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Irrational Numbers
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 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 VideoWhy is the value you calculated for π not the same as the one in the calculator?ANSWER

The 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 nonperfect square by using the square root button on a scientific calculator.


After VideoWhat is a rational number?ANSWER

A 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 nonperfect 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.

Nonterminating decimals
DEFINE
A decimal that continues forever and whose digits do not repeat.

Irrational number
DEFINE
A number whose decimal form is nonterminating.

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
Download as PDF Download PDF View as Separate PageWHAT IS AN IRRATIONAL NUMBER?Rational numbers are numbers that can be represented with fractions (or ratios). Irrational numbers are numbers that cannot be represent with fractions. “Irrational” means “nonrational.”
To better understand irrational numbers…
WHAT IS AN IRRATIONAL NUMBER?. Rational numbers are numbers that can be represented with fractions (or ratios). Irrational numbers are numbers that cannot be represent with fractions. “Irrational” means “nonrational.” To better understand irrational numbers…LET’S BREAK IT DOWN!
π is an irrational number.
Adesina, Emily, and Amari are eating pizza. Amari wants to find the length of crust on their pizza. You can use the formula C = πd to find the circumference of the pizza, where C is the circumference and d is the diameter. The diameter is the distance across the center of the circle. The pizza has diameter 18 inches. The circumference of the pizza crust is C = πd = π(18) =56.54866776461628… The reason there are so many decimal places when you work with π is because π represents an irrational number. Try this yourself: Calculate the circumference of a circle that has diameter 17 inches.
π is an irrational number. Adesina, Emily, and Amari are eating pizza. Amari wants to find the length of crust on their pizza. You can use the formula C = πd to find the circumference of the pizza, where C is the circumference and d is the diameter. The diameter is the distance across the center of the circle. The pizza has diameter 18 inches. The circumference of the pizza crust is C = πd = π(18) =56.54866776461628… The reason there are so many decimal places when you work with π is because π represents an irrational number. Try this yourself: Calculate the circumference of a circle that has diameter 17 inches.You can use known measurements to calculate π.
Adesina, Amari, and Emily are decorating cakes, and exploring the value of π. π is the ratio between the diameter of a circle and its circumference, π = [ggfrac]C/d[/ggfrac]. To determine the value of π, they decide to measure the circumference and diameter of their cakes and use their ratio to calculate π. Amari measures her cake and finds that the cake is 31 inches around and 10 inches across. She finds that the value of π is π = [ggfrac]C/d[/ggfrac] = [ggfrac]31/10[/ggfrac] = 3.1. Emily uses a more accurate tape measure on her cake and finds that the cake is 15.7 inches around and 5 inches in diameter. She finds that the value of π is π = [ggfrac]C/d[/ggfrac] = [ggfrac]15.7/5[/ggfrac] = 3.14. Next, Adesina measures her cake using an even more accurate tape measure. Adesina’s cake is 18.87 inches around and 6 inches in diameter, so her value of π is π = [ggfrac]C/d[/ggfrac] = [ggfrac]18.87/6[/ggfrac] = 3.145. All of their answers are close, but the varying accuracy of their measurements give them values for π with different levels of accuracy. If they used more precise scientific equipment, then they would have found π = π=3.14159265358979… The decimals in the number π go on forever and never repeat, which means it is an irrational number. Irrational numbers have decimals that go on forever and never repeat. We use the symbol π because it is a much more compact way to represent this irrational number. Try this yourself: Calculate π for a circle with circumference 62.8319 inches and diameter 20 inches.
