Rational and irrational numbers

A number is described as rational if it can be written as a fraction (one integer divided by another integer). The decimal form of a rational number is either a terminating or a recurring decimal. Examples of rational numbers are 17, -3, and 12.4. Other examples of rational numbers are 54=1.25 (terminating decimal) and 23=0.6˙ (recurring decimal).

A number is irrational if it cannot be written as a fraction. The decimal form of an irrational number does not terminate or recur. Examples of irrational numbers are π = 3.14159… and √2 = 1.414213...

Surds

A surd is an expression that includes a square root, cube root or another root symbol. Surds are used to write irrational numbers precisely – because the decimals of irrational numbers do not terminate or recur, they cannot be written exactly in decimal form.

Example

This square has an area of 3 m². Write down the exact length of the side of the square.

The length of the side is √3 m.

This answer is in surd form. It is irrational and it is said to be "in exact form". A decimal answer, such as 1.73 (2 decimal places), is not exact. Even 1.732050807568877 is not exact. When an answer is required in exact form, you must write it as a surd, ideally simplifying it if possible.

Simplifying surds

Surds can be simplified if the number in the root symbol has a square number as a factor.

Learn these general rules:

• ab=a×b
• a×a=a
• ab=ab=a÷b

Examples

Simplify √12.

12=4×3, so we can write 12=(4×3)=4×3

4=2 so 12=23

Simplify 10×5.

10×5=50

50=25×2, so we can write 50=25×2=25×2=52

Simplify 126.

126 = 126 = 12÷6=2

Question

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