Friday, April 15, 2011
Thursday, April 14, 2011
Pythagoras and the Square !
Pythagoras is famous for his expertise with triangles. A famous and important mathematical formula learned in grade school is the Pythagoras' Formula.
c2 = a2 + b2
This formula is used to calculate the lengths of the sides of a special type of triangle called a "right" triangle. To understand triangles, first we need to understand squares.
If we read the formula aloud we would say,
The letters a, b, and c are used to represent numbers. That's ALGEBRA Baby!
For example:
52 = 32 + 42
where c = 5, a = 3 and b = 4
The "superscript" '2' next to each number means "squared."
The official mathematical name for this superscript is "exponent." Exponents can be any number.
When a number has an exponent that is one of the counting numbers , i.e., an "integer", such as 1, 2, 3, 4, ... etc., it means that the the number (whose official mathematical name, by the way, is the "base") is to be multiplied by itself as many times as is the value of the exponent.
Example: 52 = 5 x 5
The base value 5 is multiplied by itself twice (it appears as a multiplier or "factor", twice.)
Another example : 63 = 6 x 6 x 6 (6 is a factor 3 times)
Let's talk about the special case of exponents of value '2'.
Think of a square.
It has a length and a width that are the same value. If we wish to know the area of the square (the part colored in red), then:
area of square = length x width.
But because the length and width are the same size, we could say:
area of square = length x length
which is the same as length2
or
area of square = width x width
which is the same as width2
So the word "square" is naturally associated with the exponent value 2.
1. The Pythagorean Formula
2. Pythagoras was an expert on triangles.
3. Pythagoras knew a lot about squares too.
4. What an exponent is.
5. What a superscript is.
6. What a base number is.
7. How to calculate the area of a square.
8. Why the exponent value 2 is called "the square."
9. What an integer is.
10. What a factor is.
11. Letters can be used to represent numbers in Algebra.
12. The length and width of a square are the same.
When you visit Perfesser Wizard, you don't go home empty handed!
 |
| "Pythagoras" |
c2 = a2 + b2
This formula is used to calculate the lengths of the sides of a special type of triangle called a "right" triangle. To understand triangles, first we need to understand squares.
If we read the formula aloud we would say,
" 'c' squared is equal to the sum of 'a' squared +' b' squared."
The letters a, b, and c are used to represent numbers. That's ALGEBRA Baby!
For example:
52 = 32 + 42
where c = 5, a = 3 and b = 4
The "superscript" '2' next to each number means "squared."
The official mathematical name for this superscript is "exponent." Exponents can be any number.
When a number has an exponent that is one of the counting numbers , i.e., an "integer", such as 1, 2, 3, 4, ... etc., it means that the the number (whose official mathematical name, by the way, is the "base") is to be multiplied by itself as many times as is the value of the exponent.
Example: 52 = 5 x 5
The base value 5 is multiplied by itself twice (it appears as a multiplier or "factor", twice.)
Another example : 63 = 6 x 6 x 6 (6 is a factor 3 times)
Let's talk about the special case of exponents of value '2'.
Gee, Perfesser Wizard! Why is raising a base number to the exponent (or "the power of") 2 called "square"?
Think of a square.
 |
| A Square |
It has a length and a width that are the same value. If we wish to know the area of the square (the part colored in red), then:
area of square = length x width.
But because the length and width are the same size, we could say:
area of square = length x length
which is the same as length2
or
area of square = width x width
which is the same as width2
So the word "square" is naturally associated with the exponent value 2.
Well then, what have we learned?
1. The Pythagorean Formula
2. Pythagoras was an expert on triangles.
3. Pythagoras knew a lot about squares too.
4. What an exponent is.
5. What a superscript is.
6. What a base number is.
7. How to calculate the area of a square.
8. Why the exponent value 2 is called "the square."
9. What an integer is.
10. What a factor is.
11. Letters can be used to represent numbers in Algebra.
12. The length and width of a square are the same.
When you visit Perfesser Wizard, you don't go home empty handed!
Monday, April 11, 2011
How Smart People Get Smarter
I'm a pretty smart guy. I got that way by being very curious, reading a lot, and being around other very smart people.
I was poking around the internet today to see what other smart people knew about math.
I found a very interesting web site:
He explains calculus in a very easy to understand way.
Take a look! Click here!
I was poking around the internet today to see what other smart people knew about math.
I found a very interesting web site:
He explains calculus in a very easy to understand way.
Take a look! Click here!
Sunday, April 10, 2011
Calculus for Smart Kids who are bored in Jordan's 5th grade Math Class
So, you want to learn to read "mathonese", eh? Do you have what it takes. Can you stare at the scribbles for hours, days, maybe years on end and ponder their secrets. Are you that CURIOUS?
"Welcome to my parlor, said the spider to the fly."
