1 Week 3: Data Representation: Negative Numbers READING: Chapter 3.

Data representation

2

How do we represent data in a computer system?

•Numbers • Negative numbers

•Text

•Audio

•Images and graphics

•Video

EECS 1520 -- Computer Use: Fundamentals

Number systems

3

•Sign-Magnitude Representation in base 10:Examples : + 5 - 6

Sign indicates positive Magnitude is 5 Sign indicates negative Magnitude is 6

• In Binary Systems, we can use the most significant bit to represent the “sign”

Examples: 00101 is +5 10101 is -5

• The problem is: 00000 is + 0 10000 is - 0

• The two representations of zero within a computer can cause unnecessary complexity.

• How do we represent negative numbers in a computer system?


Number systems

4

0 1 2 3

1

…... 49

2 3 49

-1-2-3-49-50

0

…...

98975150 …... …...99 Fixed-size

Sign-magnitude

• Let’s go back to the example with base 10.

• If we allow only a fixed number of values, we can represent numbers as just integer values, where half of them represent negative numbers.

• To perform addition, we just add the numbers together and discard any carry.


Number systems

5

5- 2+

598+

103 3Signed-magnitude Fixed-size

- 46+

2

966+

102Signed-magnitude Fixed-size

• Examples:

0 1 2 3

1

…... 49

2 3 49

-1-2-3-49-50

0

…...

98975150 …... …...99 Fixed-size

Sign-magnitude

Note: “1” is the carrier and is discarded


Number systems

6

• So far we used the number line to calculate the negative representation of a number.

0 1 2 3

1

…... 49

2 3 49

-1-2-3-49-50

0

…...

98975150 …... …...99 Fixed-size

Sign-magnitude

• In general, a formula that computes the negative representation is:

IBI k B = Base; I = integer; k = number of digits

982102 2

• From the above number line, to calculate the negative representation for -2:

• Example:


Number systems

7

• Apply the same formula: 964104 2 955105 2 946106 2

50501050 2

…... 937107 2

• How about “-4”, “-5”, “-6” … “-50”?

0 1 2 3

1

…... 49

2 3 49

-1-2-3-49-50

0

…...

98975150 …... …...99 Fixed-size

Sign-magnitude


Number systems

8

0 1 2 3

1

…... 49

2 3 49

-1-2-3-49-50

0

…...

98975150 …... …...99 Fixed-size

Sign-magnitude

• This representation of negative numbers is called the Ten’s complement (i.e. the base is ten)

II k 10


Number systems: Two’s Complement

9

• From the formula: II k 2

• To represent negative numbers using binary numbers, we use “Two’s Complement” (i.e. base is 2)

• Example: how do we represent “- 5 ” using 4 digits (or bits) with “Two’s Complement”

11525 4

• (11)10 in binary number is: 1011

• So - 5 in base 10 is represented as 1011 using 4-bit binary numbers



10

15121 4

• Example: what about “-1” using 4 digits (or bits) with “Two’s Complement”

• (15)10 in binary number is: 1111

• So - 1 in base 10 is represented as 1111 using 4-bit binary numbers

• These are called “Signed binary numbers using two’s complement”

• If the leftmost bit is “0”, the number is positive• If the leftmost bit is “1”, the number is negative



11

• How do we find the two’s complement binary numbers of a negative integer?

• Method: 1) Start with the positive version of the number

2) invert all the bits (i.e. 1 0, and 0 1)

3) Add 1

• Back to the first example: what is the two’s complement binary number of “-5” in 4-bit?

Step 1) look for the 4- bit binary number for +5, which is 0101

Step 2) invert all the bits, so 0101 now becomes 1010

Step 3) Add 1, so 1010 becomes 1011



12

Natural number 4-bit binary 2’s complement

0 0000 0

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

8 1000 -8

9 1001 -7

10 1010 -6

11 1011 -5

12 1100 -4

13 1101 -3

14 1110 -2

15 1111 -1



13

• Examples of addition and subtraction

Calculate the following in 4-bit two’s complement representation.

-73+

Step 1: convert -7 and 3 to binary numbers.Answer:

7 in 4 bits representation is: 0111-7 in 4 bits two’s complement representation is: 1001

10010011+

Step 2:

1100



14

• Examples of addition and subtraction

Calculate the following in 5-bit two’s complement representation.

- 45-



15

- 4-5+

Step 1: convert -4 and -5 to binary numbers.

Answer:

4 in 5-bit representation is: 00100 5 in 5-bit representation is: 00101

1110011011+

Step 2:

1 10111

11100

11011

Invert and add 1

Invert and add 1

Invert and add 101001 = (9)10


16

Sample Test/Exam Question:


Convert -173 to 12-bit two’s complement representation. Show all your work

Step 1: convert 173 to binary by repeated division by 2.Answer:173/2 86 186/2 43 043/2 21 121/2 10 110/2 5 05/2 2 12/2 1 01/2 0 1

Step 2: expand answer in Step 1 to 12-bits.

10101101

000010101101

Step 3: invert the bits and add one 111101010010+ 1111101010011-173 in 12-bit is: 111101010011


17

Sample Test/Exam Question:


Convert the 8-bit two’s complement under 11001100 to decimal.

Answer: -52


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Unsigned VS Signed Two’s Complement


• In Ch. 2, we learned how to perform subtraction between 2 unsigned binary numbers

• Example: Compute the following by 2 methods:1) using 5-bit unsigned binary numbers2) using 5-bit two’s complement

13- 6

• Both methods will give the same answer!

19

Analog VS Digital• analog data: information represented in a continuous form• digital data: information represented in a discrete form

• Advantages of digital data:1. A binary digit is 0 or 1, which can be represented by high and low state in an electronic signal

2. Electronic signals fluctuate – digital signal is far more resistant to information loss because of distance between the two states

x(t)

tContinuous form Discrete form

x(n)

n

.... . . . ... . . . . . ..

. . . . . . .

5V

0V

1 0 1


1 Week 3: Data Representation: Negative Numbers READING: Chapter 3.

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