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What would you use other than symbols? Describing everything with words? For example: When given two numbers, if one wishes to find the quantities that give 0

by dbaupp 13y ago

What would you use other than symbols? Describing everything with words? For example: When given two numbers, if one wishes to find the quantities that give 0 when the second of the numbers is added to the product of the quantity and the first number and the quantity multiplied by itself, one should negate the first number and then either add or subtract the square root of the sum of the square of the first number minus four times the second number, and divide this summation by two. That's just the quadratic formula in disguise: Let b and c be real [or complex] numbers, then x^2 + b x + c = 0 implies x = (-b +- sqrt(b^2 - 4c)) / 2 Clearly the latter is easier to understand and digest; the same holds for higher mathematics. In fact, there are multiple interpretations of the text (admittedly I just wrote that now, and I'm not the best writer), while the symbolic mathematics itself is essentially entirely non-ambiguous (given some background in symbolic algebra). (Saying "let's do maths without the symbols" is a little like saying "let's do programming without special languages"... it is very very hard to make it work.)

aidos 13y ago
Ah yes - I used to love reading Euclid's proofs [1]. Wonderful descriptions. He used such poetic phrases too - things like describing a line as 'a length without breadth.' As wonderful as it is, mathematics needs notation, and lots of it. You can express incredibly complex ideas in mathematics, totally unambiguously, through a collection of symbols. Not to mention that they're universally recognised. The reality is that mathematics is 100% about thought. You'll struggle to put together the concepts in your mind long before the notation is the real issue. Once you have a clear picture of the abstract space you can use the notation you've learned to communicate the world you've created to others. What could be more wonderful? [1] http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI2.html http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI...
- upquark 13y ago
  Agreed. Plus, the current notation is a result of multiple iterations by some brilliant people. If you look at older attempts at formal mathematical notation, some of it is laughably bad in comparison. We are standing on the shoulder of giants :)
thetwiceler 13y ago
Pictures are worth a thousand words! Now let me put the quadratic formula in disguise as well. Suppose we wish to make a rectangle with a given area and perimeter. This is an interesting problem! Does the number of possible answers depend on the specific area and perimeter? Certainly! It all comes down to thinking about squares, since squares maximize the area given a fixed perimeter. If the area of a square with the given perimeter is LESS than the desired area, then there's no way we can make such a rectangle. If the area of the square is equal to the desired area, then that's our only answer! Now, how about if our square's area is larger than the desired area? We'll get two possible different lengths of a given side of the rectangle - one representing the rectangle's width, and the other representing it's height. Or we could also think of them as two different rectangles - a tall one, and its rotation by a quarter turn (which makes it wide). By symmetry, we know that the difference between the square's side length and the shorter side will be the same as the difference between the square's side length and the longer side. How large is that difference? Exactly enough to diminish our shape's area from the square's area to the desired area. And that difference in length is simply the square root of the difference between the square's area and the desired area! It would have been better with pictures :). Anyway, the quadratic formula is probably the greatest mistake in all of mathematics education. Somehow we use the word "quadratic" and even the phrase "complete the square," but never have I ever seen someone draw the said square! While I think notation is often great for expressing ideas concisely and precisely, I think an excess of notation not a good way to communicate concepts. Nobody should memorize the quadratic formula! We should understand instead how to think about areas and lengths, and then we solve the problems in quadrature that we want.
- dbaupp 13y ago
  I agree with you, but not everything can be done pictorially. FWIW, an animation of the quadratic formula/completing the square: http://en.wikipedia.org/wiki/File:Completing_the_square.gif http://en.wikipedia.org/wiki/File:Completing_the_square.gif
hmrmaxwell 13y ago
With more notation, some readability is lost again: Let x, b, c \epsilon C: x^2 + b x + c = 0 => x = (-b +- sqrt(b^2 - 4c)) / 2
- ReidZB 13y ago
  Well, this is highly tangential, but if you're talking about LaTeX formatting, \epsilon is not generally what you want for the "element of" symbol. Instead, use \in. The biggest difference is that \in is a binary operator and has the appropriate spacing. Example of the difference: http://i.imgur.com/gwAqirx.png http://i.imgur.com/gwAqirx.png generated by code: \textbackslash{}epsilon: $\epsilon$ \\ Example: $x \epsilon \mathbb{R}$ \textbackslash{}in: $\in$ \\ Example: $x \in \mathbb{R}$ If you use \mathbin{\epsilon} instead, you'll get proper binary operator spacing, but you'll still get odd looks from people who are accustomed to \in. Admittedly, the symbol did historically begin as an epsilon, but that notation died off a while ago.

