How does fixed point binary represent a number with a fractional part?

The binary point is placed at an agreed, unchanging position within the bits. Columns to the left of the point carry the usual integer place values 1, 2, 4, 8 and so on, and columns to the right carry the fractional place values 1/2, 1/4, 1/8, 1/16 and so on. The point itself is not stored — its position is fixed by the format.

State the place values of the first four bits after a binary point.

1/2 (0.5), 1/4 (0.25), 1/8 (0.125) and 1/16 (0.0625). Each column is worth half the one to its left, that is 2⁻¹, 2⁻², 2⁻³ and 2⁻⁴.

A fixed point format uses 8 bits with 4 bits before the binary point and 4 after. What decimal value is 0110.1010?

6.625. The integer part 0110 is 4 + 2 = 6, and the fraction 1010 is 1/2 + 1/8 = 0.625.

Using the same 4-and-4 fixed point format, represent 5.75 in binary.

0101.1100. The integer 5 is 0101, and 0.75 is 1/2 + 1/4, so the fraction bits are 1100.

Using the same 4-and-4 fixed point format, represent 0.6875 in binary.

0000.1011. Working down the fraction columns: 0.6875 − 0.5 = 0.1875, which is less than 0.25 so that column is 0; 0.1875 − 0.125 = 0.0625; and 0.0625 − 0.0625 = 0. So the bits are 1/2, 1/8 and 1/16 set, giving .1011.

In a signed fixed point format of 8 bits with 4 before the point, held in two's complement, what value is 1111.1000?

−0.5. The most significant bit carries the weight −8, so the sum is −8 + 4 + 2 + 1 + 0.5 = −0.5. Alternatively, negate by inverting to 0000.0111 and adding the smallest step 0.0625, which gives 0000.1000 = 0.5, so the original is −0.5.

AQA A-Level Computer Science 7517

Fundamentals of Programming

Topic overview Data types Programming concepts and program structure Arithmetic operations Relational Operations, Constants and Variables Boolean operations String-handling operations Random numbers and exception handling Subroutines, parameters and return values Variable scope and stack frames Recursive techniques Programming paradigms and the procedural approach Classes, Objects, Inheritance and Polymorphism Aggregation, Composition, Class Diagrams and Design Principles

Fundamentals of Data Structures

Topic overview Data Structures and Abstract Data Types Static and Dynamic Data Structures Queues and Linear Queues Circular and Priority Queues Stacks Graphs Trees Hash tables Dictionaries Vectors and Their Notation Vector Arithmetic and Geometry

Fundamentals of Algorithms

Topic overview Graph traversal Tree traversal Reverse Polish notation Searching algorithms Sorting algorithms Optimisation algorithms

Theory of Computation

Topic overview Logic Problems and Algorithm Constructs Pseudo-Code, Hand-Tracing and Program Correctness Representational Abstraction, Generalisation and Information Hiding Procedural and Functional Abstraction Data Abstraction and Problem Reduction Decomposition and Composition Automation and Finite State Machines Sets and Set Notation Subsets and Set Operations Regular expressions and regular languages Backus-Naur Form and syntax diagrams Comparing Algorithms and Functions Order of complexity and Big-O notation Limits of computation, tractability and heuristics Computable problems and the Halting problem Turing machines

Fundamentals of Data Representation

Topic overview Number systems and the sets of numbers Number bases: decimal, binary and hexadecimal Bits, bytes and units of information Unsigned binary and binary arithmetic Signed binary using two's complement Fixed point and floating point representation Normalisation of floating point numbers Rounding errors, accuracy, range and precision Character coding systems Error checking and correction Bit patterns, analogue and digital Bitmapped graphics Vector graphics Digital representation of sound MIDI and Data Compression Encryption

Fundamentals of Computer Systems

Topic overview Hardware, Software and Its Classification System Software and the Operating System Translators and the Role of the Operating System Classification of programming languages Types of program translator Logic Gates and Their Truth Tables Logic Circuits and D-Type Flip-Flops Half-Adders and Full Adders Boolean algebra

Fundamentals of Computer Organisation and Architecture

Topic overview Internal components, buses and von Neumann and Harvard architectures The Stored Program Concept and the Processor The Fetch-Execute cycle The processor instruction set Addressing modes Machine-Code Operations Shifts, Halting and Assembly Language Programs Interrupts Factors affecting processor performance Input and output devices Secondary storage devices

Fundamentals of Communication and Networking

Topic overview Communication methods Communication basics Network Topology and Types of Networking Wireless networking The Internet and Packet Switching Routing, URLs and Domain Names Firewalls, Worms, Trojans and Viruses Encryption, Digital Signatures and Certificates TCP/IP Standard application layer protocols IP Address Structure and Subnet Masking IP Standards and Public and Private Addresses DHCP and Network Address Translation Client server model Thin- versus thick-client computing

