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Binary Math
11 questions By Tony R. Kuphaldt
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Question 4 of 11
What is the one’s complement of a binary number? If you had to describe this principle to someone who just learned what binary numbers are, what would you say?
Determine the one’s complement for the following binary numbers:
- 100010102
- 110101112
- 111100112
- 111111112
- 111112
- 000000002
- 000002
Reveal answer- 100010102: One’s complement = 011101012
- 110101112: One’s complement = 001010002
- 111100112: One’s complement = 000011002
- 111111112: One’s complement = 000000002
- 111112: One’s complement = 000002
- 000000002: One’s complement = 111111112
- 000002: One’s complement = 111112
Follow-up question: is the one’s complement 111111112 identical to the one’s complement of 111112? How about the one’s complements of 000000002 and 000002? Explain.
Notes:The principle of a “one’s complement” is very, very simple. Don’t give your students any hints at all concerning the technique for finding a one’s complement. Rather, let them research it and present it to you on their own!
Be sure to discuss the follow-up question, concerning the one’s complement of different-width binary numbers. There is a very important lesson to be learned here!
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Question 5 of 11
Determine the two’s complement of the binary number 011001012. Explain how you did the conversion, step by step.
Next, determine the two’s complement representation of the quantity five for a digital system where all numbers are represented by four bits, and also for a digital system where all numbers are represented by eight bits (one byte). Identify the difference that “word length” (the number of bits allocated to represent quantities in a particular digital system) makes in determining the two’s complement of any number.
Reveal answerThe two’s complement of 01100101 is 10011011.
The two’s complement of five is 1011 in the four-bit system. It is 11111011 in the eight-bit system.
Notes:The point about word-length is extremely important. One cannot arrive at a definite two’s complement for any number unless the word length is first known!
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Question 6 of 11
In a computer system that represents all integer quantities using two’s complement form, the most significant bit has a negative place-weight. For an eight-bit system, the place weights are as follows:
Given this place-weighting, convert the following eight-bit two’s complement binary numbers into decimal form:
- 010001012 =
- 011100002 =
- 110000012 =
- 100101112 =
- 010101012 =
- 101010102 =
- 011001012 =
Reveal answer- 010001012 = 6910
- 011100002 = 11210
- 110000012 = −6310
- 100101112 = −10510
- 010101012 = 8510
- 101010102 = −8610
- 011001012 = 10110
Notes:Students accustomed to checking their conversions with calculators may find difficulty with these examples, given the negative place weight! Two’s complement notation may seem unusual at first, but it possesses decided advantages in binary arithmetic.
