## Learning Focus

Support or challenge claims about different types of numbers and the result of adding, subtracting, multiplying, and dividing.

What are the similarities and differences between the arithmetic of integers, rational numbers, real numbers, and complex numbers?

## Open Up the Math: Launch, Explore, Discuss

Now that we have been thinking about new number sets, including irrational and complex numbers, it’s time to generalize the results.

In order to find solutions to all quadratic equations, we had to extend the number system to include complex numbers.

Do the following for each of the problems below:

• Choose the best word to complete each conjecture.

• After you have made a conjecture, create at least three examples to show why your conjecture is true.

• If you find a counterexample, change your conjecture to fit your work.

### 1.

Conjecture #1: The sum of two integers is [always, sometimes, never] an integer.

More conjectures:

### 2.

The sum of two irrational numbers is [always, sometimes, never] an irrational number.

### 3.

The sum of a rational number and an irrational number is [always, sometimes, never] an irrational number.

### 4.

The difference of two whole numbers is [always, sometimes, never] a whole number.

### 5.

The difference of two rational numbers is [always, sometimes, never] a rational number.

### 6.

The product of two integers is [always, sometimes, never] an integer.

### 7.

The quotient of two integers is [always, sometimes, never] an integer.

### 8.

The product of two rational numbers is [always, sometimes, never] a rational number.

### 9.

The product of two irrational numbers is [always, sometimes, never] an irrational number.

### 10.

The product of two real numbers is [always, sometimes, never] a real number.

### 11.

The product of two complex numbers is [always, sometimes, never] a complex number.

### 12.

The ratio of the circumference of a circle to its diameter is given by the irrational number . Can the diameter of a circle and the circumference of the same circle both be rational numbers? Explain why or why not.

Write two claims of your own about number sets and use examples to demonstrate that they are true.

Claim #1:

Claim #2:

## Takeaways

Some true statements about operations on numbers:

## Vocabulary

• closure
• Bold terms are new in this lesson.

## Lesson Summary

In this lesson, we examined claims about sets of numbers and the operations of addition, subtraction, multiplication, and division. An example of such a claim is: The quotient of two whole numbers is always a whole number. A counterexample that shows this claim to be false is: . Since is a rational number, this example shows that dividing two whole numbers does not always result in a whole number.

## Retrieval

### 1.

Solve the system.

,

### 2.

Find the product.