Day 1 - Irrational Numbers

We are going to need a new number system. Your textbook uses a review of the Pythagorean Theorem to explain why.

Recall that in any right triangle, the area of the square on the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares on the other two sides:

c² = a² + b²

The followers of the Greek philosopher, Pythagoras, found what seemed to be a contradiction to his theorem about right triangles. They were so devote to their leader that they would kill any one that would expose the contradiction.

The contradiction occurred when a right triangle had sides of the right angle being of length 1. Using his formula, the hypotenuse could be found by:

c² + 1² + 1² = 2

They did not have a number that was squared that would equal 2. They could draw the square with the correct length but they could not measure it so that the area would be 2. The length could not be represented by a rational number. A new number system is needed.

Practice using the Pythagorean Theorem to find the missing side of right triangles. There are basically two types of problems. If you find the length of the longest side (it will always be opposite the right angle) it will be c in the formula.

One type of problem is to find the length of a missing side that is not the hypotenuse. For example, find the length of x in the following diagram:

The other type of problem involves finding the length of the hypotenuse if the other two sides are given. For example, find the length of x in the following diagram:

Daily Practice

Do Exercises #1 - 4 on page 99. Answer the questions in your binder or notebook and check your answers in the back of the textbook.

Use the Internet to try to find more information about the Pythagoreans. Email me with any links to sites with good information.

The Pythagoreans did not have a rational number to represent √2 but they knew such a number existed. We can even find where it is on the number line. If we use a triangle that has both side 1 unit long, the hypotenuse will be √2 units long. Using a compass to measure the hypotenuse we can draw where it will be on the number line.

Notice that we can use the first triangle to create a side that is √2 units long and then create another right triangle that has a side of 1 unit. The hypotenuse of this triangle is √3 units long.

We can use a compass set to the length of the hypotenuse to find where √3 is on the number line. We can continue to find other radicals on the number line.

Radicals of numbers that are not perfect squares (perfect squares are numbers such as 4, 9, 16 and 25) do exist but they are not rational numbers. They cannot be represented in fraction form or as repeating decimals. See page 97 for a proof.

If they are not rational numbers, they must beIrrational Numbers.Irrational numbers have decimal representations that do not terminate or repeat. Recall that the symbol for rational numbers was Q. We use the symbol to represent the irrational numbers. We use the bar over the Q to represent not being rational.

Another number that has a decimal representation that does not repeat or terminate is p. p is an irrational number. On page 101 you have the first one thousand decimal places for p. Mathematicians have calculated p to billions of decimal places. We have other special numbers that are irrational that you may come across such as e that is approximately 2.78. We use the symbols p or e to represent the numbers.

We can create irrational numbers but indicating that the decimal continues forever but does not have a set of digits that repeats. For example, 1.232 232 223 222 232

We combine the rational numbers and irrational numbers into a new number system called the Real Numbers.(Later you will find another type of number system that does not contain real number called imaginary numbers.) We can now diagram our numbers systems as:

Run your mouse over the circles above to see how the number systems relate to one another.

Daily Practice

Do Exercises #1, 5, 7, 8, 9 and 12 on page 104. Answer the questions in your binder or notebook and check your answers in the back of the textbook.

Use the Internet to try to find the number of decimal places that π has been calculated to today. Submit to me links to sites with the answer, below:

(Note: The assignment hand-in boxes don't work in the course samples.)