powers are used in digital data

8/13/2019 Powers Are Used in Digital Data

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EE487 Lesson 4:

Digital Data

There are 10 types of people in this world,

those who understand binary, and those who dont.


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A Simple Riddle You just lost a MAJOR bet, your options are:

A) Pay $1,000 per day for a month

OR

B) Pay 1 the first day, then double the paymenteach day for a month

Which do you choose?


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Topics

Number Bases

Decimal -> Base 10

Binary -> Base 2


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Positional number system1

The number system we learned in elementaryschool is an example of a positional number

system.

Numbers are composed of a weighted string ofdigits and the weightings are determined by thedigits position.

3 2 1 0 1 21492.63 1 10 4 10 9 10 2 10 6 10 3 10

1 1000 4 100 9 10 2 1 6 0.1 3 0.01

= + + + + +

= + + + + +

weighting of each digits position

1 Much of this discussion comes from John F. Wakerlys Digital Design: Principles and Practices, 2nded.


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Decimal numbers Each digit can take one of ten values.

The weighting of each digits position is a powerof 10.

Hence, decimal numbers are called base 10 orradix 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

3 2 1 0 1 2

1492.63 1 10 4 10 9 10 2 10 6 10 3 101 1000 4 100 9 10 2 1 6 0.1 3 0.01

= + + + + +

= + + + + +

weighting of each digits position


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Digital signal Digital signals are limited to a finite number of

discrete amplitudes.

In digital computers, the most basic unit storagecan take one of only two values, 1 or 0.


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Binary numbers Each binary digit (or bit) can take on one of two

values.

The weighting of each bit position in a binarynumber is a power of 2.

Binary numbers are called base 2 or radix 2.

Binary numbers are denoted by the subscript 2.

0 or 1

3 2 1 0 1 22

10

1101.01 1 2 1 2 0 2 1 2 0 2 1 2

1 8 1 4 0 2 1 1 0 0.5 1 0.25 13.25

= + + + + +

= + + + + + =


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Binary numbers The leftmost bit is called the most significant bit

(MSB) because it has the largest weighting.

The rightmost bit is called the least significant bit(LSB) because it has the smallest weighting.

10110111

most significant bit

(MSB)

least significant bit

(LSB)


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Binary to decimal conversion We can convert from a binary number to a

decimal as previously indicated.

It is helpful to be familiar with powers of 2.

5 4 3 2 1 02

10

100010 1 2 0 2 0 2 0 2 1 2 0 2

1 32 0 16 0 8 0 4 1 2 0 0 34

= + + + + +

= + + + + + =

0 1 2 3

4 5 6 7

8 9 10 11

2 1 2 2 2 4 2 82 16 2 32 2 64 2 128

2 256 2 512 2 1024 2 2048

= = = =

= = = =

= = = =


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Convert the binary number 10111012 into decimal.

Example Problem 1


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Convert the binary number 10111012 into decimal.

Example Problem 1

1 0 1 1 1 0 164 32 16 8 4 2 1

64 + 16 + 8 + 4 + 1 = 93

sum the weight values correspondingto the 1s in the binary number


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Decimal to binary conversion To convert from decimal numbers to binary:

1. Repeatedly divide the quotients by two until the final quotient is zero.

2. The remainders read in reverse order yield the equivalent binarynumber.

Convert 20310 into binary.

Quotient Remainder203/ 2 101 1101/2 50 150/ 2 25 0

25/ 2 12 112 / 2 6 06/ 2 3 03/ 2 1 11/ 2 0 1

=

=

=

==

=

=

=

stop when quotient equals zero

110010112

Read binary equivalentin reverse order

first remainder becomesthe right-most digit


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Example Problem 2


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Example Problem 2

Quotient Remainder182 2 91 091 2 45 145 2 22 122 2 11 011 2 5 15 2 2 12 2 1 0

1

2 0 1

1 0 1 1 0 1 1 0


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Using binary addition, count in binary from 0 to 8.

Example Problem 4

0 0 0 0 0

0 0 0 1 1

0 0 1 0 2

0 0 1 1 3

0 1 0 0 4

0 1 0 1 5

0 1 1 0 60 1 1 1 7

1 0 0 0 8

8 4 2 1


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EE487 Lesson 4:

Digital Data

There are 10 types of people in this world,

those who understand binary, and those who dont.

WHICH TYPE ARE YOU???

powers are used in digital data

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