You can use known measurements to calculate π. Adesina, Amari, and Emily are decorating cakes, and exploring the value of π. π is the ratio between the diameter of a circle and its circumference, π = [ggfrac]C/d[/ggfrac]. To determine the value of π, they decide to measure the circumference and diameter of their cakes and use their ratio to calculate π. Amari measures her cake and finds that the cake is 31 inches around and 10 inches across. She finds that the value of π is π = [ggfrac]C/d[/ggfrac] = [ggfrac]31/10[/ggfrac] = 3.1. Emily uses a more accurate tape measure on her cake and finds that the cake is 15.7 inches around and 5 inches in diameter. She finds that the value of π is π = [ggfrac]C/d[/ggfrac] = [ggfrac]15.7/5[/ggfrac] = 3.14. Next, Adesina measures her cake using an even more accurate tape measure. Adesina’s cake is 18.87 inches around and 6 inches in diameter, so her value of π is π = [ggfrac]C/d[/ggfrac] = [ggfrac]18.87/6[/ggfrac] = 3.145. All of their answers are close, but the varying accuracy of their measurements give them values for π with different levels of accuracy. If they used more precise scientific equipment, then they would have found π = π=3.14159265358979… The decimals in the number π go on forever and never repeat, which means it is an irrational number. Irrational numbers have decimals that go on forever and never repeat. We use the symbol π because it is a much more compact way to represent this irrational number. Try this yourself: Calculate π for a circle with circumference 62.8319 inches and diameter 20 inches.Square roots of numbers can be irrational.
Square roots of numbers can be irrational. Adesina and the kids are trying to find the length of the long side in a right triangle. The triangle has short side lengths 3 feet and 4 feet. They can use the Pythagorean theorem to find the value of the missing side. The formula is c2 = a2 + b2 , where a and b are the lengths of the short sides. To find the value of c in the triangle, plug in the lengths of the sides for a and b: c2 = a2 + b2, c2 = 32 + 42, c2 = 5. So, the length of the longest side is 5 feet. Another triangle has two short sides with length 1 foot. The Pythagorean theorem says that the missing side length is √2. You can calculate this value using the square root button on your calculator. Adesina's calculator gives 1.414213562… This number is irrational because the decimal never terminates or repeats. In fact, the square roots of nonperfect squares are always irrational. Try this yourself: Identify whether each square root is rational or irrational. √6, √36, √17.Repeating and terminating decimals are rational.
Emily converts [ggfrac]1/3[/ggfrac] to a decimal and gets 0.333333333… Is [ggfrac]1/3[/ggfrac] irrational? The decimal is endless, but the 3 in the decimal part repeats, so 13 is not an irrational number; it is a rational number. Rational numbers are numbers that can be written as a ratio of integer values. Their decimal form either terminates or repeats. You can summarize a repeating decimal by putting a bar across the repeated values, so 0.33333333333 becomes 0. 3. Emily then finds the decimal form of [ggfrac]2/3[/ggfrac] and gets 0.66666666, or 0.6. Likewise, [ggfrac]34/99[/ggfrac] becomes 0.3434343434, or 34. These are all rational numbers. Try this yourself: Identify whether each number is rational or irrational. 0.94, 0.65475694, [ggfrac]38/99[/ggfrac], 23.573659312859362843…
Repeating and terminating decimals are rational. Emily converts [ggfrac]1/3[/ggfrac] to a decimal and gets 0.333333333… Is [ggfrac]1/3[/ggfrac] irrational? The decimal is endless, but the 3 in the decimal part repeats, so 13 is not an irrational number; it is a rational number. Rational numbers are numbers that can be written as a ratio of integer values. Their decimal form either terminates or repeats. You can summarize a repeating decimal by putting a bar across the repeated values, so 0.33333333333 becomes 0. 3. Emily then finds the decimal form of [ggfrac]2/3[/ggfrac] and gets 0.66666666, or 0.6. Likewise, [ggfrac]34/99[/ggfrac] becomes 0.3434343434, or 34. These are all rational numbers. Try this yourself: Identify whether each number is rational or irrational. 0.94, 0.65475694, [ggfrac]38/99[/ggfrac], 23.573659312859362843… 
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Is the number [ggfrac]5/7[/ggfrac] a rational number?
Why are some square roots rational and others irrational?
What is the length of the hypotenuse in a right triangle with legs that measure 7 feet and 8 feet? Is the length of the hypotenuse rational or irrational? If it is irrational, provide a rational approximation with a reasonable precision.
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