Chapter 1:
"I’ve Reached My Limit!"
Or
“How to Make your own rules, impress your teachers and meet cute girls.”
Pick a number, any number. Try again, but make it smaller, anything but zero. I said smaller, not vanishing altogether. Do it again. Again, Again. Keep going! Why'd ya stop? Are we there yet?
I told you that ε ("epsilon") is a Greek letter that represents a very peculiar number, 2.171228… . Well, Greek is a very versatile language. We sometimes use the same Greek letters to mean other things that have nothing to do with each other. Today, we’ll be generous and let you make ε any number you like. Hey, Perfesser Wizard is easy! Chill baby. In fact, ε has another Greek friend, ‘Δ’ ("delta", which means "different"). You can make ‘Δ’ be any number you like too. How about that!
Well, there might be SOME limits. HAR! HAR! That’s a pun! (This story is about “limits”, get it?)
You see you can either pick ε or Δ, but not both at the same time. Whatever you pick for ε will decide what Δ has to be. Or, if you prefer, whatever you pick for Δ will decide what ε has to be. Hey! I didn’t make the rules. It’s how stuff works in math. The Sun comes up. The Sun goes down. You can pick either ε or Δ. Live with it!
Let’s think about a typical expression in algebra. Say,
y = x + 1 .
We could compute a whole list of results, i.e, values of y for this little snippet of addition. Pick a value for x and the result is one more than that value. So, let’s say that we start with x equal to 100 and we calculate x+1, then make x = 99, then x=98, and on and on. The sum of x+1 keeps getting smaller every time we change x. How small will it get? What’s your limit?
What is the “limit” of the expression y = x+1 as we make x smaller and smaller?
If x is 100, then y becomes 101.
If x is 99, then y becomes 100.
Every time we make x a little “smaller”, or closer to zero, y follows.
Another way we can say that we are making x smaller is to ask how different it is from zero. That’s where ε comes in handy. We use ε to represent the difference between the value of x and zero. Sometimes, if we like, we can use ε to represent the difference between any two numbers we choose. Greek is versatile. Perfesser Wizard is easy! Life is good!
How small is ε? Not much, but it’s not 0!
ε = x – 0 (We can compare x to other numbers too. But this example happens to use 0.)
ε’s Greek pal Δ is sort of a copycat. He represents how close the sum of x+1 is to some number, we'll call it y, while we keep making ε smaller and smaller. How close do you want it to get? Pick a number? If you want the sum of x+1 to be Δ = ½ different than what it will be if x were 0, then you have to make ε be small enough to make that happen. Or said another way, the difference between x and the value that makes x+1 as close as Δ to the “limit” is ε.
As x – 0 gets smaller ( ε gets smaller), the difference between y and( x+1) gets smaller, i.e. Δ gets smaller.
y + Δ = (x + 1) + ε
As you make ε smaller, the difference between y and (x+1), which we call Δ, gets smaller. Or we can say,
(x + 1) becomes Δ close to y, as x becomes ε close to 0.
The fancy, schmancy, braniac way to write this so you look like you work for NASA (or came from Mars) is:
y = lim (x+1)
ε = (x-0)-> 0
"The limiting value of the sum (x+1) as ε get very small, i.e., very close to 0, is equal to a number represented by y.)
ε-> 0 is how we write “as the value of ε gets close t 0.” How close? Why ε close, of course!
How close does (x+1) get to y? Or asked another way, “what is the limit of (x+1) as x gets close to the value 0?
Well, (x+1) get Δ close to y. It will never get smaller than y because that is it’s “limit” as ε approaches 0.
But y = x +1 is kind of a lame example. Let’s try something tough like:
y = x2 / x
What do you suppose the value of y will become if we make x be equal to 0? Hmmm! I think I heard somewhere that we’re not allowed to divide by zero. In fact, dividing by zero is not even defined in the official rules of Intergalactic Math. (Okay, I made some of that up. But you can’t divide by 0, okay?)
What to do? WHAT TO DO?
Well, let’s take it in baby steps. Let’s see what y looks like it’s going to become as we slowly make x smaller and smaller.
x = 10, x2 = 100, x2/x = 10
x = 5, x2 = 25, x2/x = 5
x = 2, x2 = 4, x2/x = 2
x= 0.01, x2 = .0001, x2/x = 0.01
Hmmm! It kind of looks like, y wants to be whatever x is. In fact if x were 0 …, no wait, we can’t go there, it’s against the intergalactic rules.
What about x = .00000000000001 ? :
x2 = .000000000000000000000000001, y = x2/x = .00000000000001 (trust me, I’m a professional!)
Wow. X is getting pretty close to 0 and so is y!
Maybe the limit of y in this example is, GULP! YES IT IS! It’s 0!
