Dale & Lewis Chapter 3Data Representation

Data and computers• Everything inside a computer is stored as patterns of 0s

and 1s• Numbers, text, audio, video, images, graphics, etc.

− How do you convert these to 0s and 1s?− How do you store the digital information efficiently?

Representing numeric data

• Representing Natural numbers with a finite number of digits• General Property:

Number of digits Min Max

n 0 bn-1

• Example:− b=10, n=3 000 to 999− b=2, n=3 000 to 111 (equivalent to 0 to 7 in decimal)− b=8, n=3 000 to 777 (equivalent to 0 to 511 in

decimal)− b=16, n=3 000 to FFF (equivalent to 0 to 4,095 in decimal)

Representing negative numbers (integers)

• Basic definition− An integer is a number with no fractional part− Examples: +123 -67

• Integers (signed natural numbers)− Previously we looked at unsigned numbers, i.e. natural

numbers− Need a mechanism to represent both positive and negative

numbers− Two schemes: 1) sign-magnitude, 2) complementary

representaion

Sign-magnitude• In binary:

− Sign: left-most bit (0 = positive, 1 = negative)− Magnitude: remaining bits

• Example with 6-bit sign-magnitude representation+5 = 000101 -5 = 100101

• Ranges in binary

Unsigned Sign-magnitude

# of bits Min Max Min Max

1 0 12 0 3 -1 +13 0 7 -3 +34 0 15 -7 +75 0 31 -15 +156 0 63 -31 +31

n 0 2n-1 -(2n-1-1) +(2n-1-1)

Complementary representation

• Difficulties with Sign-magnitude− Two representations for zero

+0 = 000000 -0 = 100000− Arithmetic is awkward, especially subtraction

• Complementary representation− Positive numbers are represented by their corresponding

natural numbers− Negative numbers are represented as very large natural

numbers− Subtracting numbers reduces to performing addition

operations

negative numbers: negative(x) ≡ bn-x

Complementary representation

negative numbers: negative(x) ≡ bn-x• Example: let b = 10 and n = 2

Positive numbers Negative numbers

(+0) 00(+1) 01 (-1) 99(+2) 02 (-2) 98(+3) 03 (-3) 97 … …(+49) 49 (-49) 51

(-50) 50

Visualization-49 -50 +49

51 50 49

99 00 01

-1 0 +1

98 99 00 01 02

-2 -1 0 +1 +2

49

+49

50

-50

Examples of arithmetic in Ten’s complement

-35 plus +25 equals -10 65 plus 25 equals 90 Ten’s complement

+17 plus +25 equals +42 17 plus 25 equals 42 Ten’s complement

+20 plus -30 equals -10 20 plus 70 equals 90 Ten’s complement

-20 plus +30 equals +10 80 plus 30 equals 10 (110) Ten’s complement

Examples of arithmetic in Ten’s complement

• Subtraction reduces to addition because A – B = A + (-B)

+5 05 -4 96 + -6 + 94 + +6 + 06 -1 99 +2 02

-2 98 -5 95 95 + -4 + 96 - +3 - 03 + 97 -6 94 -8 92• Easy to convert between positive number and its negative

counterpart− This conversion can be made efficiently and

simplify logic circuitry (we will see how)

Two’s complement representation

• Negative numbers: negative(x) ≡ 2n-x• Used in computers because subtraction reduces to

addition simpler circuits• To form a negative number

− Start with the positive version of the number− Flip the bits: 0 1 and 1 0− Add 1 to the number produced in the previous step

• Same process for conversion back to positive numbers• The above steps are an indirect way of carrying our the

evaluation of 2n-x, or more conveniently [(2n-1)-x]+1

Examples of Two’s complement• +5 (0101) to -5 (1011)• -5 (1011) to +5 (0101)

• Evaluating 2n-x is 24-5 = 16-5 = 1110000

- 101• Or 1011• Evaluating [(2n-1)-x]+1 is

[(24-1)-5]+1 = 15-5+1 = 11 1111(this way you never borrow) - 101

1010 + 1

+2 = 0010+ +3 = +0011 +5 = 0101

-2 = 1110+ +3 = +0011 +1 = 10001 ↑ throw away

-2 = 1110 = 1110 +2 = 0010 = 0010- +3 = -0011 = +1101 - +3 = -0011 = +1101 -5 = = 11011 -1 = = 1111 ↑ throw away

Sample test/exam questions

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

• A: Step 1: Convert 173 to binary by repeated division by 2173 ÷ 2 86 1 1

86 ÷ 2 43 0 01 43 ÷ 2 21 1 101 21 ÷ 2 10 1 1101 10 ÷ 2 5 0 01101 5 ÷ 2 2 1 101101 2 ÷ 2 1 0 0101101 1 ÷ 2 0 1 10101101

Sample test/exam questions

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

• A: Step 1: Convert 173 to binary by repeated division by 2 10101101

• Step 2: Expand answer in Step 1 to 12-bits

000010101101• Step 3: Flip-the-bits and add one

111101010010 + 1 111101010011• Final answer: -173 = 111101010011

Sample test/exam questions

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

• Q: Convert the 12-bit two’s complement number 111101010011 to decimal.