Recognise real-number families and compare values
Classify natural, integer, prime, square, rational, irrational and real numbers · Use equality and order symbols with exact values
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Whole-number arithmetic
- Fractions and square roots
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Classify natural, integer, prime, square, rational, irrational and real numbers
- Use equality and order symbols with exact values
Families can overlap
Use natural numbers 1, 2, 3, … in this lesson; conventions about including zero must be stated. Integers also include zero and negative whole numbers. Prime numbers are positive integers greater than 1 with exactly two positive divisors. A square number is the square of an integer.
A rational number can be written p/q for integers p and q with q ≠ 0. Terminating and recurring decimals are rational. An irrational number cannot be written this way. Both rational and irrational numbers are real; a negative integer can also be rational.
Compare the actual values
A square-root sign means the non-negative root. √9 = 3 is rational, although √2 is irrational. To place √10, compare 3² = 9 and 4² = 16, giving 3 < √10 < 4.
The symbols ≤ and ≥ allow equality; < and > do not. Negative numbers farther left are smaller. A rounded display of an irrational number is an approximation, not an exact equality.
Sources for this lesson
Pause and explain
Is √25 irrational?
Worked example
Classify −4, 0.125, √16 and √7. Order them.
Show the worked solution
- −4 is an integer and rational; 0.125 = 1/8 is rational.
- √16 = 4 is a natural, integer, square and rational number. √7 is irrational and lies between 2 and 3.
- All four are real. Thus −4 < 0.125 < √7 < √16.
Answer −4 < 0.125 < √7 < √16; only √7 is irrational.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Classify −6, 3/4 and √3 as rational or irrational, then order them.
Give me a hint
Place √3 between two consecutive integers.
Compare my reasoning
- −6 and 3/4 are rational.
- 1² < 3 < 2², so 1 < √3 < 2; √3 is irrational.
−6 < 3/4 < √3; the first two are rational, the last irrational.
Look for these in your work
- Justify each family
- Use exact inequalities
Which of 1, 2, 9 and 15 are prime? Which are square numbers?
Give me a hint
Count positive divisors; check integer squares.
Compare my reasoning
- 2 has divisors 1 and 2. 1 has one divisor; 9 and 15 have more than two.
- 1 = 1² and 9 = 3².
Only 2 is prime; 1 and 9 are square numbers.
Look for these in your work
- Exclude 1 from primes
- Allow overlapping families
A calculator shows √5 as 2.2360679. Is that display an exact rational value for √5? Bound √5 between integers.
Give me a hint
Separate the stored decimal from the quantity it approximates.
Compare my reasoning
- The finite display is rational; √5 is irrational and is only approximated by the display.
- 2² < 5 < 3².
No. √5 ≈ 2.2360679 and 2 < √5 < 3.
Look for these in your work
- Distinguish exact and rounded values
- Support the bound by squaring
Common mistakes
- A root symbol does not automatically mean irrational.
- 1 is not prime; it has only one positive divisor.
- Do not write an approximate decimal as an exact equality.
What can you explain now?
Is √25 irrational?
Compare with the explanation
No; √25 = 5 is rational.
The root is exactly an integer, which can be written 5/1.
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Make it stick
Is √25 irrational?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.