Thanks! Shortly after posting this I wrote a Python script to confirm my suspicions.
import random students =  # Mr. A's Class for i in range(20): if i < 12: students.append('AF') else: students.append('AM') # Mr. B's Class for i in range(25): if i < 15: students.append('BM') else: students.append('BF') for s in ['AM', 'AF', 'BM', 'BF']: print '%s: %s' % (s, students.count(s)) abel_count = girl_count = 0.0 for i in range(1000000): student = random.choice(students) if student == 'F': # If a student is a girl... girl_count += 1 if student == 'A': # If that student is in Mr. A's class... abel_count += 1 print abel_count/girl_count
From a precalculus text:
There are two precalculus sections at West High School. Mr. Abel's class has 12 girls and 8 boys, while Br. Bonitz's class has 10 girls and 15 boys. If a West High precalculus student chosen at random happens to be a girl, what is the probability she is from Mr. Abel's class?
This problems also comes with the hint: The answer is not 12/22. The answer in the book is 3/5. I'm lost.
Here's my (presumably wrong) logic.
Let E = The student is a girl
and F = The student is in Mr. Abel's class
Then, P(F|E) = P(E and F)/P(E) = (12/45)/(22/45) = 12/22 which not only doesn't match the book's answer, but also directly contradicts the hint the book gives.
What am I not seeing?
I am teaching an Algebra I course in a community college. Many of these students are non-traditional, returning students (i.e. haven't been in a math class in 20+ years), or students that didn't take high school terribly seriously.
I would like to assign a research project in order to make them have a better understanding of how they might be able to apply math in their everyday lives. I'd like to give them examples of possible projects. Some that I've thought of are:
Calculate the gas millage of three different vehicles. Look up the claimed millage and attempt to explain any discrepancies you find.
Use special triangles to find the height of three different trees or buildings.
Estimate how many books are in the NHTI library. There are a variety of techniques for estimation, try to use at least three.
Choose an item that is available at at least five stores, and determine which is the cheapest. Be sure to factor in the cost of travel.
Often times when you order a package online, you have the opportunity to track it through the delivery company's website. Calculate the speed your package is moving at any time and try to determine what kind of vehicle is delivering it. Contact the delivery company to determine if you were correct.
Graph the relationship between the height and weight of at least 10 people. Try to find a ``line of best fit" and see if you can predict the weight of someone with your model.
I would love to hear ideas on possible research projects. The more the merrier!
The subjects covered in this class are:
Exponents, Roots, Powers of Ten
Order of Operations and Problem Solving
Multiples and Factors
Equivalent Fractions and Mixed Numbers
Adding, Subtracting, Multiplying, and Dividing Fractions and Mixed Numbers
Percent and Number Equivalents
Percent Increase and Decrease
Adding Signed Numbers
Subtracting Signed Numbers
Multiplying and Dividing Signed Numbers
Signed Rational Numbers
Powers of Ten
Solving Linear Equations (with or without fractions and decimals)
Inequalities and Sets
Solving Linear and Compound Inequalities
Graphical Representation of Linear Equations and Functions
Graphing Linear Equations
Linear Equation of a Line
Laws of Exponents
Basic Operations with Polynomials
Lines and Angles
Volume and Surface Area
Special Triangle Relationships
This time next year I'll have a bachelor of science in mathematics. What sorts of jobs might I be able to get, and what sorts of things should I be doing now to help ensure I get said job?
Right now I'm thinking about:
But I'm curious about other ideas people have. Thanks!
I'm interested in putting together a math dependency tree, and I'm interested in your help. I'm interested in including branches of mathematics as well as non-math like (theoretical) computer science and physics. My chart so far:
My idea is if someone is interested in quantum computing, or measure theory, or Galois theory, etc., they will be able to look at the chart to see what they need to know first.
Unless anyone has any objections, as this chart expands I will edit the image in this post so that most current chart appears in the first post.
Any and all input is most welcome.
