What is the value of ?

Question

Homi · Accepted Answer

The square root of a number is a value that, when multiplied by itself, gives the original number.

Step 1 — Understanding the nature of $\sqrt{2}$

Let $x$ be the value of $\sqrt{2}$ . This means $x \times x = 2$ . Let us test some whole numbers. We know that $1 \times 1 = 1$ . We also know that $2 \times 2 = 4$ . So, the value of $\sqrt{2}$ must be between 1 and 2. It is not a whole number. We will find its approximate decimal value.

Step 2 — Finding the value using the long division method

We use the long division method. We write 2 as $2.000000$ . We pair digits from the decimal point. We find the largest number whose square is $\le 2$ . This number is 1, because $1^2 = 1$ . We write 1 as the first digit of the square root. We subtract 1 from 2.

$2 - 1 = 1$

We bring down the next pair of zeros (00). Now we have 100. We double the quotient (1), which gives 2. We place a blank digit next to 2, forming $2\_$ . We multiply $2\_$ by the same digit. We want $2\_ \times \_ \le 100$ . We find that $24 \times 4 = 96$ . So, the next digit is 4. We write 4 after the decimal point in the quotient. We subtract 96 from 100.

$100 - 96 = 4$

We bring down the next pair of zeros (00). Now we have 400. We double the current quotient (14), ignoring the decimal point. This gives $14 \times 2 = 28$ . We place a blank digit next to 28, forming $28\_$ . We multiply $28\_$ by the same digit. We want $28\_ \times \_ \le 400$ . We find that $281 \times 1 = 281$ . So, the next digit is 1. We write 1 in the quotient. We subtract 281 from 400.

$400 - 281 = 119$

We bring down the next pair of zeros (00). Now we have 11900. We double the current quotient (141), ignoring the decimal point. This gives $141 \times 2 = 282$ . We place a blank digit next to 282, forming $282\_$ . We multiply $282\_$ by the same digit. We want $282\_ \times \_ \le 11900$ . We find that $2824 \times 4 = 11296$ . So, the next digit is 4. We write 4 in the quotient. We subtract 11296 from 11900.

$11900 - 11296 = 604$

The process continues indefinitely. So, $\sqrt{2}$ is approximately 1.414.

$\boxed{\sqrt{2} \approx 1.414}$

Diagram 1

Answer

(i) The value of $\sqrt{2}$ is approximately 1.414.

Question 9

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