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Irrational Numbers
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What you will learn from this videoWhat you will learn
- We'll learn that irrational numbers are endless and do NOT repeat.
- We will also learn how to use approximations of irrational numbers to solve real world problems.
- And we'll see how this knowledge can help us decorate a cake, make a bunny enclosure, and share a feast!
- Discussion Questions
Before Video
Why is the value you calculated for π not the same as the one in the calculator?ANSWERThe measurements used to get the calculator value are more precise, and so there are more decimal places on the calculator.
The square root I needed to calculate was not a perfect square, so finding the number that multiplies by itself to give the answer was challenging. I could have kept going forever without finding the exact right answer.
No, the decimal form of π has infinitely many places after the decimal. It never ends.
The decimal form of one third is 0.3333333333… or 0.3, where the bar above the 3 means the 3 is repeated infinitely many times.
I can calculate the square root of a non-perfect square by using the square root button on a scientific calculator.
After Video
What is a rational number?ANSWERA rational number is a number that can be written as a fraction of two integers. In decimal form, the decimals either terminate or form a repeating pattern.
An Irrational number is one whose decimal form never terminates and never repeats. They include special numbers like π, φ, or e, as well as square roots of non-perfect squares.
In practice, I can approximate an irrational number with a rational number that has enough decimal places to be useful in the problem. For example, if I need to know roughly how big √2 is, I may round to 1 decimal place to get my rational approximation. If I am an engineer and I need to be very precise, I might round to 6 decimal places. It is not practical or possible to work with infinitely many decimal places.
3.758375839203…≈3.76, √22≈4.69, [ggfrac]√20/4[/ggfrac]≈1.12, 3√2≈4.24, so the order from least to greatest is [ggfrac]√20/4[/ggfrac], 3.758375839203…, 3√2, √22.
Write the decimal as a fraction with denominator 1. Then multiply the numerator and denominator by a power of 10 until the numerator is an integer.
Example: [ggfrac]2.55676/1.00000[/ggfrac] = [ggfrac]2.55676x100,000/1.00000x100,000[/ggfrac] = [ggfrac]255,676/100,000[/ggfrac]
- Vocabulary
- Whole number DEFINE
Any of the numbers {0, 1, 2, 3, …}. They have no decimal part and do not include negative numbers.
- Integer DEFINE
Any of the numbers {…–3, –2, –1, 0, 1, 2, 3, …}. They have no decimal part and include both positive and negative numbers.
- Terminating decimal DEFINE
A decimal with a finite number of decimal places.
- Repeating decimal DEFINE
A decimal that continues forever but whose digits repeat.
- Non-terminating decimals DEFINE
A decimal that continues forever and whose digits do not repeat.
- Irrational number DEFINE
A number whose decimal form is non-terminating.
- Rational number DEFINE
A number that can be represented by a fraction or integer ratio. The decimal form of a rational number is always a terminating or repeating decimal.
- π (pi) DEFINE
A special constant used to calculate the area and circumference of circles.
- Square root DEFINE
The square root of a number, n, is a number that, when multiplied by itself, has a product of n.
- Whole number DEFINE
- Reading Material
- Practice Word Problems
- Practice Number Problems
- Lesson Plan
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