I'm moving tomorrow, and I'm not sure I'll have internet there so I'm trying to download as many Calculus and Linear Algebra videos by Khan Academy as I can. If you haven't read about Khan Academy or seen Khan's videos already, and you want to understand math, biology, and even chemistry, then go check him out on Youtube.
I read a little today about the difference between a sequence and a series. A sequence is a list of numbers with or without a relationship between the numbers, and a series is the sum of all the numbers or terms in a sequence. There are two main types of sequences: there are algebraic sequences and there are geometric sequences. The difference between geometric and algebraic sequences is in the difference (pun intended).
In an algebraic sequence, you have a list of numbers in an order, each having a common difference. Common difference means you can subtract any term in the list with the term after or before it and get the same difference no matter what term you choose. In geometric sequences, there is a common ratio. Instead of getting the difference, the ratio is the same for any term before or after whatever term you choose in the list.
I hope to learn in more detail why geometric sequences, in particular, are important to Calculus. I think that the answer lies within the integral being an infinite sum of a product. I'm not sure my choice of words is correct but I think I'm onto something.
Thursday, December 30, 2010
Monday, December 27, 2010
Multiplication in a Vertical Fashion
I was doing some multiplications 'in my head' the other day. No big deal, just some easy multiplications. In school, elementary or middle school, I was taught how to multiply in a vertical representation. I'll have to go on Google Docs or something for illustrations. However, today I realized that doing math this way is less intuitive than doing math in a horizontal fashion.
In the vertical fashion of performing the multiplication of big numbers, you are required to place a zero after a certain row is completed at the end of some numbers. I don't remember ever being explained the logical background of placing a zero after the the first row of this 'large' multiplication is completed. I think the horizontal approach to solving a multiplication is more intuitive.
Say you have to multiply 34 by 42. There's only one way that you can perform with the vertical method or fashion where you stack these two numbers and multiply the bottom right number with the top right and then the top left numbers placing the product in that order under the stack. I'm sure anyone who has read this far knows enough about multiplication to know the next steps.
But, if you think of the multiplication horizontally, there are many ways that this multiplication can be performed by taking advantage of the commutative, associative, and distributive properties of operations. You could see the multiplication as 34(40 + 2) = (30 + 4)(40 + 2) = 1200 + 60 + 160 + 8 You don't even have to think about 'placing a zero' or how many zeros you should place.
In the vertical fashion of performing the multiplication of big numbers, you are required to place a zero after a certain row is completed at the end of some numbers. I don't remember ever being explained the logical background of placing a zero after the the first row of this 'large' multiplication is completed. I think the horizontal approach to solving a multiplication is more intuitive.
Say you have to multiply 34 by 42. There's only one way that you can perform with the vertical method or fashion where you stack these two numbers and multiply the bottom right number with the top right and then the top left numbers placing the product in that order under the stack. I'm sure anyone who has read this far knows enough about multiplication to know the next steps.
But, if you think of the multiplication horizontally, there are many ways that this multiplication can be performed by taking advantage of the commutative, associative, and distributive properties of operations. You could see the multiplication as 34(40 + 2) = (30 + 4)(40 + 2) = 1200 + 60 + 160 + 8 You don't even have to think about 'placing a zero' or how many zeros you should place.
Philosophy of Math Blog
I like math quite a bit, so I decided to make this blog for my ideas about math. I have posted about math on some of my other blogs, but I realize that nobody who likes math will be interested in subscribing to those blogs if there is a huge interval of posts that have nothing to do with math between every post about math. I was surprised the Philosophy of Math blogger domain wasn't already taken. I think Math Philosophy was taken, though.
I'm no expert on any topic in math or math in general (I may be considered an expert in arithmetic by some). This is only my attempt at explaining math with many words. I will use words that you never thought could describe phenomena in math. See, I just described something in math as a phenomena. I crack myself up. My goal with this blog is to talk only about topics in math, things on the internet about math, discuss problems involving math.
I'm no expert on any topic in math or math in general (I may be considered an expert in arithmetic by some). This is only my attempt at explaining math with many words. I will use words that you never thought could describe phenomena in math. See, I just described something in math as a phenomena. I crack myself up. My goal with this blog is to talk only about topics in math, things on the internet about math, discuss problems involving math.
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