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Irrational Numbers

By Cheryl Lewandoski

Learn about the difference between rational and irrational numbers. Find the definition, example problems, and practice problems at Thinkster Math.

Why is this concept useful?

Irrational numbers provide a continuum of real numbers. They make physics, geometry, and engineering possible to do.

Where does this concept fit into the curriculum?

6th Grade Rational numbers are extended from the understanding of fractions. 8th Grade students' understanding of the number system is expanded to include Irrational numbers.

What is an irrational number?

Rational numbers are integers that can be written as simple fractions in the form of p/q where q is not zero. Irrational numbers are real numbers that can not be written as a simple fraction. Irrational numbers as decimals neither terminate or repeat.

How can we use the concept?

  • Estimating values of non-perfect squares and cubes
  • Finding area and perimeters of circles and circular objects
  • Auto manufacturers - turning circumference, tires, calibration of speedometers
  • Pendulum clocks for accurate time keeping
  • Trigonometric functions
  • Physics
  • Force Calculations
  • Calculus
  • Space travel

Identifying irrational numbers:

5,3,12,18,123:\sqrt{5} ,\sqrt{3} ,\sqrt{12} ,\sqrt{18} ,\sqrt{123} :

Irrational numbers - not perfect squares

4.58276… irrational

π\pi

Pi is irrational - does not repeat, does not terminate

Identifying rational numbers.

7.84: rational, terminating decimal, can be written as a ratio

45\frac{4}{5}
: rational number - written as a ratio

16\sqrt{16}

is a rational number because 16 is a perfect square. The square root of 16 is 4.

Sample Math Problems

1.

Choose the irrational numbersA.  25    B.   1.7834…       C.  273      D.   1.47‾ \begin{array}{l} Choose\ the\ irrational\ numbers\\ A.\ \ \sqrt{25} \ \ \ \ B.\ \ \ 1.7834\dotsc \ \ \ \ \ \ \ C.\ \ \sqrt[3]{27} \ \ \ \ \ \ D.\ \ \ \overline{1.47} \end{array}

Solution: B and D are the irrational numbers

2.

Choose the irrational numbersA.  −79    B.  932       C.  72      D.   3.14127 \begin{array}{l} Choose\ the\ irrational\ numbers\\ A.\ \ -\frac{7}{9} \ \ \ \ B.\ \ \frac{\sqrt{9}}{32} \ \ \ \ \ \ \ C.\ \ \sqrt{72} \ \ \ \ \ \ D.\ \ \ 3.14127 \end{array}

Solution: C and D are the irrational numbers.

3. Carmen and Marcus are making a list of irrational numbers. Marcus says the

48\sqrt{48}

is rational because 16 x 3 = 48 and 16 is a perfect square. Carmen says it is irrational since 48 is not a perfect square.

A. Marcus is correct because the square root of 16 is 4.

B. Carmen is correct because the

48 =4 3\sqrt{48} \ =4\ \sqrt{3}

Solution: B is the correct solution

4. Select the All the correct statements:

  1. An irrational number can be written as a decimal.
  2. An irrational number has endless non-repeating numbers to the right of the decimal.
  3. A rational number can be a terminating decimal.
  4. An irrational number cannot be written as a ratio.

Solution: 2, 3, 4

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Practice Math Problems

1. Select from the list below all the irrational numbers

A. 1.23‾    B. π    C.  40      D.  121A.\ 1.\overline{23} \ \ \ \ B.\ \pi \ \ \ \ C.\ \ \sqrt{40} \ \ \ \ \ \ D.\ \ \sqrt{121}

2. Aiden, Sophie, and Chuck are designing a border for the school’s square garden. Aiden measured the garden. He said the area of the garden is 269 sq. ft. Sophie and Chuck think he didn’t calculate the area correctly as they say that

269is an irrational number and the garden wouldn’t be square. They think the area is 289 as each side of the square would measure 17ft..
A. Aiden is correct when he calculated the square feet as 269.
B. Sophie and Chuck are correct as

269\sqrt{269}

is an irrational number.

3. Choose the group of irrational numbers

A. 3.33‾  , 47 ,−23 , 36B. 643 , −56, 125 , 97C. 48, 1.2127243…, 363, 7.372461... D. 21, π, −23, 36 \begin{array}{l} A.\ 3.\overline{33} \ \ ,\ \frac{4}{7} \ ,-23\ ,\ \sqrt{36}\\ B.\ \sqrt[3]{64} \ ,\ -56,\ \sqrt{125} \ ,\ \frac{9}{7}\\ C.\ \sqrt{48} ,\ 1.2127243\dotsc ,\ \sqrt[3]{36} ,\ 7.372461...\ \\ D.\ \sqrt{21} ,\ \pi ,\ -23,\ \sqrt{36} \end{array}

4. Select the All the correct statements


A. A rational number can be a terminating decimal.

B. An irrational number has endless non-repeating numbers to the right of the decimal.

C. An irrational number cannot be written as a ratio.

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