Question

Choose three intervals on the real number line that contain both rational and irrational numbers. Do you think that any given inte number line contains both rational and irrational numbers? Explain.

Answer 1

Any real number line ranges from negative infinity to positive infinity. A real number number line consists of all the rational and irrational numbers. Let us take three intervals which contain both the rational and irrational numbers.

First interval:  [3,4]

Since every integer is a rational number, 3 and 4 are both rational. In this interval there occurs the value of π (3.14159..) which is an irrational number.

Second interval :  [0,2]

0 and 2 are integers and hence are rational. In this interval, occurs √2 (1.41421...) is an irrational number.

Third interval : [2,3]

In this interval, Eulers number 'e' lie whose value is (2.718281.. )

Hence we can conclude that, there occurs an irrational number between any two rational number.