4.1.14

Floating Points in Binary

Test yourself on Floating Points in Binary

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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

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

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=122^{-1}=\frac{1}{2}.
  • Each subsequent binary digit corresponds to 2−2, 2−3, 2−42^{-2},\,2^{-3},\,2^{-4}.. and so on.
Example

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.

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.
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1

Components of a Computer

1.1

Structure & Function of the Processor

1.1.1Von Neumann Architecture1.1.2Registers1.1.3Buses1.1.4Fetch, Decode, Execute Cycle1.1.5Factors Affecting CPU Performance1.1.6Pipelining1.1.7CPU Architecture
1.2

Types of Processors

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1.3

Input, Output & Storage

1.3.1Elements of Computer Systems1.3.2Types of Computer Systems1.3.3How Magnetic Storage Works1.3.4Properties of Magnetic Storage1.3.5Examples of Magnetic Storage1.3.6How Optical Storage Works1.3.7Properties of Optical Storage1.3.8Examples of Optical Storage1.3.9Types of Optical Disc1.3.10Random Access Memory1.3.11Read Only Memory1.3.12Uses of Flash Memory1.3.13Properties of Flash Memory1.3.14What to do When We Run Out of Memory1.3.15How Virtual Memory Works
2

Software & Software Development

2.1

Systems Software

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2.2

Applications Generation

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2.3

Software Development

2.3.1Algorithmic Thinking2.3.2Waterfall Lifecycle2.3.3Waterfall Lifecycle - Strengths & Weaknesses2.3.4Agile Methodology2.3.5Agile Methodology - Strengths & Weaknesses2.3.6Extreme Programming2.3.7Extreme Programming - Strengths & Weaknesses2.3.8Spiral Methodology2.3.9Spiral Methodology - Strengths & Weaknesses2.3.10Rapid Application Development2.3.11RAD - Strengths & Weaknesse
2.4

Types of Programming Language

2.4.1High Level Programming Languages2.4.2Low Level Programming Languages2.4.3Programming Paradigms2.4.4Procedural Programming Languages2.4.5Assembly2.4.6Machine Code2.4.7Modes of Addressing Memory2.4.8Object-Oriented Languages2.4.9Classes2.4.10Objects2.4.11Encapsulation2.4.12Polymorphism
3

Exchanging Data

3.1

Compression, Encryption & Hashing

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3.2

Databases

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3.3

Networks

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3.4

Web Technologies

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4

Data Types, Data Structures & Algorithms

4.1

Data Types

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4.2

Data Structures

4.2.13D Arrays4.2.2Linked List4.2.3Queues4.2.4Graphs4.2.5Hash Tables4.2.6Stacks4.2.7Trees4.2.8Binary Search Trees
4.3

Boolean Algebra

4.3.1Binary Systems4.3.2Logic Circuits4.3.3Boolean Operators4.3.4Evaluating Expressions4.3.5Boolean Logic4.3.6Karnaugh Maps4.3.7Boolean Identities4.3.8Boolean Rules4.3.9Flip Flops4.3.10Adder Circuits
5

Legal, Moral, Cultural & Ethical Issues

5.1

Computing Related Legislation

5.1.1Data Protection Act (1998)5.1.2Computer Misuse Act (1990)5.1.3Copyright Designs & Patents Act (1988)5.1.4Creative Commons Licensing5.1.5Regulation of Investigatory Powers Act (2000)
5.2

Moral & Ethical Issues

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6

Elements of Computational Thinking

6.1

Thinking Abstractly

6.1.1Abstraction
6.2

Thinking Ahead

6.2.1Inputs & Outputs
6.3

Thinking Procedurally

6.3.1Decomposition
6.4

Thinking Logically

6.4.1Pseudocode6.4.2Flow Diagrams
6.5

Thinking Concurrently

6.5.1Concurrent Processing
7

Problem Solving & Programming

7.1

Programming Techniques

7.1.1Variables & Constants7.1.2Comments7.1.3Inputs & Outputs7.1.4Subroutines7.1.5Scope
7.2

Programming Construction

7.2.1Integrated Development Environments7.2.2The Purpose of Testing7.2.3Types of Testing7.2.4Test Plans
8

Algorithms

8.1

Algorithms

8.1.1Interpreting Algorithms8.1.2Correcting Algorithms8.1.3Completing Algorithms8.1.4Sort Algorithms8.1.5Bubble Sort8.1.6Insertion Sort8.1.7Merge Sort8.1.8Search Algorithms8.1.9Linear Search8.1.10Binary Search8.1.11Quick Sort Algorithm8.1.12A* Algorithm

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Converting Denary & Hexadecimal

Normalisation of Floating Points