Number Bases

Mathematical background necessary to understand binary logic. We will study:
Decimal, base 10
Binary, base 2
Octal, base 8
Hexadecimal, base 16

You will probably find this base conversion tool helpful in this lesson.

Decimal:
You may have noticed that many things are focused on the number 10 or powers of 10. One of these things is our common number system. You probably know that:

	10,000	1,000	100	10	1
238 =	0	0	2*100+	3*10+	8
5179 =	0	5*1000+	1*100+	7*10+	9
20661 =	2*10000+	0*1000+	6*100+	6*10+	1

The decimal system is based upon the number 10. Each digit in a number symbolizes that digit multiplied by the base raised to the power of the digit's position. 238 means 2*10²+3*10¹+8*10⁰.
Note that when we assign numbers to position we begin with 0, and that a decimal number consists only of the following digits: 0,1,2,3,4,5,6,7,8,9

Binary:
The binary system is based upon the number 2. There are only two digits, 0 and 1

	64	32	16	8	4	2	1
4 =	0	0	0	0	1	0	0
19 =	0	0	1	0	0	1	1
113 =	1	1	1	0	0	0	1

1110001 means 1*2^6 + 1*2^5 + 1*2^4 + 0*2^3 + 0*2^2 + 0*2^1 + 1*2^0

Octal:
The octal system is based upon the number 8. There are eight digits, 0,1,2,3,4,5,6,7

	4096	512	64	8	1
238 =	0	0	3	5	6
5179 =	1	2	0	7	3
20661 =	5	0	2	6	5

50265 means 5*8^4 + 0*8^3 + 2*8^2 + 6*8^1 + 5*8^0

Hexadecimal:
The hexadecimal system is based upon the number 16. There are sixteen digits, 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

A=10
B=11
C=12
D=13
E=14
F=15

	4096	256	16	1
238 =	0	0	E	E
5179 =	1	4	3	B
20661 =	5	0	B	5

50B5 means 5*16^3 + 0*16^2 + 11*16^1 + 5*16^0