Lesson 2.06--Roots
Approximate completion time: 2 hours
Do you believe in something so strongly that you would die for it?
Many people believe it was Hippasus of Metapontum (ca. 500 BC), an ancient Greek philosopher who discovered something important. So earth shattering was his discovery that the members of an elite mathematical society killed him! What he discovered completely obliterated what was commonly held:
"The Pythagoreans in ancient Greece believed in a religious way that all was number, and all physical reality could be expressed and understood through numbers, and in particular that all lengths could be expressed in terms of ratios of one another." [Shelley Walsh]
So when Hippasus discovered that the square root of 2 is irrational -- it can't be expressed as a fraction of 2 integers -- his colleagues became infuriated and put him to death. This fundamental number, which is the length of a diagonal of a square with side lengths of 1, couldn't be infinitely long and nonrepeating--or could it? Hippasus knew it would take literally forever to write out this number in decimal form without any repeating patterns. The other Pythagoreans refused to believe it and, not wanting it to be true, killed Hippasus.
Would you be willing to die for something you believe to be true?
Do you believe in what Hippasus did - the square root of 2 is irrational?
Let's take a look at what that means.
The length of the side of a square whose area is 2? Let's think about that.
Here are some squares and their areas:
In table form, here's what we have:
Hmmm, so what is the side length if the area were 2? That's still our problem.
Answer: "the square root of 2" which we write as
. Obviously it is some number between 1 and 2 (from the table).
Do you know how big that is in decimal form? Try to guess. Use a calulator to guess (try not to use the
button yet). You'll know if you're correct if you square your number and get exactly 2. EXACTLY TWO. It's pretty difficult to do. See if you can find it. If you think you've got it, try it out on a bigger calculator that won't round so soon.
If you can find it, the number in decimal form which is the exact value of
BONUS: If you like big and strange words, then find out what "incommensurable" means. You can use it to talk with your parents about the irrationality of
BIG POINT: We write
Let's try some other square roots.
A baseball diamond typically measures 90 feet between consecutive bases. That is, the bases form a square whose edge length is 90 feet.
So how far is it from home plate to second base? It is the length of the diagonal of a square whose side measures 90 feet. I'm thinking it will be 90 times as long as Hippasus's square's diagonal. So I'm thinking it will be. Let's see if that's true.
The area of the original square is 902 = 8,100 sq feet.
Double that area and we get 16,200 sq feet.
Now what number when squared is 16,200?
That would be. Using my handy-dandy calculator, that's approximately 127.279220613579.
I don' think the groundskeepers will need it to be that exact.Is that the same as
Wow, that's interesting. We'll have to study these properties of square roots more closely another time. (Here's a clue for now: 16,200=902·2)
For now, let us focus only on roots in their basic understanding. We'll investigate their properties and uses more thoroughly later.
"square root of x"
or
The positive number which when squared is x.Most of us do not write the small 2 that is really there in a square root.
is really the same thing as
.
For higher order roots, we leave the small number there. Like in cube roots.
The length of a side of a cube whose volume is x.
or
The number which when cubed is x.
Examples:
1.
. This is because 23=8.
2.
. This is because 43=64.
3.
. This is because 92=81.
Bonus: Did you know that cube roots can also be irrational?
