Chapter 2

Signed and Unsigned Numbers

Representation of integers base 2, 10, and 16, conversion between bases. Binary addition and subtraction

 �Signed and unsigned numbers�

• Humans think in base 10, but numbers may be represented in any base. For example, 123₁₀ = 1111011₂.
• Numbers are kept in computer hardware as a series of high and low electronic signals, and so they are considered base 2 numbers.
• A single digit of a binary number is thus the "atom" of computing, since all information is composed of binary digits or bits.
• Binary digit: Also called a bit. One of the two numbers in base 2 (0 or 1) that are the components of information.
• Just as base 10 numbers are called decimal numbers, base 2 numbers are called binary numbers.

�Binary Numbers(Review)�

• In assembly language it is important to remember that the actual hardware to be used only understands binary values 0 and 1.
• Binary values 0 and 1 are called binary numbers
• The numbering system that everyone learns in school is called decimal or base 10.
• The numbering system is called decimal because it has 10 digits, [0..9].
• Binary Numbers uses base 2 numbering system
• In binary, there are only two digits, 0 and 1.

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How does a computer represent numbers? (Review)

• Numbers are kept in computer hardware as a series of high and low electronic signals, and so they are considered base 2 numbers
• Computers use switches that can be either on (1) or off(0), and so computers use the binary numbering system (base 2 numbering system)
• A single digit of a binary number is thus the "atom" of computing, since all information is composed of binary digits or bits. 
• Binary digit: Also called a bit. One of the two numbers in base 2 (0 or 1) that are the components of information.

�How to convert binary to decimal?�

• decimal = d₀×2⁰ + d₁×2¹ + d₂×2² + ...
• Generalizing the point, in any number base, the value of i th digit d is

d×Baseⁱ

^Example:

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Example 1:

• What is 1110_two in base ten?
• Solution:

(1 * 2³) + (1 * 2²) + (1 * 2¹) + (0 * 2⁰)

(1 * 8) + (1 * 4) + (1 * 2) + (0 * 1)

8 + 4 * 2 + 0

14

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Example 2:

• What is 1011_two in base ten?
• Solution:

(1 * 2³) + (0 * 2²) + (1 * 2¹) + (1 * 2⁰)

(1 * 8) + (0 * 4) + (1 * 2) + (1 * 1)

8 + 0 * 2 + 1

11

• What is 111001_two in base ten?
• What is 100011_two in base ten?

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Example 3:

• What is 111001_two in base ten?
• Solution:

(1 * 2⁵) + (1 * 2⁴) + (1 * 2³) (0 * 2²) + (0 * 2¹) + (1 * 2⁰)

(1 * 32) + (1 * 16) + (1 * 8) + (0 * 4) + (0 * 2) + (1 * 1)

32 + 16 * 8 + 0 + 0 + 1

57

• What is 100011_two in base ten?

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Example 4:

• What is 100011_two in base ten?
• Solution:

(1 * 2⁵) + (0 * 2⁴) + (0 * 2³) (0 * 2²) + (1 * 2¹) + (1 * 2⁰)

(1 * 32) + (0 * 16) + (0 * 8) + (0 * 4) + (0 * 2) + (1 * 1)

32 + 0 * 0 + 0 + 2 + 1

25

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�Decimal to Binary Conversions�

• Conversion steps:
• Divide the number by 2.
• Get the integer quotient for the next iteration.
• Get the remainder for the binary digit.
• Repeat the steps until the quotient is equal to 0.

Example 1

• Convert 13₁₀ to binary:

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Writing the remainders from bottom to top, we have: 13₁₀ = 1101₂

Example 2

• Convert 147₁₀ to binary:

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Writing and reading the remainders from bottom to top, we have: 147₁₀ = 10010011₂

�Converting Fractions�

• Fractions in any base system can be approximated in any other base system using negative powers of a radix.
• Radix points separate the integer part of a number from its fractional part.
• In the decimal system, the radix point is called a decimal point. Binary fractions have a binary point

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�Conversion steps for fractions:�

• Multiply the fractional decimal number by 2.
• Integral part of resultant decimal number will be first digit of fraction binary number.
• Repeat step 1 using only fractional part of decimal number and then step 2.

