Slide 5.2: Numbering systems

Slide 5.1: Data representation
Slide 5.3: Binary numbers
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Numbering Systems

A couple of numbering systems are developed for the convenience of human readers. Below is the table showing the possible digits for the numbering systems used most commonly in computer literature.

System	Base	Possible Digits
Binary	2	0 1
Octal	8	0 1 2 3 4 5 6 7
Decimal	10	0 1 2 3 4 5 6 7 8 9
Hexadecimal	16	0 1 2 3 4 5 6 7 8 9 A B C D E F

Binary Numbers
They are base 2 numbers in which each binary digit is either a 0 or 1.

Unsigned binary integers: Starting with the least significant bit, each bit in an unsigned binary integer represents an increasing power of 2.

Bit Value ( `D_j` )	1	1	1	1	1	1	1	1
Position Value ( `D_j×2^j` )	128=2⁷	64=2⁶	32=2⁵	16=2⁴	8=2³	4=2²	2=2¹	1=2⁰
Bit Number ( `j` )	7	6	5	4	3	2	1	0

Translating unsigned binary integers to decimal: Weighted positional notation represents a convenient way to calculate the decimal value of an unsigned binary integer having n digits:
```
   dec = (D_n-1×2^n-1) + (D_n-2×2^n-2) + … +(D₁×2¹) + (D₀×2⁰)
```
Each digit D_j is multiplied by 2^j . For example, the decimal equivalent of the binary number (11010)₂ can be found from the multiplication of the digits by powers of 2:
(11010)₂ =
= ( )₁₀

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Slide 5.1: Data representation
Slide 5.3: Binary numbers
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“Sometimes it’s not enough to know what things mean,
sometimes you have to know what things don’t mean.”
― Bob Dylan