Drag or tap the number line to explore a real number.x = √2 · irrational

Real Numbers

Number line, sets, irrationals, intervals, and distance

Workspace

Real-number explorer

Selected number

Value√2

Simulation speed1×

Zoom1×

Interaction sounds

Fraction builder

Numerator p

Denominator q

x = p ÷ q, where p and q are integers and q ≠ 0

Interval laboratory

A start−4

A end3

B start0

B end7

Live classification

Exact form√2

Decimal1.414214…

Smallest setIrrational

Distance from 01.414214

Object focus

Tap a marker, interval, set region, or construction line.

Teacher presets

Teacher narration

Language

Narrator

Playback speed1×

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Chapter 1 of 80%

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Chapters

Learning objectives

Place and compare real numbers on a number line.
Recognize N ⊂ W ⊂ Z ⊂ Q ⊂ R.
Distinguish rational and irrational decimals.
Construct √2 geometrically.
Interpret intervals and absolute value.

Key relationships

Rational: x = p ÷ q, q ≠ 0.

Pythagorean: c² = a² + b².

Midpoint: m = (a + b) ÷ 2.

Absolute value: |x| = distance from zero.

Suggested experiments

Plot 1/3 and compare exact and decimal forms.
Zoom around √2 and observe its fixed position.
Change interval endpoints and compare union with intersection.
Move x from −4 to +4 and watch |x|.

Questions for exploration

Why is every integer also rational?
Can two different real numbers occupy one point?
Why does an open endpoint not belong to an interval?

Seven-question check

Answer all seven questions.

Teacher whiteboard

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