Number Systems (AQA A-Level Computer Science): Revision Notes

Number Systems

Introduction

A number is a fundamental unit of mathematical data that we use to count, quantify, label, and measure. In everyday life, we naturally use the decimal system (also called base 10) without thinking about it. However, in computer science, you'll discover that different number systems exist, including binary and hexadecimal, which are covered in later chapters.

In this chapter, we explore various types or sets of numbers and how they're used in computing. Understanding these different categories is essential because computers need to handle numbers in specific ways, and programmers must choose the appropriate number type for their data and operations.

Natural numbers

Natural numbers are the most familiar numbers we encounter. They're the counting numbers we use every day for ordering and basic arithmetic. These are the numbers formed from the decimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The term "decimal" comes from the fact that we use base 10, meaning there are ten different digits available. With a single digit, we can represent a maximum value of 9. To create larger numbers, we simply add more digits: 10–99 uses two digits, 100–999 uses three digits, and so on, continuing indefinitely.

An important concept to understand is positional value. Each additional digit we add is worth ten times as much as the previous digit because we're working in base 10. This principle is crucial when you later study the binary system, where each position is worth twice the previous one (base 2).

The mathematical symbol for natural numbers is ℕ or N, and we can represent them as follows:

$\mathbb{N} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...\}$

The three dots (...) indicate that the sequence continues infinitely.

Integer numbers

An integer is a whole number that can be positive, negative, or zero. Unlike natural numbers, integers extend in both directions from zero. The key characteristic is that integers contain no fractional part and no decimal places - they're complete, whole numbers.

Negative values are indicated by the minus sign (−). There's an important mathematical relationship between negative and positive integers: if you add a number and its negative counterpart, you always get zero. For example:

$-3 + 3 = 0$

or equivalently:

$3 + (-3) = 0$

Integers are one of the standard data types available when defining variables and constants in programming languages. In theory, there's an infinite number of integers that can be represented. However, in practice, different programming languages have variations of integer types depending on the size of the number that needs to be stored.

The mathematical symbol for integer numbers is ℤ or Z, represented as:

$\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$

Again, the dots at each end indicate the sequence continues infinitely in both directions.

Rational numbers

A rational number is one that can be expressed as a fraction, also known as a quotient. Fractions consist of two integers, with the value being the ratio between them. This ratio can be expressed either as a fraction or as its decimal equivalent.

For example, we understand $\frac{1}{2}$ as half, which can also be represented as the decimal 0.5. The integer on top is called the numerator, and the integer on the bottom is called the denominator.

The numerator can be any integer, meaning it can be positive, negative, or zero. This flexibility means that the fraction's value can be negative, positive, greater than one, less than one, or equal to zero. Fractions can also result in the value of one. For example:

$\frac{3}{3} = 1$

$\frac{25}{25} = 1$

An important consequence of this definition is that all integers are rational numbers because any integer can be expressed as a fraction with a denominator of 1. For instance, $5 = \frac{5}{1}$.

The mathematical symbol for rational numbers is ℚ or Q.

Irrational numbers

An irrational number is any number that cannot be represented as a ratio of integers because the decimal equivalent would continue forever without repeating. A classic example of an irrational number is $\pi$ (pi). As a fraction, a widely used approximation is $\frac{22}{7}$. In decimal form, the number is infinite and must be truncated (cut off) to a set number of decimal places. For example, $\pi$ might be rounded to 3.14, or for greater accuracy, to 3.1415926535.

Some square roots and cube roots are also irrational numbers. A square root can be represented as $x = n^2$. Starting with a simple example, the square root of 16 is 4. However, if you try to calculate the square root of 2, 3, or 99, you'll find it's impossible to identify a number that, when squared, gives exactly these values. Hence, these square roots are irrational.

A defining feature of irrational numbers is that the values after the decimal place do not repeat in a pattern. In contrast, some fractions produce recurring decimals that do have a pattern and are therefore classed as rational numbers. For example, one-third = 0.333... recurring (the dots indicate the 3 repeats forever).

Real numbers

A real number is any positive or negative value that can include a fractional part. Integers, rational numbers, and irrational numbers are all categorised as real numbers. The defining feature of a real number is that the fractional part can be any length, allowing the number to represent a measurement to any level of precision and accuracy required.

For example, on a scale between 1 and 10, real numbers enable us to break that down into units of 0.1. On a scale between 0 and 0.1, real numbers allow us to work in units of 0.01, and so on. There's an infinite range of real numbers that can be generated, providing unlimited granularity for measurements.

Real numbers are essential for measuring continuous or infinitely changing values. Consider these computing applications where real numbers are necessary:

For programmers, the challenge is defining how accurate and precise their numbers need to be for each application. As with irrational numbers, computers handle real numbers using fixed or floating point techniques that are described in detail in Chapter 25.

Ordinal numbers

Ordinal numbers are those that identify the position of something within a list. For example, we use terms like "first, second, and third." Ordinal numbers are often used alongside cardinal numbers, which identify the size of the list. For instance, you might say "you were third out of 20" - here, "third" is the ordinal number (position), and 20 is the cardinal number (total size).

In computing, ordinal numbers are used to identify the position (or location) of data within an ordered set. Consider the following set:

$S = \{\text{'Anne', 'Asif', 'John', 'Mary', 'Wanda', ...}\}$

This set is described as well-ordered because it has an internal structure that defines the relationship between the data items. In this example, the data consists of names arranged in ascending alphabetical order. The ordinal number indicates the order of the data, so in this case:

• The first item $S(1) = \text{Anne}$
• The second item $S(2) = \text{Asif}$
• The third item $S(3) = \text{John}$, and so on

This technique is particularly useful for locating and manipulating data within certain data structures such as lists, queues, and arrays. Some programming languages support an ordinal data type, which contains a value that can be counted and ordered.

Arrays and ordinal numbers

A one-dimensional array called Register shows a list of names in a register:

Ordinal numbers are used to assign the data as follows:

Register(1) = "Allen"
Register(2) = "Brown"

A table called Exam shows the results for each student:

Exam(1,2) = 50
Exam(2,2) = 75

This demonstrates how ordinal numbers provide a systematic way to access specific elements within data structures, making it easier to write programs that process collections of information.

Counting and measurement

As we've explored, the basic use of numbers is to count and measure. We employ different types of numbers in different ways depending on the task at hand.

When counting, we use natural numbers because we only need positive whole numbers. Some common applications in programming include:

• A counter may track how many times a loop statement is repeated
• The program counter in the processor keeps track of which instruction needs to be processed next
• A natural number identifies the location of data within a data structure
• A variable may keep count - for example, of the number of items in a stock control system, or the score in a computer game

When measuring, we use real numbers because the range of numbers may be positive or negative and may require a fractional part. Some examples in programming include:

• CNC machines handle measurements that vary from millimetres to metres and must work to a high degree of accuracy
• Microwave cookers control and measure both time and temperature
• Power stations use data control systems to optimise the production of electricity
• Robotics engineers use real-time measurements of the environment in which the robot is working

Remember!