Number Systems

Tom Kelliher, CS 220

Aug. 31, 2001

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Assignment

Read 1.3 of Mano.

Outline

1. Using voltages to represent information

2. Number systems: decimal, binary, hexadecimal.

3. Number ranges.

Coming Up

Base conversion and character representation.

Information Representation

Why are computers digital?

What number system do they use?

How do we represent 0 and 1? Voltage.

Logic levels for SN7400 5 V part (see class home page for data sheet):

1. VoH: 2.4 V min

2. VoL: 0.4 V max

3. ViH: 2.0 V min

4. ViL: 0.8 V max

Example: 7400 two-input NAND gate waveform. Note voltage vs. time.

Why is digital logic binary?

Number Systems

Positional, weighted number systems. Example of a non-weighted number system: Roman numerals. Example of non-positional non-weighted number system: unary.

Radix, digit set.

Decimal Numbers

1. Radix 10, digit set 0--9.

2. .

   Radix point.

3. General case:

   We can substitute any radix and digit set we want.

Binary Numbers

1. Radix 2, digit set: 0, 1.

2. Decimal values of , , and ?

   Binary point

3. Powers of two. K, M, and G.

Hexadecimal Numbers

(Octal is similar --- study on your own.)

1. Radix 16, digit set: 0--9, A--F.

2. Generally used as compact binary form.

   How? --- Powers of two.

3. C Constants: 0XFE23, 0x89ab. (0567).

4. Relationship to binary and decimal.

   

   .

Number Ranges

1. Range of values which can be represented.

   Meaning of unsigned.

2. Consider an n-bit number. How many values can be represented? What is the range of values for an n-bit unsigned number?

3. What is the range of values of an 8-bit unsigned number?

Exercises

Convert the following binary numbers to hexadecimal and decimal:

1. 11100111

2. 00111010

3. 11000111

4. 00011111

5. 11101110