1. CS334: Number Systems Lab 1

2. General Instructions Quizzes/Participation/Absence 5%
Logisim/MIPS Programming 5%
CPU Design Project 10%
Website:
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3. Contents:
Review
Exercises
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4. 1-1 Introduction
The decimal number system uses 10 different digits (symbols), (0,1, 2, 3, 4, 5, 6 ,7 ,8 ,9). The binary number system uses only two digits, 0 and 1, which are represented by two different voltage levels within a computer’s electrical circuits. Any value can be represented in either number system as long as there is no limit on the number of digits we can use. In the case of the MIPS architecture, values in registers and in memory locations are limited to 32 bits. The range of values that can be stored in a register or a memory location is -2,147,483,648 to +2,147,483,647 (Assuming the use of the two’s complement number system).
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5. (2) binary (10) (8) decimal octal (16) hexadecimal
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6. 1-2Positional Notation
The polynomial representation of 2056 in the base ten number system is:
Therefore, 2056 in the base eight number system is equivalent to 1070 in the base ten number system
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7. 1-3Converting Binary Numbers to Decimal Numbers2 1 3 4 5 6 7 8 9 10 16 32 64
128
256
512
1024
Therefore, the binary number is equivalent to 26 in the decimal number system.
1-4 Detecting if a Binary Number is Odd or Even
. If the right most digit in a binary number is a one, then the number is odd. For example
is an even number, which is the value 28 in the decimal number system. The value is an odd number, specifically the value 9 in decimal.
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8. 3x4 =Sift by tow bit to left
Multiplication by Constants that are a Power of Two
Another important feature of the binary number system is that multiplication by two (2) may be accomplished by shifting the number left one bit. Multiplication by four (4) can be accomplished by shifting left two bits.
0011
0100 x
0000 +
0000
0011
0000
1100
3x4 =Sift by tow bit to left
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9. 1-6 The Double and Add Method
A quick and efficient method for converting binary numbers to decimal involves visually scanning the binary number from left to right, starting with the left most 1. As you visually scan to the right, double the value accumulated so far, and if the next digit to the right is a 1, add 1 to your accumulating sum.
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10. 1-7Converting Decimal Numbers to Binary Numbers
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11. 1-8 The Two’s Complement Number System
The major benefit of the two’s complement number system is that it simplifies the design of the hardware to perform addition and subtraction
In the two’s complement number system, all numbers that have a one in the most significant digit (MSD) are negative numbers.
two’s complement would be=
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12. 1-9The Two’s Complement Operation
When we take the two’s complement of a number, the result will be the negative of the value we started with. One method to find the two’s complement of any number is to complement all of the digits in the binary representation and then add one to this new binary value.
Complement every Bit
Add one
This is the value minus
1-10 A Shortcut for Finding the Two’s Complement of any Number
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13. 1-11 Sign Extension
For example, the value 6 represented as an 8-bit binary number is:
“ ”, and the value 6 as a 32-bit binary number is
“ ”.
The same rule applies for negative numbers. For example the value minus 6 represented as an 8-bit binary number is:
, and the value -6 as a 32-bit binary number is
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14. 1-12Binary Addition 1-13 Binary Subtraction Ins.Ebtesam AL-Etowi
Decimal
Sum
1-13 Binary Subtraction
Minuend is
Subtrahend is Take two’s complement and add
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15. 1- 14 Overflow Detection
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16. 1-15 Hexadecimal Numbers
symbols used as digits in hexadecimal numbers are (0,1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). A convention has been adopted to identify a hexadecimal number. The two characters “0x” always precede a hexadecimal number.
Decimal
Binary
Hexadecimal
0000
0x0 1
0001
0x1 2
0010
0x2 3
0011
0x3 4
0100
0x4 5
0101
0x5 6
0110
0x6 7
0111
0x7 8
1000
0x8 9
1001
0x9 10
1010
0xA 11
1011
0xB 12
1100
0xC 13
1101
0xD 14
1110
0xE 15
1111
0xF
For example, 0xA2F0 in hexadecimal corresponds to in binary.
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17. Exercises Convert the decimal number 32 to an 8-bit binary number
Using the double and add method convert to a decimal number.
Explain why the Least Significant digit of a binary number indicates if the number is odd or even
Convert the binary number to a hexadecimal number.
Convert the hexadecimal number 0x15 to a decimal number.
Convert the decimal number -32 to an 8-bit two’s complement binary number.
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 = =
Y = =
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18. Convert the base 5 number 204 to decimal
Assuming the use of the two’s complement number system find the equivalent decimal values for the following 8-bit binary numbers:
(a)
Convert the base 5 number 204 to decimal
You are given the following two numbers in two’s complement representation.
Perform the binary addition. Did signed overflow occur?
determined whether or not overflow occurred.
Convert the base 10 number 81 to a base 9 number
Sign extend the 2 digit hex number 0x88 to a 4 digit hex number. 0x .
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19. You are given the following two numbers in two’s complement representation.
Perform the binary subtraction. Did signed overflow occur? determined whether or not overflow occurred.
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20. H.W Ins.Ebtesam AL-Etowi 16 Bit Binary Hexadecimal Decimal
0xFF88
-128
0x0011
-25
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