Lim x2/x = Lim (x times x)/x = Lim x = 0 EUREKA!
ε = x->0 ε = x->0 ε = x->0
Notice how I got rid of one of the x’s in the numerator by dividing (also called canceling)? You can’t do that if you let x = 0 because YOU CAN’T DIVIDE BY ZERO!
But if you sneak up on it with a “limit”, you can see what will happen if you make x as close as you like to zero, you know, ε close. Go ahead, pick a number, any number!
Calculus is a very powerful method of mathematical calculation based on the properties of limits, It’s so powerful, it can break the Intergalactic Rules of Mathematics. Well, it can bend them anyway! But there's lots more to limits. The Intergalactic Congress has been very busy!
It’s all Greek to me!
By the way, about that promise of meeting cute girls, it helps if they are as smart as you are and just as bored with 5th grade math.
So, you want to learn to read "mathonese", eh? Do you have what it takes. Can you stare at the scribbles for hours, days, maybe years on end and ponder their secrets. Are you that CURIOUS?
"Welcome to my parlor, said the spider to the fly."
Chapter 1:
"I’ve Reached My Limit!"
Or
“How to Make your own rules, impress your teachers and meet cute girls.”
Pick a number, any number. Try again, but make it smaller, anything but zero. I said smaller, not vanishing altogether. Do it again. Again, Again. Keep going! Why'd ya stop? Are we there yet?
I told you that ε ("epsilon") is a Greek letter that represents a very peculiar number, 2.171228… . Well, Greek is a very versatile language. We sometimes use the same Greek letters to mean other things that have nothing to do with each other. Today, we’ll be generous and let you make ε any number you like. Hey, Perfesser Wizard is easy! Chill baby. In fact, ε has another Greek friend, ‘Δ’ ("delta", which means "different"). You can make ‘Δ’ be any number you like too. How about that!
Well, there might be SOME limits. HAR! HAR! That’s a pun! (This story is about “limits”, get it?)
You see you can either pick ε or Δ, but not both at the same time. Whatever you pick for ε will decide what Δ has to be. Or, if you prefer, whatever you pick for Δ will decide what ε has to be. Hey! I didn’t make the rules. It’s how stuff works in math. The Sun comes up. The Sun goes down. You can pick either ε or Δ. Live with it!
Let’s think about a typical expression in algebra. Say,
y = x + 1 .
We could compute a whole list of results, i.e, values of y for this little snippet of addition. Pick a value for x and the result is one more than that value. So, let’s say that we start with x equal to 100 and we calculate x+1, then make x = 99, then x=98, and on and on. The sum of x+1 keeps getting smaller every time we change x. How small will it get? What’s your limit?
What is the “limit” of the expression y = x+1 as we make x smaller and smaller?
If x is 100, then y becomes 101.
If x is 99, then y becomes 100.
Every time we make x a little “smaller”, or closer to zero, y follows.
Another way we can say that we are making x smaller is to ask how different it is from zero. That’s where ε comes in handy. We use ε to represent the difference between the value of x and zero. Sometimes, if we like, we can use ε to represent the difference between any two numbers we choose. Greek is versatile. Perfesser Wizard is easy! Life is good!
How small is ε? Not much, but it’s not 0!
ε = x – 0 (We can compare x to other numbers too. But this example happens to use 0.)
ε’s Greek pal Δ is sort of a copycat. He represents how close the sum of x+1 is to some number, we'll call it y, while we keep making ε smaller and smaller. How close do you want it to get? Pick a number? If you want the sum of x+1 to be Δ = ½ different than what it will be if x were 0, then you have to make ε be small enough to make that happen. Or said another way, the difference between x and the value that makes x+1 as close as Δ to the “limit” is ε.
As x – 0 gets smaller ( ε gets smaller), the difference between y and( x+1) gets smaller, i.e. Δ gets smaller.
y + Δ = (x + 1) + ε
As you make ε smaller, the difference between y and (x+1), which we call Δ, gets smaller. Or we can say,
(x + 1) becomes Δ close to y, as x becomes ε close to 0.
The fancy, schmancy, braniac way to write this so you look like you work for NASA (or came from Mars) is:
y = lim (x+1)
ε = (x-0)-> 0
"The limiting value of the sum (x+1) as ε get very small, i.e., very close to 0, is equal to a number represented by y.)
ε-> 0 is how we write “as the value of ε gets close t 0.” How close? Why ε close, of course!
How close does (x+1) get to y? Or asked another way, “what is the limit of (x+1) as x gets close to the value 0?
Well, (x+1) get Δ close to y. It will never get smaller than y because that is it’s “limit” as ε approaches 0.
