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The way I estimate square roots, is by finding the closest lowest perfect square, then adding decimals to the number to determine the estimation. How do I estimate the square root of a number with decimals using this method using the number $3.51$ for example?

I know how to estimate for whole numbers using this method but I'm confused on the decimal numbers. When I used this same method on $3.51$, I tested $1.8^2$ and $1.9^2$ to see which one was closer. $1.9^2$ $(3.61)$ was closer to $3.51$ than $1.8^2$ $(3.24)$. But in my math book, the answer was that $1.8$ was the correct estimate.

How and why?

Haseen Siddiqui
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jul 16 '23 at 04:54
  • There is a method for finding square roots of any number upto certain digits ig – MathStackexchangeIsNotSoBad Jul 16 '23 at 05:05
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    If you are presenting the problem accurately, and if your math book didn't explicitly say it wanted an underestimate, then the book is wrong. $\sqrt{3.51}=1.873\dotsc$, so $1.9$ is the better estimate. – Gerry Myerson Jul 16 '23 at 06:17
  • Do you have anything to say about the answer and the comments that have been posted, Haseen? – Gerry Myerson Jul 18 '23 at 13:15

1 Answers1

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You are looking for the square root of $3.51$ and you know that it's somewhere between $1.8$ (square $3.24$) and $1.9$ (square $3.61$).

As the square of $1.9$ is the closest, you start from there:

$3.51 = (1.9 - \alpha)^2 = 1.9^2 - 3.8 \alpha + \alpha^2$

As $\alpha$ is quite small, let's drop it and continue the approximation:

$$3.51 \approx 3.61 - 3.8 \alpha$$ $$\alpha \approx \frac{3.61 - 3.51}{3.8}$$ ($3.8$ is close to $4$) $$\alpha \approx \frac{0.10}{4} \approx 0.025$$ Hence: $1.9 - 0.025 = 1.875$ is a better approximation than $1.9$.

Dominique
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