Question 10 of 58beginner💡 UnderstandShort Answer2 marks

Can you explain why we need q≠0q \neq 0 in the definition of a rational number?

Correct Answer

A rational number is defined in the form pq\frac{p}{q}, and the set of rational numbers is denoted by Q\mathbb{Q}, which stands for quotient. Since a quotient represents division, having q=0q=0 would make the division undefined.

Moreover, to represent a rational number pq\frac{p}{q} on the number line, we must divide the unit interval into qq equal parts. It is logically impossible to divide a line segment into 0 equal parts. Therefore, the condition q≠0q \neq 0 is essential.

Exercise: Think and Reflect | Q: 1 | (Page 7)
For More Understanding

Explanation

The textbook explains that the symbol Q\mathbb{Q} for rational numbers stands for 'quotient', which inherently requires a non-zero divisor. Additionally, the text explicitly details the method for representing a rational number on the number line, which involves dividing the unit interval into qq equal parts. Both of these context points directly rely on qq being a non-zero integer to be mathematically and logically valid.

Solution Steps

  1. Step 1: Identify that Q\mathbb{Q} stands for quotient, which implies division by qq.

  2. Step 2: Note that representing pq\frac{p}{q} on a number line requires dividing the unit interval into qq equal parts.

  3. Step 3: Conclude that since division by 0 is undefined and we cannot divide an interval into 0 parts, we must have q≠0q \neq 0.

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