Binary Arithmetic

Question 1

What symbols does the binary number system use?

Answer 1

It uses 2 symbols: 0 and 1

eg. 19 in binary is 10011

Question 2

What is the right-most digit called?

Answer 2

Least Significant Bit (LSB)

Question 3

What is the left-most digit called?

Answer 3

Most Significant Bit (MSB)

Question 4

What is the highest number that we can represent in binary with N digits?

Answer 4

2N - 1

Question 5

What does Q3 format mean?

Answer 5

It has 3 bits after the binary point (binary equivalent of decimal point)

Question 6

How do you convert from decimal to binary?

Answer 6

Do repeated division by 2 until you get 1. The remainders are the bits starting from the LSB

Question 7

What is a flag?

Answer 7

A binary number only having 1 bit

Question 8

What is a nibble?

Answer 8

A binary number having 4 bits

Question 9

What is a byte?

Answer 9

A binary number having 8 bits

Question 10

What is a word?

Answer 10

A binary number having 16 or more bits

Question 11

How do you carry out binary addition?

Answer 11

110 + 101
Going from LSB
0+1 = 1, carry out = 0
1+0 = 1, carry out = 0
1+1 = 0, carry out = 1

SUM = A=OB
C = A.B

Question 12

What does the half adder do?

Answer 12

• Generates the sum and carry-out for one column of an addition sum
• BUT it does not allow for a carry-in from a previous column (this can be done with a full adder circuit)

Question 13

Describe how a full adder works

Answer 13

• The full adder uses 2 half adders; one to form the sum of A and B and another to add any carry in to this intermediate sum.
• If EITHER of the two additions generates a carry-out, we must indicate that by setting the final carry out to 1.

[HALF ADDER then HALF ADDER then OR gate]

Question 14

How do you built a multi-bit full adder?

Answer 14

• A row of N full adders would be required and the result has to be (N+1 bits in the case of overflow).
  (this is catered for by carry out being the MSB

Question 15

How do you represent negative numbers in binary?

Answer 15

2’s complement

[Half the possibilities for positive numbers and negative numbers, let the MSB have a negative weighting}

Question 16

How do you differentiate between positive and negative signed numbers?

Answer 16

• Signed numbers are positive if the MSB is 0

- Signed numbers are negative if the MSB is 1

Question 17

How do you carry out binary subtraction using 2’s complement?

Answer 17

1) Invert all the bits (eg. 0011 turns to 1100)
2) Add 1 (1100 + 1 = 1101)

NOTE: doesn’t work for the maximum negative number

Question 18

Calculate 4 minus 3 in binary

Answer 18

Convert to binary
4 = 0100
3 = 0011

Convert negative number to 2’s complement
-3 = 1101

Perform binary addition
0100 + 1101 = 10001

Discard carry out
=0001 (final answer)

Question 19

How do you prevent overflow when adding two N-bit signed numbers?

Answer 19

1) Sign extend to N + 1 bits

2) Use an N + 1 bit adder

Question 20

How do you extend a number of have more bits in signed binary?

Answer 20

Duplicate the MSB

eg.
- Converting 0011 into 8 bits gives 00000011
- Converting 1011 into 8 bits gives 11111011

This is called signed extension

Question 21

How do you build a subtraction machine?

Answer 21

Use an array of full adders but before that, pass all the B-input bits through inverters.
Then set the carry-in to 1 on the least significant full adder.

Question 22

What is a use of XOR gates?

Answer 22

They can be used as controllable inverters

Question 23

What is the difference between Unsigned and Signed values?

Answer 23

Unsigned

• MSB has a positive value
• The carry out from the MSB of an adder can be used as an extra bit of the answer to avoid overflow
• To increase the umber of bits, add zeros to the left hand side
• Complementing and adding 1 converts X to (2N - X)

Signed

• MSP has a negative value
• To avoid overflow, we need to sign extend then use and adder with one more bit than the numbers to be added
• TO increase the number of bits, sign extend by duplicating the MSB
• Complementing and adding 1 converts X to - X