0.4 Irrational numbers and rounding off

Introduction

You have seen that repeating decimals may take a lot of paper and ink to write out. Not only is that impossible, but writing numbers out to many decimal places or a high accuracy is very inconvenient and rarely gives practical answers. For this reason we often estimate the number to a certain number of decimal places or to a given number of significant figures , which is even better.

Irrational numbers

Irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers. This means that any number that is not a terminating decimal number or a repeating decimal number is irrational. Examples of irrational numbers are:

If you are asked to identify whether a number is rational or irrational, first write the number in decimal form. If the number is terminated then it is rational. If it goes on forever, then look for a repeated pattern of digits. If there is no repeated pattern, then the number is irrational.

When you write irrational numbers in decimal form, you may (if you have a lot of time and paper!) continue writing them for many, many decimal places. However, this is not convenient and it is often necessary to round off.

Investigation : irrational numbers

Which of the following cannot be written as a rational number?

Remember : A rational number is a fraction with numerator and denominator as integers. Terminating decimal numbers or repeating decimal numbers are rational.

1. $\pi =3,14159265358979323846264338327950288419716939937510...$
2. $\mathrm{1,4}$
3. $1,618033989...$
4. $100$

Rounding off

Rounding off or approximating a decimal number to a given number of decimal places is the quickest way to approximate a number. For example, if you wanted to round-off $2,6525272$ to three decimal places then you would first count three places after the decimal and place a $|$ between the third and fourth number after the decimal.

All numbers to the right of the $|$ are ignored after you determine whether the number in the third decimal place must be rounded up or rounded down. You round up the final digit if the first digit after the $|$ was greater than or equal to 5 and round down (leave the digit alone) otherwise. In the case that the first digit before the $|$ is 9 and you need to round up, then the 9 becomes a 0 and the second digit before the $|$ is rounded up.

So, since the first digit after the $|$ is a 5, we must round up the digit in the third decimal place to a 3 and the final answer of $2,6525272$ rounded to three decimal places is

Summary

• Irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers.
• For convenience irrational numbers are often rounded off to a specified number of decimal places

End of chapter exercises

1. Write the following rational numbers to 2 decimal places:
   1. $\frac{1}{2}$
   2. 1
   3. $0,11111\overline{1}$
   4. $0,99999\overline{1}$
2. Write the following irrational numbers to 2 decimal places:
   1. $3,141592654...$
   2. $1,618033989...$
   3. $1,41421356...$
   4. $2,71828182845904523536...$
3. Use your calculator and write the following irrational numbers to 3 decimal places:
   1. $\sqrt{2}$
   2. $\sqrt{3}$
   3. $\sqrt{5}$
   4. $\sqrt{6}$
4. Use your calculator (where necessary) and write the following numbers to 5 decimal places. State whether the numbers are irrational or rational.
   1. $\sqrt{8}$
   2. $\sqrt{768}$
   3. $\sqrt{100}$
   4. $\sqrt{0,49}$
   5. $\sqrt{0,0016}$
   6. $\sqrt{0,25}$
   7. $\sqrt{36}$
   8. $\sqrt{1960}$
   9. $\sqrt{0,0036}$
   10. $-8\sqrt{0,04}$
   11. $5\sqrt{80}$
5. Write the following irrational numbers to 3 decimal places and then write them as a rational number to get an approximation to the irrational number. For example, $\sqrt{3}=1,73205...$ . To 3 decimal places, $\sqrt{3}=1,732$ . $1,732=1\frac{732}{1000}=1\frac{183}{250}$ . Therefore, $\sqrt{3}$ is approximately $1\frac{183}{250}$ .
   1. $3,141592654...$
   2. $1,618033989...$
   3. $1,41421356...$
   4. $2,71828182845904523536...$