You, π, are completely irrational!

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Presentation transcript:

You, π, are completely irrational! Defining rational and irrational numbers

Real Numbers What do you remember about the following types of numbers? Whole numbers Integers Rational numbers Tell me as much as you can about these!

Real Numbers Whole numbers: 0, 1, 2, 3, 4… We might call these the counting numbers. Integers: …, -3, -2, -1, 0, 1, 2, 3… The counting numbers and their opposites (negatives).

Real Numbers Rational! Rational! Rational! Rational numbers Any number that can be written as a ratio of integers in the form of Rational! Rational! Rational!

Rational numbers repeat! Real Numbers Previously we discovered another definition of rational numbers. What is it? Use the following examples to help you. Rational numbers repeat!

But there’s a bigger picture! Real Numbers So far, we have this picture of how numbers fit together: Rational Numbers But there’s a bigger picture! Integers Whole Numbers

Real Numbers No repeating pattern! No repeating pattern! Are there decimals that don’t eventually repeat a pattern? No repeating pattern! No repeating pattern! Why is this not a repeating pattern? Discuss this with a partner, then share with the class what you think. No repeating pattern!

Real Numbers Since numbers that eventually have a repeating pattern of digits when written as a decimal are called rational, we call numbers that don’t eventually have a repeating pattern of digits irrational numbers.

Real Numbers Now we have new picture: Rational Numbers Irrational Numbers Integers Whole Numbers

Real Numbers And together they make: THE REAL NUMBERS! Real Numbers Rational Numbers Irrational Numbers Integers Whole Numbers

Irrational Numbers Where do irrational numbers most often occur? Discuss the following numbers with a partner and decide if they are rational or irrational. You may use a calculator if you would like. Make sure you can explain why you think so!

Irrational Numbers Rational! Irrational! Rational! Irrational! Rational or irrational? Why? Rational! Irrational! Rational! Irrational! Rational! Rational!

Irrational Numbers Would a calculator help us identify these? Why or why not? While you could punch in the square root of 7 or convert the fraction 11/7 to decimal through division on a calculator, these don’t bring us any closer to knowing whether it is rational or irrational because we can’t see the full decimal expansion.

Irrational Numbers With your partner, discuss what type of numbers are commonly irrational? Why? Now share your thoughts with the class. Are all numbers written with a square root symbol irrational? This slide should lead students to the idea that most square roots are irrational.

Irrational Numbers Is the following number rational or irrational? Why? What about this one? What’s the difference between them? We often deal with positive irrational numbers, but they can just as easily be negative. Students should familiarize themselves with the concept of negative irrational numbers just as they did with negative rational numbers in previous grade levels.

Irrational Numbers Now for bonus fun, ask about this number!