Number Systems (Leaving Cert Mathematics): Revision Notes

Number Systems

What are number systems?

Number systems are different collections of numbers that mathematicians use to solve various types of problems. As you progress through mathematics, you discover that some equations require different types of numbers for their solutions. Understanding these number systems forms the foundation for working with complex numbers.

Natural numbers (N)

Natural numbers are the positive counting numbers that we use in everyday life.

Definition: $N = \{1, 2, 3, 4, 5, ...\}$

These are the numbers you naturally use when counting objects. Natural numbers do not include zero or negative numbers.

Examples: 1, 7, 25, 100, 1000

Integers (Z)

Integers extend natural numbers to include zero and negative numbers.

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

Integers allow us to solve equations like $x + 5 = 3$, which has the solution $x = -2$. This solution cannot be found within natural numbers alone.

Examples: -5, -1, 0, 3, 42

Rational numbers (Q)

Rational numbers are numbers that can be expressed as fractions.

Definition: Rational numbers are numbers that can be written in the form $\frac{a}{b}$ where $a, b \in Z$ and $b \neq 0$.

Rational numbers include all integers (since any integer $n$ can be written as $\frac{n}{1}$), all terminating decimals, and all recurring decimals.

Examples: $\frac{1}{2}$, $\frac{3}{4}$, $-\frac{2}{3}$, 0.5, 0.333..., 2.4

The diagram above shows how these number systems relate to each other. Natural numbers are contained within integers, which are contained within rational numbers.

Converting decimals to fractions

Terminating decimals can be easily converted to fractions by considering their place value.

Recurring decimals require a special algebraic method.

Irrational numbers

Irrational numbers are numbers that cannot be expressed as fractions.

When we try to solve equations like $x^2 = 2$, we get $x = \pm\sqrt{2}$. Using a calculator, $\sqrt{2} = 1.4142135...$. This is a non-terminating, non-repeating decimal that cannot be written as a fraction, making it irrational.

Examples of irrational numbers:

• $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ (but not $\sqrt{4} = 2$ or $\sqrt{16} = 4$)
• $\pi = 3.141592...$ (note: $\frac{22}{7}$ is just an approximation)

Real numbers (R)

Real numbers include all the number types we have studied so far.

Definition: Real numbers are the combination of all rational and irrational numbers.

The set of real numbers is denoted by the letter R. Every point on the number line corresponds to a real number.

The complete relationship: $N \subset Z \subset Q \subset R$

This means:

• All natural numbers are integers
• All integers are rational numbers
• All rational numbers are real numbers

Identifying number types

To determine what type of number you're dealing with, follow this systematic approach:

1. Check if it's a natural number: Is it a positive counting number?
2. Check if it's an integer: Does it include zero or negatives?
3. Check if it's rational: Can it be written as a fraction $\frac{a}{b}$?
4. If not rational, it's irrational

Exam tips

Remember!