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M68K Assembly Language Programming
Bob Britton
Chapter 4
Number Systems
Number Systems• Introduction
• Polynomial Expansion
• Binary Numbers
• Hexadecimal Numbers
• Two’s Complement Number System
• Arithmetic & Overflow Detection
• American Standard Code for Information Interchange (ASCII)
Polynomial Expansion of a Decimal Number (Base 10)
496 = 4 x 10 + 9 x 10 + 6 x 10
Polynomial Expansion of a Binary Number (Base 2)Polynomial Expansion of a Binary Number (Base 2)
2 1 0
10
00101101 = 1 x 2 + 0 x 2 + 1 x 2 + 1 x 2 + 0 x 2 + 1x 2
5 4 3 2 1 0
2
A Faster Method ------ Double and AddA Faster Method ------ Double and Add
00101101 = 45 (1, 2, 5, 11, 22, 45)
Conversion of Decimal Numbers to Binary
Divide by 2 and extract the remainder
45 Remainder 1 0 1 1 0 1
22 1
11 0
5 1
2 1
1 0
0 1
Practice - Convert 25 to BinaryDivide by 2 and record the remainder
25 Remainder
12
To represent binary values in the positive and negative domains we use the
Two’s Complement Number SystemHere is the polynomial expansion of a two’s
complement 8-bit binary number N:
N = - d7x2 + d6x2 + d5x2 + d4x2 + d3x2 + d2x2 + d1x2 +d0x2
Notice the Minus sign
*** You need to memorize powers of 2 ***
7 56 4 3 2 1 0
The Two’s Complement Operation
When we take the two’s complement of a binary number, the result will be the negative of the value we started with.
For example, the binary value 00011010 is 26 in decimal.
To find the value negative 26 (-26) in binary we perform the two’s complement operation on 00011010.
Scan the binary number from right to left leaving all least significantzeros (0) and the first one (1) unchanged, and then complementing theremaining digits to the left: 11100110
The result is the value negative 26 (-26) in binary.
Binary Arithmetic & Overflow Detectionin the Two’s Complement Number System
Here is an addition example where we assume we are limited to 8 binary digits.
01000100 = 68
+ 00111100 = 6010000000 = -128 Overflow Occurred
Overflow0000
0010
0001
0011
0100
0101
0110
0111
10001001
1010
1011
1101
1110
1111
1
2
0
3
4
5
6
7-8
-7
-6
-5
-4
-3
-2
-1
1100
Binary Arithmeticin the Two’s Complement Number System
Here is a subtraction example where we assume we are limited to 8 binary digits. To subtract in binary we always add the two’s complement of the subtrahend.
01000100 = 01000100 68
-00111100 = +11000100 60
00001000 = 00001000 = 8
The Rule for Detection of Overflow
#################################################Adding numbers of opposite signs, overflow is impossible.
When adding numbers of the same sign, if the result is not the same as the operands then overflow occurred.
#################################################Here is an example:You are given the following two numbers in two’s complement representation.Perform the binary subtraction and indicate if there is signed overflow. ______Explain Why:
11101000-00010011
11101000 = -24+11101101 = -19 11010101 Correct
Result = -43
Sign Extension
The value – 43 as an 8-bit binary number is: 11010101The value – 43 as an 32-bit binary number is: 11111111111111111111111111010101In Hexadecimal – 43 appears as: 0xFFFFFFD5
###############################################
The value 68 as an 8-bit binary number is: 01000100 The value 68 as an 32-bit binary number is: 00000000000000000000000001000100In Hexadecimal 68 appears as: 0x00000044
The Hexadecimal Number SystemDecimal Hex
Binary0 0 00001 1 00012 2 00103 3 00114 4 01005 5 01016 6 01107 7 01118 8 10009 9 100110 A 101011 B 101112 C 110013 D 110114 E 111015 F 1111
Here is an example of how wecompactly represent binary numbers in hexadecimal:
| | | | | 001111001000111101111110
$ 3 C 8 F 7 E
Multiplication by constants that are a Power of 2 is accomplished much more efficiently using the Arithmetic Shift Left instruction*.
ASL.L #3, D0 * D0*8 --> D0bvs overflow
*See Appendix B for a description of the ASL instruction
Division by constants that are a Power of 2 is accomplished much more efficiently using the Arithmetic Shift Right instruction
ASR.L #1, D0 * D0/2 --> D0
*See Appendix B for a description of the ASR instruction
MIPS Assembly Language Programming
Bob Britton
Chapter 4 Exercises
Number Systems
Exercises4.1 Convert the decimal number 35 to an 8-bit binary number.
