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Real Numbers

25 دقيقة

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🔢 What you'll learn

The real numbers are the address of every point on the number line. Sorting them into rational and irrational — and knowing how to place a root — is the foundation of all later algebra.

Rational number

Any number that can be written as a fraction

\dfrac{a}{b}

with integers

a,b

and

b\neq0

. Its decimal either stops or repeats (e.g.

0.75

0.\overline{3}

Irrational number

A number that cannot be written as such a fraction; its decimal never stops and never repeats — for example

\sqrt2

and

\pi

The set of real numbers

\mathbb{R}

Together the rationals and irrationals fill the whole number line. Every real number is exactly one point on it.

\mathbb{R}=\mathbb{Q}\cup(\mathbb{R}\setminus\mathbb{Q})

Placing a root

To locate

\sqrt n

, trap

n

between the two nearest perfect squares. For example

9<11<16

Compare and order

To compare unlike forms, use decimal approximations — but keep exact radical forms in your final answer whenever you can.

Draw the number line around the value.

▶ Watch a root get trapped between two consecutive integers.

Use the labels and colours to connect the rule with the picture.

🧩 Worked examples

أمثلة محلولة

Worked exampleClassify a number

Classify $\sqrt{49}$ and $\sqrt{10}$ .

Worked exampleBracket a root

Between which integers does $\sqrt{27}$ lie?

Worked exampleCompare exactly

Compare $3\sqrt2$ and $4$ .