Checking a two's complement conversion

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Quick Questions

Shade in the A or B circle for each question.

1. Which check proves that 11111011 is the 8-bit two's complement of -5?

A 64+32+16+8+2+1 = 123, which matches

B -128+64+32+16+8+2+1 = -5, which matches

2. You convert 5 to 8-bit two's complement by flipping 00000101. You write 11111010. What went wrong?

A You forgot to add 1 after flipping

B You should have added 1 before flipping

3. A round-trip check on 10000001 uses -128 + 1. What denary value does it give?

A -127

B -1

4. You convert -20 to binary, then convert your answer back and get -19. What should you do?

A Treat it as an error and redo the conversion

B Accept it, because -19 is close enough

5. The check on 11110100 gives a negative result. Flipping gives 00001011 and adding 1 gives 00001100 (12). What is the value?

A 12

B -12

Match the Pairs

Draw a line connecting each item on the left to its match on the right.

Left column		Right column
1-128 64 32 16 8 4 2 1		A11111011 -> 00000100 -> 00000101 = 5
211111011 = -128+64+32+16+8+2+1 = -5		BThe student used only 4 bits instead of all 8 bits.
3If the checked result is negative, flip the bits and add 1 to get the positive value and confirm the size		CA round-trip check: converting the binary back gives -5, which matches the starting number
4A 1 in the sign bit adds -128, so add the other place values to it to check a negative number		DThe 8 place values used to convert an 8-bit two's complement number back to denary for checking
5-3 in 8 bits: 00000011 -> flip -> 11111100		EThe student forgot to add 1 after flipping the bits.
6-5 stored as 1011		F10000011 = -128 + 2 + 1 = -125

Definitions

-128 64 32 16 8 4 2 1

Fill in the Blanks

Complete each line of code by filling in the missing parts.

Place values: -128 64 32 16 8 4 2 _

11111011 = ____ + 64 + 32 + 16 + 8 + 2 + 1

-128 + 64 + 32 + 16 + 8 + 2 + 1 = __

11111011 flip = 00000100 00000100 + 1 = ________ So 11111011 is -5

Always write _ bits: 00000101, not 101

Spot the Error

Shade in the letter of the line that contains an error.

Question 1

A Start: -45

B Binary: 11010010

C Check: -128+64+16+2+1

D Total: -45

Question 2

A Start: -20

B Binary: 11101100

C Check: 128+64+32+8+4

D Total: -20

Question 3

A Start: -6

B Binary: 111010

C Check: -128+64+32+16+8+2

D Matches: yes

Question 4

A Binary: 1111000

B Check: -128+64+32+16

C Total: -16

Question 5

A Binary: 11010011

B Sign bit is 1: negative

C Flip: 11010100

D Add 1: 00101101

E Value: -45

Answer Key

Checking a two's complement conversion

Section A: Quick Questions

B — -128+64+32+16+8+2+1 = -5, which matches

A — You forgot to add 1 after flipping

A — -127

A — Treat it as an error and redo the conversion

B — -12

Section B: Match the Pairs

-128 64 32 16 8 4 2 1 → The 8 place values used to convert an 8-bit two's complement number back to denary for checking

11111011 = -128+64+32+16+8+2+1 = -5 → A round-trip check: converting the binary back gives -5, which matches the starting number

If the checked result is negative, flip the bits and add 1 to get the positive value and confirm the size → 11111011 -> 00000100 -> 00000101 = 5

A 1 in the sign bit adds -128, so add the other place values to it to check a negative number → 10000011 = -128 + 2 + 1 = -125

-3 in 8 bits: 00000011 -> flip -> 11111100 → The student forgot to add 1 after flipping the bits.

-5 stored as 1011 → The student used only 4 bits instead of all 8 bits.

Section C: Definitions

-128 64 32 16 8 4 2 1 — Check a two's complement answer by converting it back to denary with these place values, then compare the result with your starting number. They must match.

Section D: Fill in the Blanks

Place values: -128 64 32 16 8 4 2 1

11111011 = -128 + 64 + 32 + 16 + 8 + 2 + 1

-128 + 64 + 32 + 16 + 8 + 2 + 1 = -5

11111011 flip = 00000100
00000100 + 1 = 00000101
So 11111011 is -5

Always write 8 bits: 00000101, not 101

Section E: Spot the Error

B — The +1 step was forgotten. Flipping 00101101 gives 11010010, and adding 1 gives 11010011. The check would give -46, which does not match -45.

C — The leftmost place value is -128, not +128. It should be -128+64+32+8+4, which gives -20.

B — Only 6 bits were used. Two's complement needs all 8 bits, so -6 is 11111010. With fewer bits the sign bit is lost.

A — This has only 7 bits. It needs 8 bits, 11110000, to match the place values -128 64 32 16 8 4 2 1 used in the check.

C — This line added 1 instead of flipping the bits. Flipping each bit of 11010011 gives 00101100. Then you add 1 on the next line.