number system

Question

The most common numeration system used in this era is base 10 system, however this is not the only base. Other bases widely used include base 2 and base 60. Discuss a creative way to introduce this idea of working with different bases including their decimals rather than saying one’s place, tens’ place, hundreds place, etc…

Answer 1

Well in the field of computer science, base 2 and base 16 are used a lot.

In the area of computer programming and software development, the programmer

may have to view data stored in the memory of the machine as binary (base 2)

or hexidecimal (base 16).

 

 The digits are always 0-k-1 where K is the base of the numbering system.

So for decimal (base 10) the numbering system we use all the time, the digits are 0-9.

 

In binary (base 2), the digits are zero (0) and (1)

 

In hexadecimal (base 16) the digits are 0-9, A=10, B=11, C=12, D=13, E=14, F=15

 

A decimal number is converted to binary and hexidecimal

by writing it as the sum of powers of 2 and 16 respectively.

 

Before an actual example, we

First need powers of 2 and powers of 16.

 

2^0 =1

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

2^5 = 32

2^6 = 64

2^7 = 128

 

16^0 = 1

16^1 = 16

16^2 = 256

16^3 = 4096

 

 

Now as an example, let's change 125 to binary and hexidecimal.

 

First dividing each remainder by the next power of 2:

125/128 = 0 with remainder 125

125/64  = 1 with remainder 61

61/32 = 1 with remainder 29

29/16 = 1 with remainder 13

13/8 = 1 with remainder 5

5/4 = 1 with remainder 1

1/2 = 0 with remainder 1

1/1 = 1 with remainder 0

 

So the binary representation of 125 is 01111101

 

Likewise, dividing by powers of 16:

125/4096 = 0 remainder 125

125/ 256 = 0 remainder 125

125/16 = 7 with remainder 13

13/1 = 13 with remainder 0

 

So the hexidecimal representation of 125 is 007D