Real Numbers
All rational and irrational numbers; the continuous number line.
Key idea
All rational and irrational numbers; the continuous number line.
In depth
The real numbers include all rationals and irrationals. They fill the number line completely, with no gaps. Every point on the line corresponds to exactly one real number.
Worked example
−2, 0, 3/5, √2, and π are all real numbers.
Formula
Real-world uses
- Modelling continuous quantities such as time, temperature and length in science and engineering
Common mistake
Forgetting that square roots of negative numbers are not real.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Real Numbers?
All rational and irrational numbers; the continuous number line. It is a topic at the advanced level in Arithmetic & Number Sense.
What is the formula for Real Numbers?
The key formula is ℝ.
Where is Real Numbers used in real life?
Real-world examples: Modelling continuous quantities such as time, temperature and length in science and engineering.
What should I learn before Real Numbers?
Make sure you are comfortable with Irrational Numbers, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Real Numbers?
Forgetting that square roots of negative numbers are not real.
How can I practise Real Numbers?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Real Numbers?
Move on to Square Roots of Non-squares next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Square Roots of Non-squares – Approximate or simplify surds like √50 = 5√2.