Number Systems

Number Systems.

Cliff Strahan, 20 February 2001

1. ref Floyd: Chapter 2, p38 - p71.

2. Historically, many number systems have been used. These include systems with a base of 12, 16, 20 and 60. There are remnants of these systems in our languages today.

the words eleven and twelve in English.

the words for the numbers from 11 to 16 in French.

our use of dozens, 12 inches in a foot.

60 seconds in a minute.

3. A variety of methods were used to write numbers down. The best known is the "Roman" system: I, II, III, IV, V, VI - - -

4. The Arabs developed three special concepts:

A "place value" notation, where a number's value changes with its position.

The number "zero", with a specific place in the number line between 1 and -1.

Negative numbers.

5. In our society, we use the "decimal" system.

There are 10 different symbols used.

Each symbol has a basic value in the units column.

Each symbol's value is increased by a factor of 10 for each position to the left.

Each symbol's value is decreased by a factor of 10 for each position to the right.

6. Because of the on/off nature of electronic switching, computers use the binary system of numbering.

There are 2 different symbols used.

Each symbol has a basic value in the units column.

Each symbol's value is increased by a factor of 2 for each position to the left.

Each symbol's value is decreased by a factor of 2 for each position to the right.

7. The human brain is poor at recognising and remembering binary numbers. The display of the binary information in computers is done by converting them into another system: decimal, octal or hexadecimal.

8. The hexadecimal (Hex.) number system is another place value system:

There are 16 different symbols used.

Each symbol has a basic value in the units column.

Each symbol's value is increased by a factor of 16 for each position to the left.

Each symbol's value is decreased by a factor of 16 for each position to the right.

9. To effectively use these systems, you must be able to convert between them:

decimal to binary.

binary to decimal.

Hex. to binary.

binary to Hex.

Hex to decimal.

decimal to Hex.

10. To be able to check on electronic operations, you must be able to do basic arithmetic in binary and Hex.