Bit Shifting, 2s Complement Intro to Assembly Language, Lecture notes of Design

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Binary Arithmetic: Bit Shifting, 2s Complement

Intro to Assembly Language

CS 64: Computer Organization and Design Logic Lecture # Winter 2019 Ziad Matni, Ph.D. Dept. of Computer Science, UCSB

Why do CPU programmers celebrate

Christmas and Halloween

on the same day?

Because Oct-31 = Dec-

Any Questions From Last Lecture?

5-Minute Pop Quiz!!!

YOU MUST SHOW YOUR WORK!!!

1. Calculate and give your answer in hexadecimal:

~(0x3E | 0xFC)

2. Convert from binary to decimal AND to hexadecimal. Use any technique(s) you like: a) 1001001 b) 10010010

Lecture Outline

• Bit shift operations
• Two’s complement
• Addition and subtraction in binary

Bit Shift Left

• Move all the bits N positions to the left
• What do you do the positions now empty?
  • You put in N number of 0s
• Example: Shift “1001” 2 positions to the left

1001 << 2 = 100100

• Why is this useful as a form of multiplication?

Bit Shift Right

• Move all the bits N positions to the right , subbing-in either N number of 0s or N 1s on the left
• Takes on two different forms
• Example: Shift “1001” 2 positions to the right 1001 >> 2 = either 0010 or 1110
• The information carried in the last 2 bits is lost.
• If Shift Left does multiplication, what does Shift Right do?
• It divides, but it truncates the result

Two Forms of Shift Right

• Subbing-in 0s makes sense
• What about subbing-in the leftmost bit with 1?
• It’s called “arithmetic” shift right:

1100 (arithmetic) >> 1 = 1110

• It’s used for twos-complement purposes
  • What?

Twos Complement Method

• This is how Twos Complement fixes this.
• Let’s write out -6(10) in 2s-Complement binary in 4 bits :

So, –6(10) = 1010(2) according to this rule

0110 1001 1010

First take the unsigned (abs) value (i.e. 6) and convert to binary: Then negate it (i.e. do a “NOT” function on it): Now add 1:

Let’s do it Backwards… By doing it THE SAME EXACT WAY!

• 2s-Complement to Decimal method is the same!
• Take 1010 from our previous example
• Negate it and it becomes 0101
• Now add 1 to it & it becomes 0110 , which is 6(10)

Another View of 2s Complement

NOTE : Opposite numbers show up as symmetrically opposite each other in the circle.

NOTE AGAIN : When we talk of 2s complement, we must also mention the number of bits involved

Ranges

• The range represented by number

of bits differs between positive and negative binary numbers

• Given N bits, the range represented is:

0 to

and

+2N^ – 1 for positive numbers –2N-1^ to +2N-1^ – 1 for 2’s Complement negative numbers

Addition in Binary

• Same mathematical principal applies

Q: What’s being assumed here??? A: That these are purely positive numbers

Theoretically , I can add any binary no. with N1 digits to any other binary no. with N2 digits. Practically , a CPU must have a defined no. of digits that it’s working with. WHY???

Exercises

Implementing an 8-bit adder:

• What is (0x52) + (0x4B)?
  • Ans: 0x9D, output carry bit = 0
• What is (0xCA) + (0x67)?
  • Ans: 0x31, output carry bit = 1