No answers given here - compare with your classmates!

Notes:
In order to familiarize students with these ßtrange" numeration systems, I like to begin each day of digital circuit instruction with counting practice. Students need to be fluent in these numeration systems by the time they are finished studying digital circuits!
One suggestion I give to students to help them see patterns in the count sequences is "pad" the numbers with leading zeroes so that all numbers have the same number of characters. For example, instead of writing "10" for the binary number two, write "00010". This way, the patterns of character cycling (especially binary, where each successively higher-valued bit has half the frequency of the one before it) become more evident to see.

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Notes:
Ask your students to describe what differences exist between manually adding binary numbers and manually adding decimal numbers, if any.

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No. The form of numeration used to represent numbers has no bearing on the outcome of mathematical operations.

Notes:
Although this may seem like a trivial question, I've met electronics technicians who actually believed that the form of numeration affected the outcome of certain mathematical operations. In particular, I met one fellow who believed the number p was fundamentally different in binary form than it was in decimal form: that a binary "pi" was not the same quantity as a decimal "pi". I challenged his belief by applying some Socratic irony:

Me: How do you use a hand calculator to determine the circumference of a circle, given its diameter? For example, a circle with a diameter of 5 feet has a circumference of . . .
Him: By multiplying the diameter times "pi". 5 feet times "pi" is a little over 15 feet.
Me: Does a calculator give you the correct answer?
Him: Of course it does.
Me: Does an electronic calculator use decimal numbers, internally, to do math?
Him: No, it uses binary numbers, because its circuitry is made up of logic gates . . . (long pause) . . . Oh, now I see! If the type of number system mattered in doing math, digital computers and calculators would arrive at different answers for arithmetic problems than we would doing the math by hand!

Of course, those familiar with computer programming and numerical analysis understand that digital computers can introduce ärtifacts" into computed results that are not mathematically correct. However, this is not due to their use of binary numeration so much as it is limited word-widths (leading to overflow conditions), algorithmic problems converting floating-point to integer and visa-versa, and such.

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10001010₂: One's complement = 01110101₂

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11010111₂: One's complement = 00101000₂

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11110011₂: One's complement = 00001100₂

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11111111₂: One's complement = 00000000₂

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11111₂: One's complement = 00000₂

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00000000₂: One's complement = 11111111₂

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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 öne'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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The 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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01000101₂ = 69₁₀

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01110000₂ = 112₁₀

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11000001₂ = -63₁₀

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10010111₂ = -105₁₀

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01010101₂ = 85₁₀

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10101010₂ = -86₁₀

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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.

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Largest (most positive): 01111111₂ = 127₁₀
Smallest (most negative): 10000000₂ = -128₁₀

Notes:
The most important concept in this question is that of range: what are the limits of the representable quantities, given a certain number of bits. Two's complement just makes the concept a bit more interesting.

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Notes:
Have your students do some of these problems on the board, in front of class for all to see. Ask students what happens to the left-most "carry" bit, if it exists in any of these problems. Ask them why we do what we do with that bit, when we would usually place it in our answer.

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Follow-up question: Why are some of these answers incorrect? Hint: perform the additions in decimal form rather than binary form, and then explain why those answers are not represented in the binary answers.

Notes:
This question introduces students to the phenomenon of overflow. This is a very important principle to understand, because real computer systems must deal with this condition properly, so as not to output incorrect answers!

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Check the sign bit of the answer, and compare it to the sign bits of the addend and augend.
Challenge question: under what condition(s) is overflow impossible? When can we add two binary numbers together and know with certainty that the answer will be correct?

Notes:
Later, this concept of overflow checking should be applied to a real circuit, with students designing logic gate arrays to detect the presence of overflow. First, though, they must learn to recognize its presence analytically.

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"Floating-point" numbers are the binary equivalent of scientific notation: certain bits are used to represent the mantissa, another collection of bits represents the exponent, and (usually) there is a single bit representing sign. Unfortunately, there are several different ßtandards" for representing floating-point numbers.

Notes:
Ask your students why computer systems would have need for floating point numbers. What's wrong with the standard forms of binary numbers that we've explored thus far?

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