Approximate Irrational Numbers

Videos, examples, solutions, and lessons to help Grade 8 students learn how to use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²).

For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Common Core: 8.NS.2

Suggested Learning Targets

• I can approximate irrational numbers as rational numbers
• I can approximately locate irrational numbers on a number line
• I can estimate the value of expression involving irrational numbers using rational numbers.
  (Examples: Being able to determine the value of the √2 on a number line lies between 1 and 2, more accurately, between 1.4 and 1.5, and more accurately, etc.)
• I can compare the size of irrational numbers using rational approximations. The following table gives the perfect squares of numbers from 1 to 15. Scroll down the page to learn how to approximate irrational numbers using the perfect squares.

 

How to approximately locate irrational numbers on a number line
Learn the perfect squares for the numbers 1 to 15.
Examples:
Approximate the square root to the nearest integer and place your answer on the number line.
√23
-√10
Evaluate the expression
2√(a + b²) when a = 11 and b = 5.

• Show Step-by-step Solutions

8NS2 Approximating the Values of Irrational Numbers
Know that there are numbers that are not rational, and approximate them by rational numbers.
Example:
Approximate the value of the following irrational numbers
√3
√11
√17

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Estimating Irrational Numbers
An instructional math video on how to make estimations about the value of irrational square roots.
An irrational number is any number that cannot be written as a fraction. Irrational numbers have decimals that keep on going forever without a repeating pattern.
How to make estimations when roots are irrational?

1. Count up until you hit a square root that works.
   
2. Count down until you hit a square root that works.
   
3. Square root the high and low number, then graph their points on a number line.
   
4. Your estimate should be somewhere between those two numbers.
   Example:
   Estimate √5 and then graph your estimate on the number line

• Show Step-by-step Solutions

Estimating Square Roots
The square root of a number n is a number whose square is equal to n, that is, a solution of the equation x² =n. The positive square root of a number n, written √n, is the positive number whose square is n.
Estimate each of the following square roots
√40
√150
-√75
√75
√93
√119
√30
√45
√63

• Show Step-by-step Solutions

8NS2 Estimating the Value of Irrational Expressions part 1
Example:
Approximate the value of the following irrational number
√5

• Show Step-by-step Solutions

8NS2 Estimating the Value of Irrational Expressions part 2
Example:
Approximate the value of the following irrational number
√7

• Show Step-by-step Solutions

Estimating the Value of Irrational Numbers
A rational number is any number that can be written as a fraction: positive or negative
An irrational number is a number that cannot be written as a fraction. It is a non-repeating, non-terminating decimal.
Examples:

1. Simplify the following square roots
   √32
   √18
   √20
   √75
   √56
   √40
   √99
   
2. Estimate the value of each irrational number
   √15
   √130
   √2

• Show Step-by-step Solutions

8.NS.2 Rational Approximations of Irrational Numbers
Approximate square root of numbers that are not perfect squares and put them on the number line.
A number is a perfect square if you can take that many 1 × 1 unit squares and form them into a square.
When a number is a perfect square, the side length of the square it forms is called its square root.
Example:
√11 is between what two integers?
Which number in the number line below best represents the location of √122?

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