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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:

2 , 3 , 4 3 , π , 1 + 5 2 1 , 618 033 989 2 , 3 , 4 3 , π , 1 + 5 2 1 , 618 033 989
(1)

Tip:

When irrational numbers are written in decimal form, they go on forever and there is no repeated pattern of digits.

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. π = 3 , 14159265358979323846264338327950288419716939937510 ... π = 3 , 14159265358979323846264338327950288419716939937510 ...
  2. 1,4
  3. 1 , 618 033 989 ... 1 , 618 033 989 ...
  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 2,65252722,6525272 to three decimal places then you would first count three places after the decimal.

2 , 652 | 5272 2 , 652 | 5272
(2)

All numbers to the right of || 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 or equal to 5 and round down (leave the digit alone) otherwise. In the case that the first digit before the || is 9 and the || you need to round up 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 2,65252722,6525272 rounded to three decimal places is

2 , 653 2 , 653
(3)

Exercise 1: Rounding-Off

Round-off the following numbers to the indicated number of decimal places:

  1. 12099=1,2121212121˙2˙12099=1,2121212121˙2˙ to 3 decimal places
  2. π=3,141592654...π=3,141592654... to 4 decimal places
  3. 3=1,7320508...3=1,7320508... to 4 decimal places

Solution

  1. Step 1. Determine the last digit that is kept and mark the cut-off point with ||. :
    1. 120 99 = 1 , 212 | 121212 1 ˙ 2 ˙ 120 99 = 1 , 212 | 121212 1 ˙ 2 ˙
    2. π = 3 , 1415 | 92654 ... π = 3 , 1415 | 92654 ...
    3. 3 = 1 , 7320 | 508 ... 3 = 1 , 7320 | 508 ...
  2. Step 2. Determine whether the last digit is rounded up or down. :
    1. The last digit of 12099=1,212|1212121˙2˙12099=1,212|1212121˙2˙ must be rounded-down.
    2. The last digit of π=3,1415|92654...π=3,1415|92654... must be rounded-up.
    3. The last digit of 3=1,7320|508...3=1,7320|508... must be rounded-up.
  3. Step 3. Write the final answer. :
    1. 12099=1,21212099=1,212 rounded to 3 decimal places
    2. π=3,1416π=3,1416 rounded to 4 decimal places
    3. 3=1,73213=1,7321 rounded to 4 decimal places

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. 1212
    2. 1
    3. 0,111111¯0,111111¯
    4. 0,999991¯0,999991¯
    Click here for the solution
  2. Write the following irrational numbers to 2 decimal places:
    1. 3,141592654...3,141592654...
    2. 1,618033989...1,618033989...
    3. 1,41421356...1,41421356...
    4. 2,71828182845904523536...2,71828182845904523536...
    Click here for the solution
  3. Use your calculator and write the following irrational numbers to 3 decimal places:
    1. 22
    2. 33
    3. 55
    4. 66
    Click here for the solution
  4. Use your calculator (where necessary) and write the following irrational numbers to 5 decimal places:
    1. 88
    2. 768768
    3. 100100
    4. 0,490,49
    5. 0,00160,0016
    6. 0,250,25
    7. 3636
    8. 19601960
    9. 0,00360,0036
    10. -80,04-80,04
    11. 580580
    Click here for the solution
  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, 3=1,73205...3=1,73205.... To 3 decimal places, 3=1,7323=1,732. 1,732=17321000=11832501,732=17321000=1183250. Therefore, 33 is approximately 11832501183250.
    1. 3,141592654...3,141592654...
    2. 1,618033989...1,618033989...
    3. 1,41421356...1,41421356...
    4. 2,71828182845904523536...2,71828182845904523536...
    Click here for the solution

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