Real Numbers — Part 2: Irrational numbers and proofs: visual lesson, CBSE Class 10 Mathematics (Standard 041 / Basic 241)

Lesson

Real Numbers, Part 2: Irrational numbers

$\sqrt2$ is a real point on the number line — but no fraction lands exactly on it.

In this lesson you will

Distinguish rational and irrational numbers with examples
Use the fact that if a prime divides a², it divides a
Prove by contradiction that √2, √3 and √5 are irrational
Prove that numbers such as 4 − 3√2 are irrational
Decide whether sums and products involving surds are rational

The circle swings the hypotenuse onto the line: $\sqrt2 \approx 1.414$.

Concept

Rational and irrational numbers

A number is rational if it can be written as $\dfrac pq$ with $p, q$ integers and $q \ne 0$ . A real number that cannot be written this way is irrational.

Rationals and irrationals together make up the real numbers.
Number	Rational?	Why
$-7$	yes	$-7 = \dfrac{-7}{1}$
$0.75$	yes	$\dfrac34$
$\sqrt{49}$	yes	equals $7$
$\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$	no	proved in this lesson
$\pi$	no	$\dfrac{22}{7}$ is only an approximation

Diagram

Where does √2 come from?

The diagonal of a unit square is $\sqrt2$ — a length you can draw but not write as a fraction.

A square tile of side $1$ m has a diagonal of $\sqrt{1^2 + 1^2} = \sqrt2$ m.

Cookies