Number Systems

What is a Positional Number System?

• A way to represent a number and its magnitude using the "position" a digit a placed.
• Base determines the power base.
• Decimal 10^p
• Binary 2^p
• Hexadecimal 16^p
• Octal 8^p
• Place starts with power 0
• Base determines available symbols. The number of symbols is the base. Avaiable symbols (digits) are typically represented by 0 through (base - 1). Past "9", we typically use letters starting with A.
• Decimal: 0-9
• Binary: 0-1
• Hexadecimal: 0-F
• Octal: 0-7

Decimal (Base 10)

The weight of each digit of a decimal number depends on its relative position within the number.

• (Ex.) Decoding of the baae-10 number 1234

• 1234:
• 1*1000 + 2*100 + 3*10 + 4*1
• Decimal consists of powers of 10
• 1000(10³), 100(10²), 10(10¹), 1(10⁰)
• For each ascending place, its magnitude grows 10x

Fractions: Descending place, its magnitude is divided by 10.

• 27.212 = 2(10) + 7(1) + 2(10^-1) + 1(10^-2) + 2(10^-3)

Binary (Base 2)

A digit in binary(0 or 1) is also called a bit. Only these two digits are used in binary.

Like a decimal number, the "weight" of each binary digit in a binary number depends on its relative position.

For each ascending place, magnitude grows 2x.

• (ex.) 1101
• 1*8 + 1*4 + 0*2 + 1*1
• Binary consists of powers of 2
• 1*(2³) + 1*(2²) + 0*(2²) + 1*(2⁰)
• For each ascending place, its magnitude grows 2x

Distinguishing Base 2

• 1101...
• Base 10: One thousand, one hundred, one
• 1(1000) + 1(100) + 0(10) + 1(1)
• Base 2: 13
• 1(8) + 1(4) + 0(2) + 1(1)
• Distinguish from base 10 by prefix or suffix
• 0b1101 = Binary 1101 (0b means binary) -- how you represent binary in C++ and assembly
• 1101₂ = Binary 1101 (subscript 2 means binary)

Fractions in Base 2

• Descending place, its magnitude is divided by 2.
• Basically how we do base 10 but instead of 10 its 2.
• (ex.) 10.110₂ = 2.75₁₀
• 1(2) + 0(1) + 1(2^-1) + 1(2^-2) + 0(2^-3) = 2.75

Why Binary?

• A binary system just requires two distinct states. These can be represented by: on (1) or off(0), true or false, yes or no, etc.
• The basic unit of information that a computer uses consists of 8 bits, which is called a "byte"
• 0b0101_0011 = 1 byte

Hexadecimal Digits

• The hexadecimal number system has a base of 16
• digits>9 are represented by a letter A-F
• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

Hexadecimal (Base 16)

• 19AC = 1*(16³) + 9*(16²) + A*(16¹) + C*(16⁰)
• For each place, its magnitude grows 16x

Distinguishing Base 16

• hex 16 = 1(16¹) + 6(16⁰) = 16 + 6 = 22
• 0x16 = Hex 16 (0x means hex)
• This is how you represent hex in C++ and assembly
• 16₁₆ = Hex 16 (superscript 16 means hex)

Fractions (Base 16)

• Descending place, its magnitude is divided by 16
• (ex.) 2E.1AA₁₆ = 46.10400...
• 2(16) + E(1) + 1(16^-1) + A(16^-2)
• = 2(16) + 14(1) + 1(16^-1) + 10(16^-2)
• = 46.1...

Number System Comparison

Reference

• Vacumn tubes first before transistors
• ENIAC was first true general all purpose machine
• Moore's Law: The number of transistors on a microchip roughly doubles about every two years, increasing computing power while lowering cost per transistor.
• Rock's Law: The cost of building semiconductor manufacturing plants doubles roughly every four years, making chip production increasingly expensive.