But y = x +1 is kind of a lame example. Let’s try something tough like:
y = x2 / x
What do you suppose the value of y will become if we make x be equal to 0? Hmmm! I think I heard somewhere that we’re not allowed to divide by zero. In fact, dividing by zero is not even defined in the official rules of Intergalactic Math. (Okay, I made some of that up. But you can’t divide by 0, okay?)
What to do? WHAT TO DO?
Well, let’s take it in baby steps. Let’s see what y looks like it’s going to become as we slowly make x smaller and smaller.
x = 10, x2 = 100, x2/x = 10
x = 5, x2 = 25, x2/x = 5
x = 2, x2 = 4, x2/x = 2
x= 0.01, x2 = .0001, x2/x = 0.01
Hmmm! It kind of looks like, y wants to be whatever x is. In fact if x were 0 …, no wait, we can’t go there, it’s against the intergalactic rules.
What about x = .00000000000001 ? :
x2 = .000000000000000000000000001, y = x2/x = .00000000000001 (trust me, I’m a professional!)
Wow. X is getting pretty close to 0 and so is y!
Maybe the limit of y in this example is, GULP! YES IT IS! It’s 0!
Lim x2/x = Lim (x times x)/x = Lim x = 0 EUREKA!
ε = x->0 ε = x->0 ε = x->0
Notice how I got rid of one of the x’s in the numerator by dividing (also called canceling)? You can’t do that if you let x = 0 because YOU CAN’T DIVIDE BY ZERO!
But if you sneak up on it with a “limit”, you can see what will happen if you make x as close as you like to zero, you know, ε close. Go ahead, pick a number, any number!
Calculus is a very powerful method of mathematical calculation based on the properties of limits, It’s so powerful, it can break the Intergalactic Rules of Mathematics. Well, it can bend them anyway! But there's lots more to limits. The Intergalactic Congress has been very busy!
It’s all Greek to me!
By the way, about that promise of meeting cute girls, it helps if they are as smart as you are and just as bored with 5th grade math.
Saturday, April 9, 2011
Amazing Numbers in a Complex World
- (This is for Jordan because he appreciates really cool, weird looking math writing. I do too!)
εiπ = -1
What the heck does that mean? What is ‘ε’ ? Why are ‘i’ and ‘π’ written above it? This is so strange. Did Martians write this? When were they here? Oh, let’s worry about that another day shall we?
ε is a letter in the Greek alphabet. It’s name is pronounced “epsilon.” Sounds like a good name to give your dog.
π is another Greek letter. It’s name is pronounced “Pi”, sort of like “pie” but pie are round and
πr2 . (That’s a dorky math joke.)‘i’ is just the 9th letter of the good old American (actually Arabic) alphabet.
Each of these letters is used to represent some very peculiar numbers. We use letters because the numbers we want to talk about are hard to write. For example, take π. We all learn this number in grade school. It’s value is 3.14. Well actually, it’s more like 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 …. PHEW! Are you kidding me? Hey I have an idea. Let’s call it π.
But what’s this ε ? Another whopper. 2.71828 18284 59045 23536 02874 71352 66249 …
JEEZ LOUISE! You’re killing me already!
But the real puzzle, the real ENIGMA" is ‘i’. (I like that word "enigma". It's kind of sexy and it makes me sound smart because lots of people don't use it much.) Its value is simple to write. The value of the number it represents is √(-1).
What’s so puzzling about that?
Don’t you know anything? Don’t you pay attention in school? Everyone KNOWS that you just can’t take the square root of a negative number. Why? Because any number multiplied by itself will always have a positive number for a result. A negative number times the same negative number will be a positive number. So, if there is no number that you can “square” and get a negative number, how can you have a negative number to take a square root from? Hey, go get a glass of juice or take a bike ride or something and stop hyper-ventilating. Your mother always said it was not healthy to be so "negative", (Yuk! Yuk!) Ahem! Sorry.
You're back? GOOD!
Oh! By the way, we use ‘i’ because it’s stands for “imaginary.” That would be a very good description of a number like √(-1)! But, trust me, ‘i’ is very real (shouldn't that be "'i' am very real?" Never Mind!), which is to say, it truly does exist. Actually, “real” and “imaginary” numbers are two sub-classes of something called “complex” numbers. That’s not too surprising after what you’ve read so far! But I digress. In fact εiπ is a complex number.
These numbers are just mind boggling. They have “magical” properties.(1)
If you take ε and raise it to the power of i multiplied by π, the result is a very reasonable , easy to understand and to WRITE, -1 ! Wow! How does that work?
Well, grasshopper. Trust me. It does. I was first told about this mystery when I was 18 years old in my first year of college calculus. I did not understand how this worked until many years later.
But IT IS SO COOOOOOOL!
1 "Any sufficiently advanced technology is indistinguishable from magic." Arthur C. Clarke, “Profiles of the Future”, 1961 (Clarke’s third law)
Subscribe to:
Posts (Atom)