Google Classroom
GeoGebraGeoGebra Classroom
GeoGebra

IM Alg1.7.20 Lesson: Rational and Irrational Solutions

Author:GeoGebra Classroom Activities, IM Algebra 1, Geometry, Algebra 2
Numbers like -1.7, , and  are known as rational numbers. Numbers like  are known as irrational numbers.

Here is a list of numbers. Sort them into rational and irrational.

Graph each quadratic equation using graphing technology below. Identify the zeros of the function that the graph represents, and say whether you think they might be rational or irrational. Be prepared to explain your reasoning.

Find exact solutions (not approximate solutions) to each equation and show your reasoning. Then, say whether you think each solution is rational or irrational. Be prepared to explain your reasoning.

Here is a list of numbers:


Here are some statements about the sums and products of numbers. For each statement, decide whether it is always true, true for some numbers but not others, or never true. Experiment with sums and products of two numbers in the given list to help you decide. Sums:
  • The sum of two rational numbers is rational.

Select all that apply
  • A
  • B
  • C
Check my answer (3)

  • The sum of a rational number and an irrational number is irrational.

Select all that apply
  • A
  • B
  • C
Check my answer (3)

  • The sum of two irrational numbers is irrational.

Select all that apply
  • A
  • B
  • C
Check my answer (3)

Products:

  • The product of two rational numbers is rational.

Select all that apply
  • A
  • B
  • C
Check my answer (3)

  • The product of a rational number and an irrational number is irrational.

Select all that apply
  • A
  • B
  • C
Check my answer (3)

  • The product of two irrational numbers is irrational.

Select all that apply
  • A
  • B
  • C
Check my answer (3)

It can be quite difficult to show that a number is irrational. To do so, we have to explain why the number is impossible to write as a ratio of two integers. It took mathematicians thousands of years before they were finally able to show that  is irrational, and they still don’t know whether or not is irrational. Here is a way we could show that  can’t be rational, and is therefore irrational.

  • Let's assume that  were rational and could be written as a fraction , where  and  are non-zero integers.
  • Let’s also assume that  and  are integers that no longer have any common factors. For example, to express 0.4 as , we write  instead of  or . That is, we assume that  and  are 2 and 5, rather than 4 and 10, or 200 and 500.
If , then .

Explain why  must be an even number.

Explain why if  is an even number, then  itself is also an even number. (If you get stuck, consider squaring a few different integers.)

Because  is an even number, then  is 2 times another integer, say, . We can write . Substitute  for  in the equation you wrote in the first question. Then, solve for .

Explain why the resulting equation shows that , and therefore , are also even numbers.

We just arrived at the conclusion that  and  are even numbers, but given our assumption about  and , it is impossible for this to be true. Explain why this is.

If  and  cannot both be even,  must be equal to some number other than . Because our original assumption that we could write  as a fraction  led to a false conclusion, that assumption must be wrong. In other words, we must not be able to write  as a fraction. This means  is irrational!

New Resources

Discover Resources

Discover Topics