Floating Points in Binary

Floating Point Representation

Floating point is a method of representing numbers in binary, which makes use of a binary point placed after the ‘most significant bit’ (MSB).

Floating point

A floating point binary number is made up of two parts:
- The mantissa is the main number, containing a binary point after the MSB.
- The exponent determines where the binary point is positioned before the number is read.
If the exponent is negative (i.e. it begins with a ‘1’), the binary point is moved left instead of right.

Converting to denary

After the binary point has been moved, we can read the value of the mantissa.
The denary value of the binary digit to the right of the binary point corresponds to:
- $2^{-1}=\frac{1}{2}$ .
Each subsequent binary digit corresponds to $2^{-2},\,2^{-3},\,2^{-4}$ .. and so on.

Example

The example below is a 12-bit floating point number, with an 8-bit mantissa and a 4-bit exponent:
- 0 . 1 0 1 1 0 0 0 0 0 1 1
The exponent 0011 (which is equal to 3) identifies where the binary point should move in order to read the number.

Example cont.

In this example, the binary point is moved 3 places to the right, which in turn subtracts 3 from the exponent:
- 0 1 0 1 . 1 0 0 0 0 0 0 0
This represents the decimal number 5.5.

Normalisation of Floating Points

Floating point binary numbers should be normalised to ensure they are as precise as possible.

Normalisation

The same decimal number can be represented in a number of different ways using floating point, depending on the number of bits available.
To ensure the mantissa is as precise as possible (normalised), the binary point and exponent should be adjusted so that:
- A positive number has a ‘0’ before the binary point and a ‘1’ after it.
- A negative number has a ‘1’ before the binary point and a ‘0’ after it.

Example

The example shown is an un-normalised 12-bit floating point number, with an 8-bit mantissa and a 4-bit exponent:
- 0 . 0 0 1 0 1 1 0 0 1 0 1
It is un-normalised because the bits either side of the binary point are the same.
The binary point needs to move two places to the right, ensuring it remains a positive number (the MSB is still a ‘0’), but the bit after the point is a ‘1’.

Example cont.

Moving the point two places to the right reduces the value of the exponent by two, resulting in:
- 0 . 1 0 1 1 0 0 0 0 0 1 1
Notice the two excess bits to the left of the MSB in the mantissa have been moved to the end of the mantissa, ensuring it remains 8 bits in size.

Negative floating point

Normalising negative floating point binary numbers follows the same procedure, except the point is moved until the MSB is a ‘1’ and the bit following the point is a ‘0’.

1Components of a Computer

1.1Structure & Function of the Processor

1.1.1Structure of the CPU

1.1.2Fetch, Decode, Execute Cycle

1.1.3Factors Affecting CPU Performance

1.1.4CPU Architecture

1.2Types of Processors

1.2.1CISC & RISC

1.2.2Multicore & Parallel Cores

1.3Input, Output & Storage

1.3.1Computer Systems

1.3.2Magnetic Storage

1.3.3Optical Storage

1.3.4Types of Optical Storage

1.3.5Types of Memory

1.3.6Flash Memory

1.3.7Virtual Memory

2Software & Software Development

3Exchanging Data

4Data Types, Data Structures & Algorithms

4.1Data Types

4.1.1Data Types

4.1.2Arrays & Strings

4.1.3Representing Numbers

4.1.4Binary Arithmetic

4.1.5Hexadecimal

4.1.6Converting Binary Numbers

4.1.7Floating Points in Binary

4.1.8Floating Point Arithmetic

4.1.9Bitwise Manipulation

4.1.10Character Sets

4.2Data Structures

4.2.1Arrays, Linked Lists & Queues

4.2.2Graphs, Hash Tables & Stacks

4.2.3Trees

4.3Boolean Algebra

4.3.1Computational Logic

4.3.2Boolean Logic

4.3.3Karnaugh Maps

4.3.4Boolean Identities

4.3.5Flip Flops & Adder Circuits

5Legal, Moral, Cultural & Ethical Issues

5.1Computing Related Legislation

5.1.1Data Protection Act (1998)

5.1.2Computer Misuse Act (1990)

5.1.3Copyright Designs and Patents Act (1988)

5.1.4Regulation of Investigatory Powers Act (2000)

5.2Moral & Ethical Issues

5.2.1Online Activity Tracking

5.2.2Censorship

5.2.3Cultural Issues

5.2.4Environmental Issues

5.2.5Layout, Colour Paradigms & Character Sets

6Elements of Computational Thinking

6.1Thinking Abstractly

6.1.1Abstraction

6.2Thinking Procedurally

6.2.1Decomposition

6.3Thinking Logically

6.3.1Pseudocode & Flow Diagrams

7Problem Solving & Programming

7.1Programming Techniques

7.1.1Programming Fundamentals

7.1.2Modularity

7.2Programming Construction

7.2.1Integrated Development Environments

7.2.2Testing

8Algorithms

8.1Algorithms

8.1.1Interpreting, Correcting & Completing Algorithm

8.1.2Sort Algorithms

8.1.3Insertion & Merge Sort

8.1.4Search Algorithms

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