by dbaupp 13y ago

aidos 13y ago

Ah yes - I used to love reading Euclid's proofs [1]. Wonderful descriptions. He used such poetic phrases too - things like describing a line as 'a length without breadth.' As wonderful as it is, mathematics needs notation, and lots of it. You can express incredibly complex ideas in mathematics, totally unambiguously, through a collection of symbols. Not to mention that they're universally recognised. The reality is that mathematics is 100% about thought. You'll struggle to put together the concepts in your mind long before the notation is the real issue. Once you have a clear picture of the abstract space you can use the notation you've learned to communicate the world you've created to others. What could be more wonderful? [1] http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI2.html http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI...

upquark 13y ago
Agreed. Plus, the current notation is a result of multiple iterations by some brilliant people. If you look at older attempts at formal mathematical notation, some of it is laughably bad in comparison. We are standing on the shoulder of giants :)

thetwiceler 13y ago

Pictures are worth a thousand words! Now let me put the quadratic formula in disguise as well. Suppose we wish to make a rectangle with a given area and perimeter. This is an interesting problem! Does the number of possible answers depend on the specific area and perimeter? Certainly! It all comes down to thinking about squares, since squares maximize the area given a fixed perimeter. If the area of a square with the given perimeter is LESS than the desired area, then there's no way we can make such a rectangle. If the area of the square is equal to the desired area, then that's our only answer! Now, how about if our square's area is larger than the desired area? We'll get two possible different lengths of a given side of the rectangle - one representing the rectangle's width, and the other representing it's height. Or we could also think of them as two different rectangles - a tall one, and its rotation by a quarter turn (which makes it wide). By symmetry, we know that the difference between the square's side length and the shorter side will be the same as the difference between the square's side length and the longer side. How large is that difference? Exactly enough to diminish our shape's area from the square's area to the desired area. And that difference in length is simply the square root of the difference between the square's area and the desired area! It would have been better with pictures :). Anyway, the quadratic formula is probably the greatest mistake in all of mathematics education. Somehow we use the word "quadratic" and even the phrase "complete the square," but never have I ever seen someone draw the said square! While I think notation is often great for expressing ideas concisely and precisely, I think an excess of notation not a good way to communicate concepts. Nobody should memorize the quadratic formula! We should understand instead how to think about areas and lengths, and then we solve the problems in quadrature that we want.

dbaupp 13y ago
I agree with you, but not everything can be done pictorially. FWIW, an animation of the quadratic formula/completing the square: http://en.wikipedia.org/wiki/File:Completing_the_square.gif http://en.wikipedia.org/wiki/File:Completing_the_square.gif

hmrmaxwell 13y ago

With more notation, some readability is lost again: Let x, b, c \epsilon C: x^2 + b x + c = 0 => x = (-b +- sqrt(b^2 - 4c)) / 2

ReidZB 13y ago
Well, this is highly tangential, but if you're talking about LaTeX formatting, \epsilon is not generally what you want for the "element of" symbol. Instead, use \in. The biggest difference is that \in is a binary operator and has the appropriate spacing. Example of the difference: http://i.imgur.com/gwAqirx.png http://i.imgur.com/gwAqirx.png generated by code: \textbackslash{}epsilon: $\epsilon$ \\ Example: $x \epsilon \mathbb{R}$ \textbackslash{}in: $\in$ \\ Example: $x \in \mathbb{R}$ If you use \mathbin{\epsilon} instead, you'll get proper binary operator spacing, but you'll still get odd looks from people who are accustomed to \in. Admittedly, the symbol did historically begin as an epsilon, but that notation died off a while ago.