A Peek Beyond | Class 7 Mathematics Notes

Irrational Numbers

This section introduces irrational numbers, which are numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are non-terminating and non-repeating, unlike rational numbers. The section explains that numbers like √2, √3, and π are irrational. It discusses the historical context of irrational numbers, such as the discovery of √2's irrationality by the Pythagoreans. The section explains that irrational numbers fill the gaps on the number line that rational numbers cannot cover, making the number system continuous. It also explains that irrational numbers cannot be exactly represented as fractions or decimals but can be approximated to any desired degree of accuracy. The section includes examples of irrational numbers and their approximate decimal values. It also explains how to identify irrational numbers and distinguishes them from rational numbers based on decimal expansions. The section emphasizes the importance of irrational numbers in mathematics and real-life applications, such as geometry and measurement.

📊 Diagram: Number line showing rational numbers as points and irrational numbers filling the gaps between them. Decimal expansions of √2 and π are shown to illustrate non-terminating, non-repeating decimals.

🧪 Activity: Activity to approximate the value of √2 by successive decimal approximations and observe the non-terminating, non-repeating nature.

🔗 Connection: This section leads to the next, which discusses the real number system as a whole, combining rational and irrational numbers.