1. The University of Adelaide, School of Computer Science
19 November 2018
Lecture 0
Number Representations
& Operations
Chapter 2 — Instructions: Language of the Computer

2. Objectives
Understand that everything in digital word is stored and manipulated in binary format
Master the skill to convert representations between various formats (e.g., binary format and hexadecimal format)
Apply the conversion of unsigned and signed integer numbers between decimal format and binary format
Perform the addition and subtraction between two numbers in hexadecimal format
Detect the overflow of addition and subtraction between two numbers
Chapter 2 — Instructions: Language of the Computer — 2

3. Coverage Textbook Chapter 2.4 Online sources
Chapter 2 — Instructions: Language of the Computer — 3

4. Computer: a binary world
Everything inside a computer is saved and manipulated at binary format, i.e., in the sequence of ones and zeros
Chapter 2 — Instructions: Language of the Computer — 4

5. Hexadecimal format
For the convenience of demonstration, binary format is typically displayed in hexadecimal format
Each binary digit is called a bit
A bit can be one of two possible values, i.e., 0 or 1
Eight consecutive bits are called a byte
Byte is the default unit of capacity in computer world
Each hexadecimal digit can be one of 16 possible values
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
“0x” is the typical prefix to indicate hexadecimal format
Chapter 2 — Instructions: Language of the Computer — 5

6. Conversion between binary and hexadecimal formats
Convert 4 consecutive binary bits to 1 hexadecimal digit H B 8 1 9 2 A 3 4 C 5 D 6 E 7 F
Chapter 2 — Instructions: Language of the Computer — 6

7. How to represent integers in a computer?
The University of Adelaide, School of Computer Science
19 November 2018
How to represent integers in a computer?
Covered in Chapter 2.4
Unsigned integer
No sign, i.e., 0 or positive
Signed integer
Can be negative, 0, or positive
No explicit ‘+’ or ‘-’ symbol to tell the sign of the number
Two’s complement
Incorporate the sign in the representation
Sign extension
Before we discuss the details of instruction set, we need to talk about two things.
How are numbers are represented in a computer?
How to store them and address them in the memory?
Chapter 2 — Instructions: Language of the Computer — 7
Chapter 2 — Instructions: Language of the Computer

8. Generic numbering system
N=cn-1…c3c2c1c0
N is an unsigned number represented by n digits
What is the value of N?
N=Cn-1…+C3+C2+C1+C0
The value of N is the sum of the values of all columns
Ci=ci×qi
q: is the base value
i: is the column index
Chapter 2 — Instructions: Language of the Computer — 8

9. Unsigned Binary Integers
The University of Adelaide, School of Computer Science
19 November 2018
Unsigned Binary Integers
Given an n-bit number
§2.4 Signed and Unsigned Numbers
Range: 0 to +2n – 1
Example
0000,0000,0000,0000,0000,0000,0000,10112 = 0 + … + 1×23 + 0×22 +1×21 +1×20 = 0 + … = 1110
Using 32 bits
0 to +4,294,967,295
The value of ith column is the value of the digit times its weight.
The value of the n-bit number is the sum of the values of all columns.
Chapter 2 — Instructions: Language of the Computer — 9
Chapter 2 — Instructions: Language of the Computer

10. Convert unsigned decimal integer to binary format
Example: 87
87 = =
8710 = 0101,01112
Chapter 2 — Instructions: Language of the Computer — 10

11. Two’s-Complement Signed Integers
The University of Adelaide, School of Computer Science
19 November 2018
Two’s-Complement Signed Integers
Given an n-bit number
Range: –2n – 1 to +2n – 1 – 1
Example
1111,1111,1111,1111,1111,1111,1111,11002 = –1× ×230 + … + 1×22 +0×21 +0×20 = –2,147,483, ,147,483,644 = –410
Using 32 bits
–2,147,483,648 to +2,147,483,647
Chapter 2 — Instructions: Language of the Computer — 11
Chapter 2 — Instructions: Language of the Computer

12. Two’s-Complement Signed Integers
The University of Adelaide, School of Computer Science
19 November 2018
Two’s-Complement Signed Integers
Most significant bit is the sign bit
1 for negative numbers
0 for non-negative numbers
2n – 1 can’t be represented
Non-negative numbers have the same unsigned and two’s-complement representation
Some specific numbers
0: 0000,0000,…,0000
–1: 1111,1111,…,1111
Most-negative: 1000,0000,…,0000
Most-positive: 0111,1111,…,1111
Chapter 2 — Instructions: Language of the Computer — 12
Chapter 2 — Instructions: Language of the Computer

13. The University of Adelaide, School of Computer Science
19 November 2018
Signed Negation
Complement and add 1
Complement means 1 → 0, 0 → 1
Complement and add 1 → counterpart with opposite sign
Two uses
Given a negative decimal number, find its binary representation
Given a negative binary number, find its positive counterpart
Chapter 2 — Instructions: Language of the Computer — 13
Chapter 2 — Instructions: Language of the Computer

14. The University of Adelaide, School of Computer Science
19 November 2018
Examples
Binary representation of -210
+2 = 0000,00102
–2 = 1111, = 1111,11102
Decimal value of 1111,10112
-(1111,10112)=0000, =0000,01012
1111,10112=-510
Chapter 2 — Instructions: Language of the Computer — 14
Chapter 2 — Instructions: Language of the Computer

15. Range of 2’s complement A demo Start with 0
Keep adding 1 to the n-bit number
Chapter 2 — Instructions: Language of the Computer — 15

16. Sign Extension of Signed Numbers
The University of Adelaide, School of Computer Science
19 November 2018
Sign Extension of Signed Numbers
Representing a number using more bits
Preserve the numeric value
In MIPS instruction set
addi: extend immediate value
lb, lh: extend loaded byte/halfword
beq, bne: extend the displacement
Replicate the sign bit to the left
Examples: 8-bit to 16-bit
+2: 0000,0010 => 0000,0000,0000,0010
–2: 1111,1110 => 1111,1111,1111,1110
Chapter 2 — Instructions: Language of the Computer — 16
Chapter 2 — Instructions: Language of the Computer

17. Extension of Unsigned Numbers
Simply zero extension
Add leading zeros
Examples: 8-bit to 16-bit
210: ,0010 => 0000,0000,0000,0010
25410: 1111,1110 => 0000,0000,1111,1110
Chapter 2 — Instructions: Language of the Computer — 17

18. Addition/subtraction in base 16
Typically both operands and result have the same number of digits
C=A+B
ci=
ai+bi+carryi-1 if ai+bi+carryi-1 < 16
ai+bi+carryi otherwise
C=A-B
ai-bi-borrowi-1 if ai-bi-borrowi-1 > 0
16+ai-bi-borrowi-1 otherwise
Chapter 2 — Instructions: Language of the Computer — 18

19. The University of Adelaide, School of Computer Science
19 November 2018
Overflow
The result of an arithmetic operation is beyond the representation range
An n-digit number has its range
How to decide if overflow occurs
Unsigned operations
Check if there is a carry (in addition) or borrow (in subtraction)
Signed operations
Check the signs of the operands and the results
Use examples
Chapter 2 — Instructions: Language of the Computer — 19
Chapter 2 — Instructions: Language of the Computer

20. Overflow Addition with signed operands C = A + B A B C + No overflow -
Don’t care
Chapter 2 — Instructions: Language of the Computer — 20

21. Overflow Subtraction with signed operands C = A - B A B C + Don’t care
No overflow -
Overflow
Chapter 2 — Instructions: Language of the Computer — 21