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Irrational Square Roots

Approximate the value of multiples of pi.

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Approximate Solutions to Equations Involving Irrational Numbers

A common place where you will see irrational numbers is when you are working with circles or spheres. Since pi is related to a circle, you will need to work with irrational numbers when solving problems involving circles. Take a look at this dilemma.

Henrietta knew that to find the circumference of a circle, she needed to multiply the diameter by \begin{align*}\pi\end{align*}. If the diameter of Henrietta’s circle is 6 inches, what is the approximate circumference of the circle?

Pay attention and you will learn how to successfully solve this problem.


There are many different ways to classify or name numbers.

All numbers are considered real numbers.

When you were in the lower grades, you worked with whole numbers. Whole numbers are counting numbers. We consider whole numbers as the set of numbers \begin{align*}\{0, 1, 2, 3, 4 \ldots \}\end{align*}.

In middle school, you may also have learned about integers. The set of integers includes whole numbers, but also includes their opposites. Therefore, we can say that whole positive and negative numbers are part of the set of integers \begin{align*}\{ \ldots -2, -1, 0, 1, 2, 3 \ldots \}\end{align*}.

We can’t stop classify numbers with whole numbers and integers because sometimes we can measure a part of a whole or a whole with parts. These numbers are called rational numbers. A rational number is any number that can be written as a fraction where the numerator or the denominator is not equal to zero.

Take a look at these definitions and then write them in your notebook.

Sometimes, you will need to find estimates of irrational numbers to solve an equation. The easiest way to do this is to find a decimal value on your calculator that is close to the irrational number. Remember that the more decimal points you include, the more accurate your answer will be. For these purposes, it is usually okay to round an irrational number to the nearest hundredth or thousandth. Once you have found the decimal approximate, solve the equation normally. It is crucial to use words or signage to show that your answer is approximate, not exact. The symbol \begin{align*}\approx\end{align*} means approximately equal to, and is more appropriate than an equals sign in these situations.

Take a look at this situation.

Solve for \begin{align*}a\end{align*}: \begin{align*}a=4 \pi\end{align*}.

First find a decimal approximation for \begin{align*}\pi\end{align*} using your calculator. The value of \begin{align*}\pi\end{align*} is 3.1415927... This can be rounded to 3.14 for these purposes.

To solve the equation, multiply 3.14 by 4. This will be the approximate value of \begin{align*}a\end{align*}.

\begin{align*}3.14 \times 4=12.56\end{align*}

So the value of \begin{align*}a\end{align*} is approximately 12.56. \begin{align*}a \approx 12.56\end{align*}

Evaluate each solution.

Example A

\begin{align*}a\end{align*}: \begin{align*}a=3 \pi\end{align*}

Solution:  \begin{align*}9.42\end{align*}

Example B

\begin{align*}x\end{align*}: \begin{align*}x=7 \pi\end{align*}

Solution:  \begin{align*}21.98\end{align*}

Example C

\begin{align*}y\end{align*}: \begin{align*}y=5 \pi\end{align*}

Solution:  \begin{align*}15.7\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

First, translate the information in the problem into an equation. Let \begin{align*}C\end{align*} equal the circumference.

\begin{align*}C=\pi \times 6\end{align*}

The value of \begin{align*}\pi\end{align*} is 3.1415927... Rounded to the nearest hundredth, this value is 3.14. You can substitute this value back into the equation to find the value of \begin{align*}C\end{align*}. Remember to use the approximately equals sign after you make this estimation.

\begin{align*}C&=\pi \times 6\\ C &\approx 3.14 \times 6\\ C &\approx 18.84\end{align*}

The circumference of Henrietta’s circle is approximately 18.84 inches.


Whole Numbers
the set of positive counting numbers.
the set of whole numbers and their opposites.
Rational Numbers
any number that can be written in fraction form including terminating and repeating decimals.
Irrational Numbers
numbers that do not have an end point when written in decimal form – the decimal values continue indefinitely. These numbers do not fit into the set of rational numbers.
\begin{align*}\pi\end{align*}, the ratio of the diameter to the circumference of a circle. We use 3.14 to represent this irrational number.
Real Numbers
the set of rational and irrational numbers make up the set of real numbers.

Guided Practice

Here is one for you to try on your own.

Solve for \begin{align*}y\end{align*}: \begin{align*}12-\sqrt{7}=y\end{align*}.


First find a decimal approximation for \begin{align*}\sqrt{7}\end{align*} using your calculator. The value of \begin{align*}\sqrt{7}\end{align*} is 2.64575... This can be rounded to 2.65 for these purposes.

To solve the equation, subtract 2.65 from 12. This will be the approximate value of \begin{align*}y\end{align*}


So, the value of \begin{align*}y\end{align*} is approximately 9.35. \begin{align*}y \approx 9.35\end{align*}

Video Review


Directions: Approximate the solution for each equation given the irrational numbers.

  1. \begin{align*}a\end{align*}: \begin{align*}a=3 \pi\end{align*}
  2. \begin{align*}x\end{align*}: \begin{align*}x=8 \pi\end{align*}
  3. \begin{align*}b\end{align*}: \begin{align*}b=9 \pi\end{align*}
  4. \begin{align*}c\end{align*}: \begin{align*}c=12 \pi\end{align*}
  5. \begin{align*}a\end{align*}: \begin{align*}a=2 \pi\end{align*}
  6. \begin{align*}y\end{align*}: \begin{align*}y=6 \pi\end{align*}
  7. \begin{align*}b\end{align*}: \begin{align*}b=7 \pi\end{align*}
  8. \begin{align*}d\end{align*}: \begin{align*}d=12 \pi-6\end{align*}
  9. \begin{align*}a\end{align*}: \begin{align*}a=14 \pi-9\end{align*}
  10. \begin{align*}x\end{align*}: \begin{align*}x=11 \pi-5\end{align*}
  11. \begin{align*}\sqrt{2}+5=x\end{align*}
  12. \begin{align*}8 = \sqrt{2} + x\end{align*}
  13. \begin{align*}t=\pi-5.3\end{align*}
  14. \begin{align*}\sqrt{h}=\sqrt{6}-\frac{3}{4}\end{align*}
  15. Mrs. DeFazio wrote the following equation on the board. \begin{align*}w=\sqrt{11}-2^2\end{align*} What is the value of \begin{align*}w\end{align*} in Mrs. DeFazio’s equation?




The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Irrational Number

Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.


\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.
rational number

rational number

A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.
Real Number

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
Whole Numbers

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...

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