2s complement arithmetic
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This document discusses 2's complement arithmetic in digital electronics. It explains that subtracting one number from another is the same as making one number negative and adding them. It then demonstrates how to represent negative numbers in binary by taking the 2's complement of a number, which involves complementing all its digits and adding 1. Various examples are provided of adding positive and negative binary numbers by taking the 2's complement of negative terms before adding. The most significant bit is identified as the sign bit that determines if a number is positive or negative.
![“Odometer” Math Examples
3
+ 2
5
003
+ 002
005
6
+ (-3)
006
+ 997
3
1]003
(-5)
+ 2
995
+ 002
(-2)
+ (-3)
998
+ 997
(-3)
997
(-5)
1]995
It Works!
Disregard
Overflow
Disregard
Overflow
7](https://image.slidesharecdn.com/2scomplementarithmetic-131013132443-phpapp02/85/2s-complement-arithmetic-7-320.jpg)
![POS + NEG → POS Answer
Take the 2’s complement of the negative number and
use regular binary addition.
9
+ (-5)
4
→
←
00001001
+ 11111011
1]00000100
8th Bit = 0: Answer is Positive
Disregard 9th Bit
00000101
↓↓↓↓↓↓↓↓
11111010
+1
11111011
2’s
Complement
Process
17](https://image.slidesharecdn.com/2scomplementarithmetic-131013132443-phpapp02/85/2s-complement-arithmetic-17-320.jpg)
![NEG + NEG → NEG Answer
Take the 2’s complement of both negative numbers and
use regular binary addition.
(-9)
+ (-5)
-14
→
→
←
11110111
+ 11111011
1]11110010
2’s Complement
Numbers, See
Conversion Process
In Previous Slides
8th Bit = 1: Answer is Negative
Disregard 9th Bit
To Check:
Perform 2’s
Complement
On Answer
11110010
↓↓↓↓↓↓↓↓
00001101
+1
00001110
19](https://image.slidesharecdn.com/2scomplementarithmetic-131013132443-phpapp02/85/2s-complement-arithmetic-19-320.jpg)
2s complement arithmetic
- 1. 2’s Complement Arithmetic Digital Electronics
- 2. 2’s Complement Arithmetic This presentation will demonstrate • That subtracting one number from another is the same as making one number negative and adding. • How to create negative numbers in the binary number system. • The 2’s Complement Process. • How the 2’s complement process can be use to add (and subtract) binary numbers. 2
- 3. Negative Numbers? • Digital electronics requires frequent addition and subtraction of numbers. You know how to design an adder, but what about a subtract-er? • A subtract-er is not needed with the 2’s complement process. The 2’s complement process allows you to easily convert a positive number into its negative equivalent. • Since subtracting one number from another is the same as making one number negative and adding, the need for a subtract-er circuit has been eliminated. 3
- 4. How To Create A Negative Number • In digital electronics you cannot simply put a minus sign in front of a number to make it negative. • You must represent a negative number in a fixedlength binary number system. All signed arithmetic must be performed in a fixed-length number system. • A physical fixed-length device (usually memory) contains a fixed number of bits (usually 4-bits, 8bits, 16-bits) to hold the number. 4
- 5. 3-Digit Decimal Number System A bicycle odometer with only three digits is an example of a fixed-length decimal number system. The problem is that without a negative sign, you cannot tell a +998 from a -2 (also a 998). Did you ride forward for 998 miles or backward for 2 miles? Note: Car odometers do not work this way. 999 forward (+) 998 997 001 000 999 998 002 001 backward (-) 5
- 6. Negative Decimal How do we represent negative numbers in this 3digit decimal number system without using a sign? +499 499 +498 498 +497 497 Cut the number system in half. +001 001 Use 001 – 499 to indicate positive numbers. 000 000 -001 999 -002 998 -499 501 -500 500 Use 500 – 999 to indicate negative numbers. Notice that 000 is not positive or negative. pos(+) neg(-) 6
- 7. “Odometer” Math Examples 3 + 2 5 003 + 002 005 6 + (-3) 006 + 997 3 1]003 (-5) + 2 995 + 002 (-2) + (-3) 998 + 997 (-3) 997 (-5) 1]995 It Works! Disregard Overflow Disregard Overflow 7
- 8. Complex Problems • The previous examples demonstrate that this process works, but how do we easily convert a number into its negative equivalent? • In the examples, converting the negative numbers into the 3-digit decimal number system was fairly easy. To convert the (-3), you simply counted backward from 1000 (i.e., 999, 998, 997). • This process is not as easy for large numbers (e.g., -214 is 786). How did we determine this? • To convert a large negative number, you can use the 10’s Complement Process. 8
- 9. 10’s Complement Process The 10’s Complement process uses base-10 (decimal) numbers. Later, when we’re working with base-2 (binary) numbers, you will see that the 2’s Complement process works in the same way. First, complement all of the digits in a number. – A digit’s complement is the number you add to the digit to make it equal to the largest digit in the base (i.e., 9 for decimal). The complement of 0 is 9, 1 is 8, 2 is 7, etc. Second, add 1. – Without this step, our number system would have two zeroes (+0 & -0), which no number system has. 9
- 10. 10’s Complement Examples Example #1 -003 ↓↓↓ 996 +1 997 Complement Digits Add 1 Example #2 -214 ↓↓↓ 785 +1 786 Complement Digits Add 1 10
