# 50 Mathematical Ideas You Really Need to Know

(Marcin) #1

number and the letter E stands for ‘exponential’. Sometimes we might want to
use bigger numbers still, for instance if we were talking about the number of
hydrogen atoms in the known universe. This has been estimated as about
1.7×10^77. Equally 1.7×10−77, with a negative power, is a very small number and
this too is easily handled using scientific notation. We couldn’t begin to think of
these numbers with the Roman symbols.

### Zeros and ones

While base 10 is common currency in everyday life, some applications require
other bases. The binary system which uses base 2 lies behind the power of the
modern computer. The beauty of binary is that any number can be expressed
using only the symbols 0 and 1. The tradeoff for this economy is that the
number expressions can be very long.

``````Powers of 2 Decimal
20 1
21 2
22 4
23 8
24 16
25 32
26 64
27 128
28 256
29 512
210 1024``````

How can we express 394 in binary notation? This time we are dealing with
powers of 2 and after some working out we can give the full expression as,
394 = 1 ×256+ 1 ×128+ 0 ×64+ 0 ×32+ 0 ×16+ 1 ×8+ 0 ×4+ 1 ×2+ 0 ×
so that reading off the zeros and ones, 394 in binary is 110001010.
As binary expressions can be very long, other bases frequently arise in
computing. These are the octal system (base 8) and the hexadecimal system