this hard $-$ just learn what you can.

Binary floating$-$point numbers can be written using scientific notation. Some examples:

$1\mathrm{.}00112^{101},1\mathrm{.}12^{11}$

A "single$-$precision" floating point number uses $32$ bits.

How many bytes is this?