Things you might have done

Without using a calculator, find a reasonable estimate of: \[1 \div \sqrt{2}\]

There are several approaches that could have been used, some of which are discussed below.

\(1 \div \sqrt{2}\) can also be written as \(\frac{1}{\sqrt{2}}\) or \(\sqrt{\frac{1}{2}}.\)

If we call our answer \(a\), then \(\sqrt{\frac{1}{2}} = a\), and \(\frac{1}{2} = a^2\), so we are looking for a number that squares to \(0.5\). Writing down some decimal versions of square numbers gives us \(0.7^2 = 0.49\) and \(0.8^2 = 0.64\), so we see the answer must be close to \(0.7\). Calculating \(0.71^2\) gives us \(0.5051\), so \(\sqrt{\frac{1}{2}} \approx 0.71.\)

As \(\sqrt{1} < \sqrt{2} < \sqrt{4}\) we know that \(\sqrt{2}\) must lie between \(1\) and \(2\). Using some square numbers will help in the estimation process as we can write down \(1.3^2 = 1.69\), \(1.4^2 = 1.96\) and \(1.5^2 = 2.25\) showing us that \(\sqrt{2}\) must be just slightly bigger than \(1.4\).

If we have an approximate value for \(\sqrt{2}\) then we can use it in a division. If we use \(\sqrt{2} \approx 1.4\) then we can write the approximation as a fraction to use in the division.

\[\frac{1}{\sqrt{2}} \approx \frac{1}{\frac{7}{5}} = \frac{5}{7}\]

and some short division gives us \(1 \div \sqrt{2} \approx 0.714.\)

Our approximation of \(\sqrt{2}\) is accurate to \(1\) decimal place. How accurate do you think the answer of \(0.714\) will be?

To improve our answer we would need to know a better approximation for \(\sqrt{2}\). If we use \(\sqrt{2} \approx 1.414\) we could write

\[\frac{1}{\sqrt{2}} \approx \frac{1}{1.414} = \frac{1}{\frac{1414}{1000}} = \frac{1000}{1414}.\]

Although our approximations improve using this method, the division we have to calculate will become harder.

Remember \(\sqrt{2}\) is an irrational number so cannot be written as a fraction. Using fractions to approximate the value of \(\sqrt{2}\) more accurately, will require larger numbers in the numerators and denominators.

We should see from the method above that we can’t rewrite \(\sqrt{2}\) in a way that gives us a simple number to divide by. What can we do to the whole fraction of \(\frac{1}{\sqrt{2}}\) so the denominator becomes an easier number to divide by?

We need to keep the division the same, so we are looking for an equivalent fraction. If we multiply by \(\frac{\sqrt{2}}{\sqrt{2}}\) then we will get a rational number in the denominator.

\[\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]

In this form we can see that our division becomes much simpler, and with an approximation of \(1.414\) for \(\sqrt{2}\) we could quickly get an answer of \(0.707\).

We have done what we call ‘rationalising the denominator’ where any fractions that involve surds in the denominator are rewritten so that the denominator only involves rational numbers. This process makes it easier to approximate the size of numbers as we have just seen, as well as being the standard way that answers are often written in mathematics.

We can rationalise algebraic expressions as well, and this can be a helpful process to give you simpler functions or equations to work with.

What can your thinking about the original problem tell you about other divisions of the form \(1 \div \sqrt{n}\)?

  • Can you select \(n\) such that you can write down the value of the division calculation more easily?

  • Can you select \(n\) such that your answer to the original problem is useful in finding the value of the division?

If we think about the different divisions we would get if we took \(n\) to be a value between \(1\) and \(9\) we see there will be some simple calculations when \(n = 1, 4\) and \(9\), i.e. when \(n\) is a square number. We have left \(n = 3, 5, 6, 7\) and \(8\). We could estimate these as we did with \(\sqrt{2}\) above, but if we look at \(\sqrt{8}\), we might recognise that it can be written as \(\sqrt{4\times 2}\) or \(2\sqrt{2}\), and then we could use our original answer to help us.

Without using a calculator, find a reasonable estimate of: \[\sqrt{50} \div 10\]

Using square numbers, we know \(\sqrt{49} = 7\), so \(\sqrt{50} \approx 7.1.\) We could improve this estimate, but already we can see that \(\sqrt{50} \div 10 \approx 0.71.\)

Compare this to your original problem: \(1 \div \sqrt{2}\).

  • What do you notice about your approach?

  • What do you notice about your answers?

\(\sqrt{50} \div 10\) was a simple calculation to do, as the square root was relatively easy to approximate, and the divisor was a whole number, i.e. this fraction is already in a rationalised form.

You might have also noticed that your answer is very similar, or the same, as your answer for \(1 \div \sqrt{2}.\) Can you see why this is? What happens if you simplify \(\frac{\sqrt{50}}{10}\)?