I'd like to start typing my homework because I really like the way it looks. I'm used to using LaTeX online, so I feel like that is the way to go.
My question is, what sort of software do I need to create printable documents that allow me to use LaTeX?
I've been working on this for a few days now and I can't for the life of me make any progress on it. I feel like I'm missing something obvious.
The task at hand is to simplify the following:
The book that this comes from (and my calculator) say that the answer is 2.
Any help (hints, a fully worked out solution, anything...) would be most appreciated.
I think I'm missing something obvious here, please help.
Towards the end of the Math World article on Schur Numbers (http://mathworld.wolfram.com/SchurNumber.html) the inequality S(n) ≤ R(n) - 2 is given, where R(n) is a Ramsey number. Everything I've read on Ramsey numbers says that R has two inputs (http://mathworld.wolfram.com/RamseyNumber.html), yet in the Schur article, it gives R as a function of one variable.
I can't make any sense of this, please help.
My understanding (which is mediocre at best) is that there's a way to define the zeta function on complex numbers. If zeta is defined on real numbers, (as you did it) I think you're correct in saying it diverges. But when it's define on complex numbers, it converges.
I think this: http://www.proofwiki.org/wiki/Equivalence_of_Riemann_Zeta_Function_Definitions might be a good place to start.
I would love for someone with some experience to shine some light on this.
I have the opportunity to be a long term substitute for eighth grade in my local middle school. Part of the curriculum calls for "academic enrichment," a 40 minute period to study something fun (but still educational). I want to spend my time doing math that isn't part of the curriculum and isn't something the students are likely to see in high school.
Yesterday I introduced the students to the Euler Characteristic of a plane by having them draw random graphs and counting the number of vertices, edges and areas for each. After about 30 minutes, half of the class discovered that V-E+F = 1 for all graphs.
Next week I plan on having the students build the Platonic Solids out of drinking straws and yarn. I also hope to work with some number sequences.
Any ideas that I might add to my curriculum? The idea is to get the students actively engaged in doing mathematics, rather than just learning mathematics.
A few months ago I wrote a random Haiku generator in Python. Here are some that it wrote:
A hungry baby
wonderfully but poorly cries
over bright crayons
The cute, little spring
eagerly but sadly walks
through cool, bouncy stars
The happy puppy
poorly, not loudly, kisses
for fractal muffins
The dumb, hungry sky
angrily and sadly thinks
about green lettuce
They're nonsensical, but I think they are funny. I hope you like them.
149 = ∫|2x|dx (-7<x<10)
148 = 37*2^(∑2^-i) (0≤i<∞)
147= 12110 (base 3)
146 = 100 + 50sin(x) + 4sin(3x) [x=pi/2]
145 = curve length from 0 to arcsec(46) of ln(cos(x))
144 = Only square Fibonacci Number > 1
143 = CXLIII (roman numerals)
142 = ∑(5+n²)² [0≤n≤2]
141 = ∑(2-|n-1|²)10^n [0≤n≤3]
140 = x²(x+3)(4x+1) [x=2]
139 = 2n+1 [n=integral from 0 to 3 of 23 dx]
138 = f'(6) [f(x) = (2/3)x³ +4x² +18x + 7]
