Binary Number System (Page 3 of 3) in pdf

The converter also offers hexadecimal, octal, quintal, and ternary versions of numbers.

Recalling the meaning of the Greek prefixes, one sees that the bases for these other systems

are 16, 8, 5, and 3, respectively. The standard decimal number system undoubtedly arose

because man’s early counting was done with his ten fingers. Other bases have also

developed historically. In particular, the Mayans had a base twenty(vigesimal) system

suggesting they developed their numbering scheme based upon the number of fingers and

toes and the Sumerians had a base sixty system probably connected with the number 360

which is close to the 365+1/4 days in the year. Vestiges of this sexagesimal numbering

system are still present in the number of seconds and minutes in an hour and the number

of degrees in a full circle. An alien civilization with four fingers on each of two hands would

probably develop an octal system but one would think that intergalactic communications

by advanced civilizations (should they exist and become detectable) will likely be in binary

code consisting of a sequence of long and short pulses reminiscent of Morse code.

Subtraction in binary makes use of the subtraction table:

1-0=1, 1-1=0, 0-0=0,

and

0-1=1 borrow

1. Thus

1100-101=111

and 101001-1110=11011. Division in binary is a bit

more complicated and best treated as the inverse of multiplication. As an example consider

the division

6/3=2

or its equivalent multiplication

2 x 3=6.

In binary we would have

110/11=10

or the equivalent multiplication

10 x 11=110.

A division such as

2048/32

becomes

trivial in binary and is equal to

1000000

(or 64 in decimal notation). A decimal point may

also be used in binary to distinguish terms involving negative powers of two. So, for

example, the binary number

101.011

corresponds to

2^2+2^0+2^(-2)+2^(-3)

=4+1+1/4+1/8=5+3/8=5.375.

Here are two famous constants expressed in binary –

11.001001000011111101101010100010001....

10.101101111110000101010001011000101…..

I have generated them using the MAPLE one liner-

convert(evalf(constant,50),binary,50);

Recognize the constants used?

Binary Number System Page 3

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