4.2 Convert the decimal number 32 to an 8-bit binary number.
4.3 Using the double and add method convert 00010101 to a decimal number.
4.4 Using the double and add method convert 00011001 to a decimal number.
4.5 Explain why the Least Significant digit of a binary number indicates if the number is odd or even.
4.6 Convert the binary number 00010101 to a hexadecimal number.
4.7 Convert the binary number 00011001 to a hexadecimal number.
4.8 Convert the hexadecimal number $15 to a decimal number.
4.9 Convert the hexadecimal number $19 to a decimal number.
4.10 Convert the decimal number -35 to an 8-bit two’s complement binary number.
00100011
00100000
21
25
LSD is a 1
$15
$19
21
25
11011101
Exercises4.11 Convert the decimal number -32 to an 8-bit two’s complement binary number.
4.12 Assuming the use of the two’s complement number system find the equivalent decimal values for the following 8-bit binary numbers:(a) 10000001(b) 11111111(c) 01010000(d) 11100000(e) 10000011
4.13 Convert the base 8 number 204 to decimal
4.14 Convert the base 7 number 204 to decimal
4.15 Convert the base 6 number 204 to decimal
4.16 Convert the base 5 number 204 to decimal
4.17 Convert the base 10 number 81 to a base 9 number.
11100000
-127 -1 80 -32-125
132
102
76
54
100
Exercises4.18 For each row of the table below convert the given 16 bit number to each of the other two bases, assuming the two’s complement number system is used.
16 Bit Binary Hexadecimal Decimal
1111111100111100
$FF88
-128
1111111111111010
$0011
-25 4.19 You are given the following two numbers in two’s complement representation.
Perform the binary addition and indicate if there is signed overflow. __Explain Why:
0110111000011010
10001000 Yes overflow occurred – Sign of the result is different from the operands
Exercises4.21 You are given the following two numbers in two’s complement representation.
Perform the binary subtraction and indicate if there is signed overflow. ______Explain Why:
4.22 Sign extend the 8 bit hex number $88 to a 16 bit number. $_________
4.23 The first subtract instruction is located at address $00001234. What are the two possible values for the contents of the PC after the branch instruction is executed? $_____________ $ ____________
loop: sub #8, D4
sub D4, D2
bne loop
11101000-00010011
11101000+ 11101101 11010101 No Overflow
FF88
00001234 00001240
Exercises
4.24 You are given the following two 8-bit binary numbers in the two’s complement number system. What values do they represent in decimal?
X = 10010100 = __________ Y = 00101100 = __________ 2 10 2 10
Perform the following arithmetic operations on X and Y. Show your answers as 8-bit binary numbers in the two’s complement number system. To subtract Y from X, find the two’s complement of Y and add it to X. Indicate if overflow occurs in performing any of these operations.
X+Y X-Y Y-X
10010100 10010100 0010110000101100
-108 44
+11010100 +0110110011000000 01101000 10011000
Exercise 4.25
The following code segment is stored in memory starting at memory location $0001234. What are the two possible values for the contents of the PC after the branch instruction has executed? $__________ $ ____________Add in line pseudocode to describe each instruction.
loop: move (A0), D0 *adda #4, A0 *and #1, D0 *beq loop *
0001234 0001240
D0 = Mem[A0]A0 = A0 + 4D0 = D0 & 1 “Extract LSD”if D0 is an even # go to loop
Example QuizYou are given the following two 8 bit binary numbers in the two’s
complement number system. What values do they represent in decimal?
X= 10010100 = 2
Y= 00101100 = 2
Perform the following arithmetic operations on X and Y. Show your answers as 8 bit binary numbers in the two’s complement number system. To subtract Y from X find the two’s complement of Y and add it to X. Indicate if overflow occurs in performing any of these operations.
X+Y X-Y Y-X
10010100 10010100 0010110000101100
Beginning of the PrintHex Function
######################################### Algorithmic Description in Pseudo Code:##########################################
.globl PrintHex
.databuffer: .asciiz " 0x00000000"
.textPrintHex:
la $a1, bufferaddi $a1, $a1, 10 # Set pointer to end of buffer
......... # Body of the Algorithm...jr $ra
