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Question

Visualise 4.¯¯¯¯¯¯26 on the number line, upto 4 decimal places, that is upto 4.2626.

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Solution

We locate 4.¯¯¯¯¯¯26 on the number line, by the process of successive magnification. This has been illustrated in Figure.Step 1: First we note that 4.¯¯¯¯¯¯26 lies between 4 and 5Step 2: Divide the portion between 4 and 5 into 10 equal parts and use a magnifying glass to visualise that 4.¯¯¯¯¯¯26 lies between 4.2 and 4.3.Step 3: Divide the portion between 4.2 and 4.3 into 10 equal parts and use a magnifying glass to visualise that 4.¯¯¯¯¯¯26 lies between 4.262 and 4.263.Step 5: Divide the portion between 4.262 and 4.263 into 10 equal parts and use a magnifying glass to visualise that 4.¯¯¯¯¯¯26 lies between 4.2625 and 4.2627.We note that 4.¯¯¯¯¯¯26 is visualised closer to 4.263 than to 4.262.The same procedure can be used to visualise a real number with a non-terminating and non-recurring decimal expansion on the number line to a required accuracy.From the above discussions and visualisations we conclude again that every real number is represented by a unique point on the number line. Further every point on the number line represents one and only one real number.

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