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# Number Systems

Updated : July 12, 2019

#### Number systems

• Integers (Positive, Zero, Negative)
• Real Numbers (Positive, Zero, Negative)
• Complex Numbers (with Real Part and Imaginary Part)

• Base 10 : Decimal
• Base 2 : Binary
• Base 3 : Ternary
• Base 4
• Base 8 : Octal
• Base 60 (e.g. Minutes & Seconds for angles & time)
• Mixed Base (e.g. most old British measures)

#### Rationale

If we use our fingers to count, then Base Ten (Decimal) seems to be convenient.

If we use our fingers and toes to count, then Base Twenty may be relevant. In French and Pali, the number 80 is represented as Four 20’s (instead of Eight 10’s).

Some theorists claim the Euler’s constant (e = 2.71828…) is optimum for use as a Base in computers. Since “e” is not an integer, the two nearest integers are 2 (for Binary) and 3 (for Ternary). The early ALU (Arithmetic Logic Units) are designed using Base 2.

The early memory units use magnetic cores that are arranged in 2-D (with an equal number of rows and columns). Since 1000 is not a perfect square, the closest approximation is 1024 = 32 x 32.

Some early computers use powers of Two. For example, IBM 360 has a word length of 32 bits and a character length of 8 bits.

Some ancient civilizations use Base 60, since it has several divisors : 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.

The British used mixed Base. For example, 1 Sterling Pound equals 20 shillings, and 1 Shilling equals 12 pennies. In the current Metric system, 1 Sterling Pound equals 100 Pence.

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