4.5.4 Binary Number System

Question 1

What are the two kinds of binary numbers?

Answer 1

Signed

Unsigned

Question 2

What are unsigned binary numbers?

Answer 2

Unsigned binary numbers can only represent positive numbers.

Question 3

What does signed binary allow for?

Answer 3

Signed binary allows for the representation of negative numbers using binary.

Question 4

What does the range of numbers that can be represented by unsigned binary number depend on?

Answer 4

The range of numbers that can be represented depends on the number of bits available.

Question 5

What is the pattern to the range of numbers that can be represented?

Answer 5

For n bits, there are 2^n possible permutations of the bits, giving a range of decimal numbers from 0 to (2^n) - 1

Question 6

What does twos complement allow for?

Answer 6

The representation of both positive and negative numbers in binary.

Question 7

How does two’s complement work?

Answer 7

• The most significant bit of a number is given a negative place value.

Question 8

What is the range of two’s complement numbers?

Answer 8

Given a certain number of bits, the range of two’s complement signed binary number includes both positive and negative values.

Question 9

What is the range of two’s complement numbers?

Answer 9

The range of a two’s complement signed binary number:

2^(n-1)) - 1 to -2^(n-1

Question 10

What are the two means of representing binary numbers with a fractional part?

Answer 10

• Fixed point form

* Floating point form

Question 11

How does fixed point work?

Answer 11

The specified number of bits are placed before a binary point and the remaining bits fall behind the binary point.

Question 12

Why may fixed point and floating point representations of decimal numbers be inaccurate?

Answer 12

There are many numbers that binary cannot accurately represent, there are some numbers which binary can only approximately represent.

Question 13

Why may you use absolute and relative errors?

Answer 13

In order to see how close a particular number is to an actual value.

Question 14

What is an absolute error?

Answer 14

Is the actual amount by which a value is inaccurate and can be calculated by finding the difference between the given value and the actual value

Question 15

What is relative error?

Answer 15

A measure of uncertainty in a given value compared to the actual value, which is relative to the size of the given value.

Question 16

What is the formula for the relative error?

Answer 16

Absolute error / Actual Value = Relative error

Question 17

What is absolute error calculated?

Answer 17

The difference between the given value and the actual value.

Question 18

How may you get the percentage from the relative error?

Answer 18

( Absolute error / Actual value ) x 100

Absolute error = difference between given value and actual value

Question 19

What is the advantage of floating point?

Answer 19

• Representation of a greater range of numbers with a given number of bits than fixed point, floating point can take advantage of an exponent which can be either positive or negative.

Question 20

What affects the number of numbers that can be represented?

Answer 20

The number bits allocated to each part of a floating point number.

• A large exponent and a small mantissa allows for a large range but little precision
• A small exponent and a large mantissa allows for good precision but only a small range.

Question 21

What does a large exponent and a small mantissa allow for?

Answer 21

Large range, little precision

Question 22

What does a small exponent and large mantissa allow for?

Answer 22

Good precision, but only a small range

Question 23

What determines the range and precision of numbers that can be represented in fixed point notation?

Answer 23

Placement of the binary point.

Question 24

What does a binary point close to the left of the number in fixed point indicate?

Answer 24

Good precision, but with a small range of numbers.

Question 25

What does a binary point close to the right in fixed point indicate?

Answer 25

An increase in the range, while decreasing precision.

Question 26

What is the purpose of normalisation?

Answer 26

• To provide the maximum level of precision for a given number of bits.
• Involves ensuring that the floating point number starts with 01 for a positive number, or 10 for a negative number.

Question 27

When does underflow occur?

Answer 27

When very small numbers are to be represented, but there are not enough bits available.

Question 28

When does overflow occur?

Answer 28

When the number is too large to be represented with the available bits.