Week 2 - Binary codes

Question 1

How do we represent unsigned numbers in binary code?

Answer 1

Unsigned binary codes are positional like the decimal system used in ordinary arithmetic.

Each digit carries a weight depending on its position in the code word.

In decimal allowed digits are 0,1,2,…,9; in binary only 1s and 0s are allowed

Question 2

What is the base number of a positional code?

Answer 2

The position of a digit is weighted using powers of the base.

The lowest weighted digit is the least significant and the highest is the most significant.

Question 3

How would you break down the decimal positional code 2053 with base 10?

Answer 3

2053 with base 10 = 2x10^3 + 0x10^2 + 5 x 10^1 + 3 x 10^0

= 2000+0+50+3

Question 4

How would you break down the binary positional code 1001 with base 2?

Answer 4

1001 with base 2 = 1x2^3 + 0x2^1 + 1x2^0

= 8+0+0+1 = 9

Question 5

How do you do decimal to binary conversion?

Answer 5

If we’re trying to convert the whole number x into binary, we need at least k bits, where k is the smallest power of 2 bigger than x.

Think: 
How many bits is it?
Write the powers for the number of bits 
Work out what can be summed to = the decmial number & put a 1 under these numbers
Then the rest should have 0s

Question 6

Convert 203 base 10 to 8-bit binary.

Answer 6

First note the largest power of 2 in 8-bit positional number is 128.

x = 203;
Is x > 128 ? If yes, write 1 and (203-128 = 75)
x = 75;
Is x>64? If yes, write 1 and (75-64 = 11)
x=11;
Is x>32? No, write 0
Is x>16? No, write 0
Is x>8? Yes, write 1 and x=11-8=3
x=3;
...
...
...
...

Question 7

Add the two 4-bit numbers (unsigned binary addition)?

x=1010 and y=1111

Answer 7

11001

look at week 2, lecture slide 7 for full solution if stuck

Question 8

What is 11001 base 2 in base 10?

Answer 8

25

Question 9

What is unsigned overflow?

Answer 9

If we add two n-bit unsigned numbers, and the sum is too big to be an n-bit number, the sum cannot be represented in the n-bit code.

Question 10

What is the necessary and sufficient condition for unsigned overflow?

Answer 10

The carry from the most significant bit (MSB) position is 1.

Question 11

What is an n-bit unsigned adder?

Answer 11

An n-bit unsigned adder is a device which takes in two n-bit unsigned binary code words and outputs an n-bit number representing the sum.

Question 12

What is a practical adder?

Answer 12

It’s an extra wire that signals whether we have a carry out and if there’s been an overflow. A practical adder should signal when there is an overflow. This can be done using the carry from the most significant bit (MSB) position. This is often referred to as the adder’s carry out (C).

Question 13

What is hexadecimal and why do we convert binary code words into them?

Answer 13

Binary codewords can be very long and difficult for humans to work with. We can shorten binary strings to base 16 number representation (called hexadecimal). Needs digits 0 up to 15. We use 0,1,2,3,4…,9,a,b,c,d,e,f

Question 14

How do you turn the hex digits to 4-bit binary?

3 a 6 e

Answer 14

0011 1010 0110 1110

See week 2, video 4, 5min for more details.

Question 15

What is the maximum number of a 16-bit unsigned code?

Answer 15

65,535

Question 16

What is the two’s complement code?

Answer 16

It’s an alternative to sign and magnitude, to represent signed numbers. It is the commonest codes for signed numbers.

Question 17

What are the downsides to sign and magnitude code?

Answer 17

You can have a codeword for -0

Question 18

In two’s complement code, how do you represent positive numbers for signed numbers?

Answer 18

Code words with most significant bit MSB = 0 represent positive numbers.

Question 19

In two’s complement code, how do you represent negative numbers for signed numbers?

Answer 19

To find the code word for -x, simply subtract x from 2^8 = 256 and convert the answer to binary in the usual way.

Question 20

What would 1111 1111 base 2 represent in signed and unsigned numbers?

Answer 20

signed: 255-256 = -1
unsigned: 255

Question 21

What is the largest number that an 8-bit two’s complement code can represent?

Answer 21

127
0111 1111 = 1+2+4+8+16+32+64
Since it must start with 0 to indicate that it’s positive

Question 22

How can we find the inverse of any two’s complement code number (with any number of bits)?

Answer 22

Flip all the bits (i.e. zeros to 1s and 1s to zeros) and add 1 to the result. This process is called negation

Question 23

Perform negation on 1111 1110

Answer 23

1) flip the digits of 1111 1110 (i.e. -2) to 0000 0001
2) add 1 to 0000 0001:
0000 0010 (i.e. 2)

Question 24

What is a two’s complement overflow? hint, try adding:

0000 0001 + 0111 1111 and see the output.

Answer 24

0000 0001 + 0111 1111 = 1000 0000 which = -128 in signed numbers. This is wrong. So we have a two’s complement overflow.

Question 25

How do we signal that there’s a two’s complement overflow?

Answer 25

We have a V line output. When using an adder with an unsigned code, look at C but ignore V.

Question 26

What is a character code?

Answer 26

It’s a table that maps the letters of the alphabet to a table.

Question 27

What is ASCII?

Answer 27

American Standard Code for Information Interchange. Each of the letters of the alphabet are represented in 7-bit or 8-bit code words.

Question 28

What is Unicode?

Answer 28

two-byte code, 16 bits to represent characters. It can also be extended to 32 bits.

Question 29

What do these give:
0+0
1+0
0+1
1+1

Answer 29

0+0=0, with no carry
1+0=1, with no carry
0+1=1, with no carry
1+1=0, and you carry 1.