IM Alg1.7.21 Lesson: Sums and Products of Rational and Irrational Numbers
Here are some examples of integers (positive or negative whole numbers):
Experiment with adding any two numbers from the list (or other integers of your choice).
Try to find one or more examples of two integers that add up to another integer.-25 -10 -2 -1 0 5 9 40
Try to find one or more examples of two integers that add up to a number that is not an integer.
Experiment with multiplying any two numbers from the list (or other integers of your choice). Try to find one or more examples of two integers that multiply to make another integer.
Try to find one or more examples of two integers that multiply to make a number that is not an integer.
Here are a few examples of adding two rational numbers.
Is this sum a rational number? Be prepared to explain how you know.
Is this sum a rational number? Be prepared to explain how you know.
Is this sum a rational number? Be prepared to explain how you know.
is an integer: Is this sum a rational number? Be prepared to explain how you know.
Here is a way to explain why the sum of two rational numbers is rational.
Suppose and are fractions. That means that and are integers, and and are not 0. Find the sum of and . Show your reasoning.
In the sum, are the numerator and the denominator integers? How do you know?
Use your responses to explain why the sum of is a rational number.
Use the same reasoning as in the previous question to explain why the product of two rational numbers, , must be rational.
Consider numbers that are of the form , where and are whole numbers. Let’s call such numbers quintegers.
Here are some examples of quintegers:
When we add two quintegers, will we always get another quinteger? Either prove this, or find two quintegers whose sum is not a quinteger. (, ) (, ) (, ) (, )
When we multiply two quintegers, will we always get another quinteger? Either prove this, or find two quintegers whose product is not a quinteger.
Here is a way to explain why is irrational.
Would be rational or irrational? Explain how you know.
Evaluate . Is the sum rational or irrational?
Use your responses so far to explain why cannot be a rational number, and therefore cannot be rational.
Use the same reasoning as in the earlier question to explain why is irrational.
Consider the equation . Find a value of so that the equation has 2 rational solutions.
Find a value of so that the equation has 2 irrational solutions.
Find a value of so that the equation has 1 solution.
Find a value of so that the equation has no solutions.
Describe all the values of that produce 2, 1, and no solutions.
Write a new quadratic equation with each type of solution. Be prepared to explain how you know that your equation has the specified type and number of solutions.
no solutions
2 irrational solutions
2 rational solutions
1 solution