Example 1

• Convert 0.34375₁₀ to binary with 4 bits to the right of the binary point.

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Reading from top to bottom, 0.3437510 = 0.0101₂ to four binary places.

�Hexadecimals�

• A Hexadecimal Number is based on the number 16
• There are 16 Hexadecimal digits. They are the same as the decimal digits up to 9, but then there are the letters A, B, C, D, E and F in place of the decimal numbers 10 to 15:�

Some Numbers to remember

Convert Binary to Hexadecimal

• Binary numbers are often expressed in hexadecimal to improve their readability.
• Example: Convert 110010011101₂ to hexadecimal.
• Solutions:

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• If there are too few bits, leading zeros can be added.

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�Signed and Unsigned Numbers�

• Computer programs calculate both positive and negative numbers, so we need a representation that distinguishes the positive from the negative
• Unsigned numbers stored only positive numbers but do not store negative numbers.
• Signed numbers use sign flag or can be distinguish between negative values and positive values.
• Number representation techniques like: Binary, Decimal and Hexadecimal number representation techniques that we have discussed above can represent numbers in both signed and unsigned ways

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signed binary and Unsigned binary

• Unsigned binary numbers do not have sign bit
• Signed binary numbers uses signed bit and magnitude
• The most obvious solution is to add a separate sign, which conveniently can be represented in a single bit; the name for this representation is sign and magnitude
• Two’s complement representation: leading 0s mean positive, and leading 1s mean negative.
• Signed integers is the set of positive and negative integers

Unsigned Numbers:

• Unsigned numbers do not have any sign as they only represent positive numbers
•  Represent decimal number 92₁₀ in unsigned binary number.

= (1x2⁶+0x2⁵+1x2⁴+1x2³+1x2²+0x2¹+0x2⁰)₁₀

= (1011100)₂ a 7-bit binary magnitude of the decimal number 92₁₀.

• You can also used division to get to the above above answer

Signed Numbers:

• Signed numbers contain sign flag ro distinguish positive and negative numbers.
• This technique contains both sign bit and magnitude of a number.
• Three types of representations for signed binary numbers
• Sign-Magnitude form
• 1’s complement form
• 2’s complement form

Sign-Magnitude form

• For n bit binary number, 1 bit is reserved for sign symbol.
• If the value of sign bit is 0, the given number is positive, else if the value of sign bit is 1, the given number is negative.
• Remaining (n-1) bits represent magnitude of the number

Sign-Magnitude form Conversions Example

• What are the decimal values of the following 8-bit sign-magnitude numbers?
• 10000011 = -3
• 00000101 = +5
• 11111111 = -127
• 01111111 = 127

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Sign-Magnitude form Conversions Example

• Represent the following in 8-bit sign-magnitude
• -15 = 10001111
• +7 = 00000111
• -1 = 1001

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�1’s complement form�

• Positive values in one's complement are the same as unsigned binary or sign-magnitude.
• To negate a one's complement value, we invert all bits. Like sign-magnitude, one's complement has two representations for zero (all zeros or all ones).

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Conversions Example

• What are the decimal values of the following 8-bit one's complement numbers?
• 00001010 = +10
• 10001010 = -(01110101) = -(1+4+16+32+64) = -117
• 11111111 = ?

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�2's Complement Form�

• Commonly used
• A positive integer in two's complement always has a 0 in the leftmost bit (sign bit) and is represented the same way as an unsigned binary integer.
• To negate a number, a process sometimes called “taking the two's complement”, we invert all the bits and add one.

Conversion Example

• Example 1
• +14₁₀ = 01110_{two's comp}

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• _{Example 2:}
• -14₁₀ = 10001 + 1
• = 10010_{two's comp}

• Convert the following 4-bit 2's comp values to decimal:
• 0111 = +(1 + 2 + 4) = +7
• 1000 = -(0111 + 1) = -(1000) = -8
• 0110 = +(2 + 4) = +6
• 1001 = -(0110 + 1) = -0111 = -(1 + 2 + 4) = -7
• 1110 = -(0001 + 1) = -0010 = -2

Readings

• Chapter 2.5 to 2.7