- 11. 8-Bit Binary Number System Apply what you have learned to the binary number systems. How do you represent negative numbers in this 8-bit binary system? Use 00000001 – 01111111 to indicate positive numbers. Use 10000000 – 11111111 to indicate negative numbers. Notice that 00000000 is not positive or negative. 01111111 +126 01111110 +125 01111101 +1 00000001 0 00000000 -1 11111111 -2 Cut the number system in half. +127 11111110 -127 10000001 -128 10000000 pos(+) neg(-) 11
- 12. Sign Bit • • • What did do you notice about the most significant bit of the binary numbers? The MSB is (0) for all positive numbers. +127 01111111 +126 01111110 +125 01111101 +1 00000001 The MSB is (1) for all negative numbers. 0 00000000 -1 11111111 • The MSB is called the sign bit. -2 11111110 • In a signed number system, this allows you to instantly determine whether a number is positive or negative. -127 10000001 -128 10000000 pos(+) neg(-) 12
- 13. 2’S Complement Process The steps in the 2’s Complement process are similar to the 10’s Complement process. However, you will now use the base two. First, complement all of the digits in a number. – A digit’s complement is the number you add to the digit to make it equal to the largest digit in the base (i.e., 1 for binary). In binary language, the complement of 0 is 1, and the complement of 1 is 0. Second, add 1. – Without this step, our number system would have two zeroes (+0 & -0), which no number system has. 13
- 14. 2’s Complement Examples Example #1 5 = 00000101 ↓↓↓↓↓↓↓↓ 11111010 +1 Complement Digits Add 1 -5 = 11111011 Example #2 -13 = 11110011 ↓↓↓↓↓↓↓↓ 00001100 +1 13 = 00001101 Complement Digits Add 1 14
- 15. Using The 2’s Compliment Process Use the 2’s complement process to add together the following numbers. POS + POS POS 9 ⇒+ 5 14 NEG + POS NEG ⇒ POS + NEG POS 9 ⇒ + (-5) 4 NEG + NEG NEG (-9) ⇒ + (-5) - 14 (-9) + 5 -4 15
- 16. POS + POS → POS Answer If no 2’s complement is needed, use regular binary addition. 9 + 5 14 → → 00001001 + 00000101 ← 00001110 16
- 17. POS + NEG → POS Answer Take the 2’s complement of the negative number and use regular binary addition. 9 + (-5) 4 → ← 00001001 + 11111011 1]00000100 8th Bit = 0: Answer is Positive Disregard 9th Bit 00000101 ↓↓↓↓↓↓↓↓ 11111010 +1 11111011 2’s Complement Process 17
- 18. POS + NEG → NEG Answer Take the 2’s complement of the negative number and use regular binary addition. (-9) + 5 -4 → ← 11110111 + 00000101 11111100 8th Bit = 1: Answer is Negative To Check: Perform 2’s Complement On Answer 11111100 ↓↓↓↓↓↓↓↓ 00000011 +1 00000100 00001001 ↓↓↓↓↓↓↓↓ 11110110 +1 11110111 2’s Complement Process 18
- 19. NEG + NEG → NEG Answer Take the 2’s complement of both negative numbers and use regular binary addition. (-9) + (-5) -14 → → ← 11110111 + 11111011 1]11110010 2’s Complement Numbers, See Conversion Process In Previous Slides 8th Bit = 1: Answer is Negative Disregard 9th Bit To Check: Perform 2’s Complement On Answer 11110010 ↓↓↓↓↓↓↓↓ 00001101 +1 00001110 19
Editor's Notes
- #2: 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #3: Introductory Slide / Overview of Presentation 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #4: This slide explains the need to negative number and why we don’t have subtract-er circuits. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #5: This slide explains why we need a way to represent negative numbers. It introduces the concept of a fixed-length number system. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #6: Before we examine negative binary numbers, let’s first look at something we understand very well: the decimal number system. This slide uses a fixed-length decimal number system to illustrate the limitation of not being able to use a minus sign. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #7: This slide explains how to represent negative numbers in a fix-length decimal number system. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #8: Here are four example of adding two numbers in the odometer number system (i.e., 3-digit decimal number system). 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #9: 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #10: This slide describes the 10’s complement conversion process. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #11: Examples of the 10’s Complement Process. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #12: Introduction to the 8-Bit Binary Number system and how negative numbers are represented. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #13: Explanation of the sign bit. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #14: This slide describes the 2’s complement conversion process. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #15: Examples of the 2’s Complement Process. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #16: This slide show that there are only four possible combinations for adding together two signed numbers. The next four slides demonstrate each of these examples. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #17: Addition of two Positive numbers. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #18: This example shows the addition of one positive and one negative numbers. Note that this is done in the same way as subtracting a positive number from a positive number. In this case, the answer is positive. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #19: This slide demonstrates the addition of one positive and one negative number. Again, this is is the same a subtracting a positive number from a positive number. In this case the answer happens to be negative. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
- #20: This slide demonstrates the addition of two negative numbers. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009