Floating Point Arithmetic

Floating Point Addition

When adding floating point binary numbers, it is important to ensure each number is represented with the same level of precision. This means that the exponents must be the same.

Addition

The example below shows the addition of two normalised 12-bit floating point numbers, with 8-bit mantissas and 4-bit exponents:
- 0.10110000011 + 0.11100000010

Matching exponents

The first step is to ensure the exponents of the two numbers match, by adding 1 to the exponent of the second number (2) to match that of the first (3).
Adding 1 to the exponent of the second number moves the binary point of the mantissa in this number one place to the left:
- 0.11100000010 $\rightarrow$ 0.01110000011

Compute the addition

The two mantissas can then be added together:
- 0.1011000 + 0.0111000 = 1.0010000
Including the exponent gives 1.00100000011.

Normalise the result

Check that the result is normalised, and move the binary point as necessary.
If the binary point gets moved, remember to change the exponent accordingly.

Floating Point Subtraction

When subtracting floating point binary numbers, it is important to ensure each number is represented with the same level of precision. This means that the exponents must be the same.

Subtraction

The example below shows the subtraction of two normalised 12-bit floating point numbers, with 8-bit mantissas and 4-bit exponents:
- 0.10110000011 − 0.11100000010

Matching exponents

The first step is to ensure the exponents of the two numbers match, by adding 1 to the exponent of the second number (2) to match that of the first (3).
Adding 1 to the exponent of the second number moves the binary point of the mantissa in this number one place to the left:
- 0.11100000010 $\rightarrow$ 0.01110000011

Two's complement

The next step is to find the twos complement of the second number:
- 1.1001000

Add mantissas

The two mantissas can then be added:
- 0.1011000 + 1.1001000 = 0.1000000
Note that the first digit of the two's complement corresponds to -1 in decimal.
Factor this in when you do the addition.

Normalise the result

The result should then be normalised by moving the binary point one place to the left.
That means we add 1 to the exponent:
- 1.00100000100

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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