A rational number is a number which can be written as a fraction where both the numerator and denominator are integers. An irrational number is a number which cannot be written as a fraction of two integers.
We can write any terminating or recurring decimal as a fraction, therefore these are rational numbers. However, decimals which are neither terminating nor recurring are irrational numbers.
You might be familiar with one irrational number already: \pi. Like all other irrational numbers, \pi really does go on forever without repeating itself. We say therefore that it doesn't terminate, or repeat.
Another number that is famously irrational is \sqrt{2}. In fact, the square root of most numbers are irrational. If a root is irrational it is called a surd. The square roots of perfect squares are rational, \sqrt{1},\, \sqrt{4},\,\sqrt{9},\,...
Is \sqrt{35} rational or irrational?
A rational number is a number which can be written as a fraction where both the numerator and denominator are integers.
An irrational number is a number which cannot be written as a fraction of two integers.
A surd is a square root which is irrational.
As mentioned before the value of \pi, square roots of non-perfect squares and cube roots of non-perfect cubes are examples of irrational numbers. The value of \sqrt 2 is a non-terminating, non-repeating decimal which makes it an irrational number. However, there are ways we can approximate the values without using a calculator.
One way is to consider the nearest integer value as a way to estimate or check our work.
\text{If } a \lt b \text{,}
\text{then } \sqrt{a} \lt \sqrt{b}
Let's say we have a square root, \sqrt{40}. If we ask ourselves what are the closest square numbers that are bigger and smaller than 40, then we'll find that they're 36 and 49. So then we have 36 \lt 40 \lt 49, which leads us to say that \sqrt{36} \lt \sqrt{40} \lt \sqrt{49}. And if we evaluate that further we get 6 \lt \sqrt{40} \lt 7, so we've managed to narrow this square root down to somewhere between 6 and 7.
Represent the following values on the number line:
We can estimate the values of irrational numbers and represent them on the number line.
To estimate square roots or cubes we can follow these steps:
Determine the closest squares or cubes that are bigger and smaller than the number
Evaluate the square root or cube root to find between what two integers the given number lies.
Estimating square and cube roots:
If a \lt b \text{,}
then, \sqrt{a} \lt \sqrt{b} and
\sqrt[3]{a} \lt \sqrt[3]{b}
We know that \sqrt{40} is estimated between 6 and 7 because \sqrt{36} <\sqrt{40} < \sqrt{49}.
What if we wanted to approximate \sqrt{40} further? There's a method for that as well. Once you know what integers the square root lies between, you can find the decimal part by using the following formula:
\text{approximation of decimal part } = \dfrac{\text{number inside square root - closest smaller square}}{\text{closest bigger square - closest smaller square}}
Let's use our \sqrt{40} example. The closest smaller square than 40 is 36 while the the closest bigger square is 49. Using the formula:
\displaystyle \text{approximation of decimal part } | \displaystyle = | \displaystyle \dfrac{40 - 36}{49 - 36} | Substite the values in the equation |
\displaystyle = | \displaystyle \dfrac{4}{13} | Evaluate the subtraction | |
\displaystyle \approx | \displaystyle 0.3 | Evaluate the division |
Since \sqrt{40} is estimated between 6 and 7 and the approximation of the decimal part is 0.3, we can say that:
\sqrt{40}\approx 6.3
If you plug this square root into a calculator, you'll see that it is indeed rounded to 6.3. However this method only works well on larger numbers, and the bigger they are the better they'll work. Try and see the difference between using this on say, \sqrt{2} and \sqrt{300}.
We can use the same process for cube roots. Let's take a closer look at the following worked question on how to approximate the decimal part of irrational numbers.
Approximate \sqrt[3]{95} to the nearest tenth without using a calculator.
We can approximate the values of irrational numbers by using the decimal approximation equation for the decimal part, especially of square roots of non perfect squares and cube roots of non perfect cubes.
\text{ approximation of decimal part} = \dfrac{\text{number inside root sign - closest smaller square/cube}}{\text{closest bigger square/cube - closest smaller square/cube}}