Find three rational numbers between and .

Question

Homi · Accepted Answer

Understand the Question

Any terminating decimal is a rational number because it can be expressed in the form $\dfrac{p}{q}$ , where $p, q \in \mathbb{Z}$ and $q \neq 0$ .
To find numbers between $3.1415$ and $3.1416$ , we can append extra decimal digits to $3.14150$ so that the resulting values remain strictly less than $3.14160$ .

Step 1 · Find Three Rational Numbers

Rewrite the given numbers with an additional decimal place: $3.1415 = 3.14150$ $3.1416 = 3.14160$

Any terminating decimals between $3.14150$ and $3.14160$ are rational numbers. Three such numbers are: $\text{First rational number} = 3.14151$ $\text{Second rational number} = 3.14152$ $\text{Third rational number} = 3.14153$

Answer

$3.14151$ , $3.14152$ , and $3.14153$

Common Mistakes

Exceeding the Upper Bound: Appending digits to $3.1416$ (e.g., writing $3.14161$ ), which makes the number strictly greater than $3.1416$ instead of lying between the two given numbers.
Rational vs. Irrational Confusion: Terminating decimals are always rational numbers. Only non-terminating and non-repeating decimals are irrational.

Question 5

More questions in Exercise 3.4