Number Systems

A number system is defined as follows:

A radix or base to define the number of numerals in the system.
A set of symbols, called numerals, to represent numbers.
A weighting and reading positional convention, called "positional-value".
A symbol to separate whole numbers from fractions expressed as decimals. In the U.S. this separation is represented by a period (  .   ), called the "decimal point".

1. Decimal (base 10) number system:

Radix 10
Numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Example: 12345 = 10000 + 2000 + 300 + 40 + 5

Note that the 1 in the 5th position (from right to left) has a weight of 10,000 times that of the 5 in the 1st position, etc.

Example: 12345.9876 = 10000. + 2000. + 300. + 40. + 5. + .9000 + .0800 + .0070 + .0006

2. Octal (base 8) number system:

Radix: 8
Numerals: 0, 1, 2, 3, 4, 5, 6, 7
Example: 12345 (decimal) = 30071 (octal)

3. Binary (base 2) number system:

Radix: 2
Numerals: 0, 1
Example: 12345 (decimal) = 11000000111001 (binary)

4. Hexadecimal (base 16) system number system:

Radix: 16
Numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Example: 12345 (decimal) = 3039 (hexadecimal)

The position of numerals in any number system can be represented by powers of its base:

Using the decimal system as an example:

12345.9876 =

10000. + 2000. + 300. + 40. + 5. + .9000 + .0800 + .0070 + .0006 =

1. x 10⁴ + 2. x 10³ + 3. x 10² + 4. x 10¹ + 5. x 10⁰ + 9. x 10^-1 + 8. x 10^-2 + 7. x 10^-3 + 6. x 10^-4

The other positional-value number systems, binary, octal, hexadecimal, etc., would use their respective bases (2, 8, 16) raised to a power representing position instead of a 10.   The period separating the integers from the decimals would be called, respectively, "binary point", "octal point", and "hexadecimal point", instead of "decimal point".

There is a modulo (also, "modular") number system where the permitted integers are limited, in contrast to our conventional numbering with unlimited integers.   Examples would be the number of degrees in a circle.   After 359, the system generates a 0 for the next whole number.   Therefore, 359 + 2 = 1 when using this system to add.   A clock uses a modulo system for the minutes and seconds (0 to 60) and a near modulo number system for the hours (1 to 12), but it omits the zero.   Therefore, on a clock, 11 + 2 = 1, in addition.   The modulo of the conventional number system is infinity; that is, it runs from 0 on indefinitely.   The modulo number system and modulo arithmetic based on it find application in computer calculations and cryptography, where, in division, the remainder is more important than the quotient because it indicates position.   In geometry, stating that an angle on a circle is 2990° is not as important as knowing that it is 110° (2990 / 360 = 8 + 110 remainder).   A more complicated use of modular arithmetic is the determination of any day of the week for any year, past or present, based on the Doomsday Algorithm.   This application of modulo arithmetic can be found on the Internet.

Numbers and number systems are chosen for their utility.   Ancient number systems were many - Mayans, Etruscans, Babylonians, Sumerians, etc. - all had number systems.   Those based on base-60 (sexagesimal) and base-12 (duodecimal) were used around 1,800 BCE by the Sumerians and Babylonians, probably because they were easier to express fractions at a time when equivalent decimal numbers to complex fractions were unknown.   We carry some of these ancient systems today in time (12 hours, 60 minutes, 60 seconds on a clock), trigonometry (360 degrees in a circle), and measures (12 eggs in a dozen).   The decimal (base 10) system may be the oldest, since it is easiest to count with the fingers and thumbs of two hands.   The binary (base 2) system, first discussed by Gottfried Liebnitz, the great German mathematician, must be used for a computer, an electrical machine, that operates only on whether the electric current is ON and OFF (thus, symbolically, 1 and 0).   Octal (base 8) is convenient when working with computer bytes (8 bits = 1 byte) and hexadecimal, (base 16) is convenient for human memories when working with the computer binary system, which is hard to remember because of the numeral length.   It is no accident that the octal and hexadecimal systems are exact powers of 2 (3 and 4, respectively; 2³ = 8 and 2⁴ = 16).   This makes translation from one system to the other simple.   By contrast, translating from and to the decimal system from the other 3 systems is complicated because 10 is not an exact power of 2.   Numbers and number systems involve complicated psychological and philosophical questions:   Are number concepts inate to humans or are they learned?   Are they real or imaginary?   The answers to those questions are another story.

Number and number systems are extremely important to daily life.   They facilitate keeping track of how many things are available and trading objects of value among people.   They make written computations possible and promote advances in science, engineering, and computer science.   Modern computers are based on the base 2 number system, and associated computer software uses the base 16 number system.   Here they are presented as an introduction to logarithms, which are important in describing natural phenomena.