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### B.1. Binary Numbers

First let's consider what you mean when you write a common everyday number such as 321 or 747. Put more precisely you mean

321 is:

`3 × 10 × 10 + 2 × 10 + 1`

and 747 is:

`7 × 10 × 10 + 4 × 10 + 7`

Because it is built around powers of ten, you call this the decimal system (derived from the Latin decimalis meaning of tithes, which was a tax of 10 percent—ah, those were the days ...).

Representing numbers in this way is very handy for people with ten fingers and ten toes, or creatures with ten of any kind of appendage for that matter. However, your PC is quite unhandy in this context, being built mainly of switches that are either on or off. This is okay for counting up to two, but not spectacular at counting to ten. For this reason your computer represents numbers to base 2 rather than base 10. This is called the binary system of counting, analogous to the bicycle (two wheels), but nothing whatever to do with bibacity, which means an inclination to drink a lot. With the decimal system, to base 10, the digits used can be from 0 to 9. In the binary system, to base 2, the digits can only be 0 or 1, ideal when you have only on/off switches to represent them; off is usually 0, and on is 1, simple. Each digit in the binary system is called a bit, which is a contraction of binary digit. In an exact analogy to the usual base 10 counting, the binary number 1101 is therefore

`1 × 2 × 2 × 2 + 1 × 2 × 2 + 0 × 2 + 1`

which, if you work it out, amounts to 13 in the decimal system. ...

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