What is a natural number, and what is the set ℕ?

A natural number is a whole, non-negative number used for counting. The set of natural numbers is ℕ = {0, 1, 2, 3, …}. It includes zero, it has no upper limit, and it contains no negative numbers and no fractions.

Does the set ℕ include zero?

Yes. ℕ is taken to be {0, 1, 2, 3, …}, so zero is a natural number. This matters when you are asked for the smallest natural number: the answer is 0, not 1.

Give a number that belongs to ℤ but not to ℕ, and explain why.

Any negative whole number, for example −4. It is a whole number so it is an integer, but ℕ contains only non-negative values, so −4 is not a natural number.

What is a rational number, and what is the set ℚ?

A rational number is one that can be written as a fraction — a ratio a/b of two integers, where b is not zero. ℚ is the set of all such numbers, for example 3/4, −7/2 and 0.25.

Explain why every integer is also a rational number.

Any integer can be written as a fraction with a denominator of 1. For example 7 = 7/1 and −3 = −3/1. Since both parts are integers and the denominator is not zero, the number satisfies the definition of a rational number, so ℤ is a subset of ℚ.

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

Number systems and the sets of numbers

3 things to know
Paper 2

Overview

Recall questions on the sets ℕ, ℤ, ℚ and ℝ, why every integer is rational, whether a recurring decimal is rational, and ordinal against cardinal numbers.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

Define an integer and state the set ℤ.

Key term

Is a recurring decimal such as 0.333… rational? Justify your answer.

Numbers

For the well-ordered set S = {'a', 'b', 'c', 'd'}, give the ordinal number of each element?

Questions from this subtopic

Each one with the model answer underneath.

1
What is a natural number, and what is the set ℕ?
Model answer
A natural number is a whole, non-negative number used for counting. The set of natural numbers is ℕ = {0, 1, 2, 3, …}. It includes zero, it has no upper limit, and it contains no negative numbers and no fractions.
2
Does the set ℕ include zero?
Model answer
Yes. ℕ is taken to be {0, 1, 2, 3, …}, so zero is a natural number. This matters when you are asked for the smallest natural number: the answer is 0, not 1.
3
Define an integer and state the set ℤ.
Model answer
An integer is a whole number, which may be positive, negative or zero. The set of integers is ℤ = {…, −3, −2, −1, 0, 1, 2, 3, …}. Every natural number is an integer, so ℕ is a subset of ℤ.
4
Give a number that belongs to ℤ but not to ℕ, and explain why.
Model answer
Any negative whole number, for example −4. It is a whole number so it is an integer, but ℕ contains only non-negative values, so −4 is not a natural number.
5
What is a rational number, and what is the set ℚ?
Model answer
A rational number is one that can be written as a fraction — a ratio a/b of two integers, where b is not zero. ℚ is the set of all such numbers, for example 3/4, −7/2 and 0.25.
6
Explain why every integer is also a rational number.
Model answer
Any integer can be written as a fraction with a denominator of 1. For example 7 = 7/1 and −3 = −3/1. Since both parts are integers and the denominator is not zero, the number satisfies the definition of a rational number, so ℤ is a subset of ℚ.

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