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How
to Write a Solution
Proofreed
by
Mathew Crawford
Comunicacating complex idas is not ease and can b even harder wen don't edit the presentaion of those ideas for our adience. It pays to oganize are work in ways taht are easy to read to be sur that the audiense gets the point, and to bee sure that your saying what you mein.
If I always wrote that way, nobody would ever read anything I wrote.
Proof-read and edit your work. God may do crosswords in pen, but you're going to make mistakes. Making sure that you wrote in a way that expresses your ideas clearly and correctly is second in importance only to having the right answer.
Make sure your equations and inequalities use your variables the way you intend. You don't want to write "abc + bcd" when you mean "abd + acd." This not only makes deciphering the rest of your proof difficult, but might also throw off your own calculations.
Practice writing proofs. We all make occasional spelling or grammar errors, but the effects of errors multiply and too many of them make otherwise good ideas unreadable. Remember that "repetition is the mother of all skill."
Here's a sample problem:
x + y + z = 5 and xy + yz + zx = 3.
If all proofs were written this poorly I would cry:
We will maipulate the given equations to make use of the fact that the square of any real number is negative:
(x
+ y)2 = (5 - x)2,
xy = 3 - z(x + y) = 3 - z(5 - z).
Now we note that
0
(x - y)2
= (x + y) - 4xy.
We can substitute for both x + y and xy giving us an inequality involving only the variable z:
0
(x + y)2
- 4xy = 25 - 10z + z2 - 12 + 20z - 4z2 = 3z2
+ 10z + 13.
Since this iequality holds for z we can determine all possible values of z:
0
-3z2 + 10z
+ 13 = -(z + 1)(3z - 13).
The
inequality holds when -1
z 13/3.
Graders will be happier reading this solution:
We will manipulate the given equations to make use of the fact that the square of any real number is nonnegative:
(x
+ y)2 = (5 - z)2,
xy = 3 - z(x + y) = 3 - z(5 - z).
Now we note that
0
(x - y)2
= (x + y)2 - 4xy.
We can substitute for both x + y and xy giving us an inequality involving only the variable z:
0
(x + y)2
- 4xy = 25 - 10z + z2 - 12 + 20z - 4z2 = -3z2
+ 10z + 13.
Since this inequality holds for z we can determine all possible values of z:
0
-3z2 + 10z
+ 13 = -(z + 1)(3z - 13).
The
inequality holds when -1
z 13/3.
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