Consequences of Uses of Computing

Topic overview The Four Kinds of Issue in Computing Legislation, Ethical Dilemmas and Case Studies

Systematic Approach to Problem Solving

Topic overview Analysis, Design and Implementation Testing and Evaluation

Fundamentals of Databases

Topic overview Conceptual data models and entity relationship modelling Relational databases Database design and normalisation techniques Structured Query Language (SQL)Client server databases

Big Data

Topic overview Volume, Velocity and Variety Why Big Data Resists Analysis Distributed Processing and Functional Programming Fact-Based Models and Graph Schemas

Fundamentals of Functional Programming

Topic overview Function Types, First-Class Objects and Function Application Partial Function Application and Composition Writing functional programs Lists in functional programming

Fundamentals of Data Representation

Fixed point and floating point representation

3 things to know
Paper 2

Overview

Recall questions on place values after the binary point, representing a given decimal in a fixed point format, and how the mantissa and exponent divide the work.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

Represent 5.75 in binary using an unsigned fixed point format with 4 bits before the point and 4 after.

Numbers

For an unsigned fixed point format with 4 bits before the point and 4 after, what are the smallest step and the largest value?

Key term

Represent 6.5 in floating point with an 8-bit two's complement mantissa, point after the sign bit, and 4 bits for the power of two.

Questions from this subtopic

Each one with the model answer underneath.

1
How does fixed point binary represent a number with a fractional part?
Model answer
The binary point is placed at an agreed, unchanging position within the bits. Columns to the left of the point carry the usual integer place values 1, 2, 4, 8 and so on, and columns to the right carry the fractional place values 1/2, 1/4, 1/8, 1/16 and so on. The point itself is not stored — its position is fixed by the format.
2
State the place values of the first four bits after a binary point.
Model answer
1/2 (0.5), 1/4 (0.25), 1/8 (0.125) and 1/16 (0.0625). Each column is worth half the one to its left, that is 2⁻¹, 2⁻², 2⁻³ and 2⁻⁴.
3
A fixed point format uses 8 bits with 4 bits before the binary point and 4 after. What decimal value is 0110.1010?
Model answer
6.625. The integer part 0110 is 4 + 2 = 6, and the fraction 1010 is 1/2 + 1/8 = 0.625.
4
Using the same 4-and-4 fixed point format, represent 5.75 in binary.
Model answer
0101.1100. The integer 5 is 0101, and 0.75 is 1/2 + 1/4, so the fraction bits are 1100.
5
Using the same 4-and-4 fixed point format, represent 0.6875 in binary.
Model answer
0000.1011. Working down the fraction columns: 0.6875 − 0.5 = 0.1875, which is less than 0.25 so that column is 0; 0.1875 − 0.125 = 0.0625; and 0.0625 − 0.0625 = 0. So the bits are 1/2, 1/8 and 1/16 set, giving .1011.
6
In a signed fixed point format of 8 bits with 4 before the point, held in two's complement, what value is 1111.1000?
Model answer
−0.5. The most significant bit carries the weight −8, so the sum is −8 + 4 + 2 + 1 + 0.5 = −0.5. Alternatively, negate by inverting to 0000.0111 and adding the smallest step 0.0625, which gives 0000.1000 = 0.5, so the original is −0.5.

See the rest and revise

Other ways to work this

The same content, four other ways.

What is inside Levo

Six ways to work the same material.

1Closed book

2Open book

3Closed book

The 3-step method, run for you

Answer cold, learn it, then prove it stuck. Levo tracks the jump between your first attempt and your last.

Fundamentals of Programming

Fundamentals of Data Structures

Fundamentals of Algorithms

Theory of Computation

Cram a whole paper in one sitting

Sort every topic into what you know and what you do not, then drill only the weak ones.

A driver is .

softwarehardwarestorage

Blank Blitz makes facts automatic

Rapid timed gap-fills on the terms and numbers you need to recall without thinking.

AThe integer 5 is 0101.

B0.75 is 1/2 + 1/4.

CThe fraction bits are 1100.

Build It gets the exam shape right

Assemble a full-mark answer from its marking points, in the order an examiner wants them.

Head means the first single element of a non-empty list.

TrueFalse

Test It catches misconceptions

Spot the fault in a statement. The traps are the mistakes students actually make.

721941163128510

Brain Breaks between sessions

Short focus games to reset attention, so the next set lands better than the last.

Every question in Fundamentals of Data Representation, on a schedule.

Attempt a question from memory, write the correct version, then attempt it again. Levo records the jump between your first attempt and your last, and brings each question back on the day you are due to forget it.

Start revising, £5 a month See how the method works

Or £46 a year, about £3.83 a month. Cancel anytime.