The World of Numbers Mind Map

The number system is a family of number sets arranged from counting numbers to all real numbers: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

When asked to classify a number, place it in the smallest correct set first and then mention the larger sets it belongs to.

Integers are numbers without fractional or decimal parts, including negative numbers, zero, and positive numbers.

In integer simplification, handle brackets and sign changes before doing the final arithmetic.

A rational number is any number that can be written as p/q, where p and q are integers and q is not zero.

For number-line questions, first locate the two nearest integers and then divide the interval according to the denominator.

Density of rational numbers means that between any two rational numbers, there are infinitely many rational numbers.

To find one rational number between two rationals, use the average method; to find many, make denominators same and insert numerators.

An irrational number is a real number that cannot be written as p/q, where p and q are integers and q is not zero.

Check whether the number under the square root is a perfect square before deciding whether the square root is rational or irrational.

Geometric construction of irrationals is a method of locating values like √2, √3, and √5 on the number line using right triangles and the Pythagoras relation.

For construction questions, mention the right angle, the unit lengths used, and the hypotenuse obtained by Pythagoras relation.

A rational number has either a terminating decimal expansion or a non-terminating repeating decimal expansion.

Before long division, reduce the fraction to lowest form and factorise the denominator.

A non-terminating non-repeating decimal is a decimal that goes on forever without a repeating block of digits, and it represents an irrational number.

Look for a fixed repeated block. If a block repeats forever, the decimal is rational; if no block repeats and it does not terminate, it is irrational.