137 = distance between (12,-5) and (100,100)
136 = zz* [z = 10+6i] [z* is the conjugate of z]
135 = 5x^x [x=3]
134 = 10n + ((n+1)(n-1)+1)/n -2cos(π)
133 = ∑11^n (0≤n≤2)
132 = 10^2 + 2^5
131 = (120x +11x -100 -31)(x+1)/(x^2-1)
130 = 104% of 125
129 = distance between (0,0,0) and (4,7,8)
128 = 2^n [n = lim (n -> 1) (7x-7)/(x-1)
127 = cuberoot(x-2)=5 [solve for x]
126 = exp(ln(2) + 2ln(3) + ln(7)) [exp(x) = e^x]
125 = (n+1)^(n-1) [n=4]
124 = Surface area of a rectangular prism with side lengths 2, 4 and 9
123 = ∑(3-n)10^n [0≤n≤3]
122 = |11+i|²
121 = f(6) [f(n) = 3*f(n-1)+1, f(0)=1]
120 = Cent et vingt
119 = a^b-(a+b) [a=2, b=7]
118 = 6!/5 - 62
117 = ∑(3n)² [2≤n≤3]
116 = (-3+3√(3)i)³-100
115 = max(228x-x²)
114 = a^b-(a*b) [a=2, b=7]
113 = x² = 15² + (x-1)² [solve for x]
112 = ∑2^n [4≤n≤6]
111 = 11² - 10
110 = [see below]
109 = |60+91i|
108 = (a^a)(b^b) [ab = 6 = a+b+1]
107 = 3 + 54 + 31 + 19
106 = zz* [z = 5+9i]
105 = Product of first three odd primes
104 = exp(3ln(2))*exp(ln(13))
103 = Sum the roots of x² - 103x + 2652]
102 = curve length from 1 to .25(205 + sqrt(42033)) of x²/2 - ln(2x)/2
101 = ∑ n!*(-1)^(n-1) [1≤n≤5]
100 = ∑ 2n+1 [0≤n≤9]
If any others need to be LaTexed let me know. It's so tedious I tried to do with out it...
What is your target demographic for this? Are you a teacher? If so, what age group are your students? If you are in fact a teacher, I'd suggest having your students do the rest (numbers 1 through 99). There's a lot more to be gained from creating these formulas than there is to solving them.
I think I'm going to pass on the remaining 99, I've been spending too much of my time on this. If you really want me to them, I will, but as I said... they're super educational (and tons of fun).
Any branch of mathematics can be used.. logarithms, trigonometry, complex numbers, lim, rot, div, integrals and derivatives etc... just anything that would give a numerical result.
200 = 5²(2³-1)(2-1)
199 = 200 + arccos(pi)
198 = √(39204)
197 = 200 - log(1000)
196 = (x-6)²(x-1)² [x = 2³]
195 = x²-390x+38020 (solve for x)
194 = Area of rectangle with sides 2 and 97
193 = 44th prime
192 = f'''(π/4) [f(x) = 3*sin(4x)]
191 = x intercept of y=2x-382
190 = 1+2+3+4+...+19
189 = distance (in meters) traveled at 7 meters/sec for 27 sec
188 = Re[(10+3i)(20+4i)]
187 = area of triangle with base 22 and height 17
186 = number of seconds in 3.1 minutes
185 = distance from (5,5) to (65,180)
184 = 5²√(16) + 80 + |6-10|
183 = x²-136x+3783 [x = 100]
182 = (100*cos(x)+80*sin(x)+2*cos(x+2π))*(√2) [x=π/4]
181 = largest prime factor of 25340
180 = 6∫(x²+2x)dx (evaluate from 2 to 5)
179 = lim (x⇒∞) (179x+3)/x
178 = x-value of the min of f(x) = (x-300)*(x-56)
177 = 175 + ∫sin(x)dx [0<x<pi]
176 = LCM(16,22)
175 = 5² + 5² + 5³
174 = 200 + (-4+6i)(2+3i)
173 = distance from the origin to (52,165)
172 = volume of block with side lengths 2,2,43
171 = n(n+1)/2 [n=18]
170 = 10*(∫(-17/x²)dx) [1<x<∞]
169 = area of circle with radius 13/√π
168 = 10*2^4 + 2^3
167 = 10100111 (base 2!)
166 = 369 mod 203
165 = 11 Choose 3
164 =The shortest possible perimeter of an isosceles triangle with two of its side-lengths 50 and 64
163 = 100*ln(e)+10*log(1,000,000)+ln(e³)
162 = length of the curve (1/3)x³-x+4 from 3 to 165
161 = cuberoot(4173281)
160 = sum first 11 primes
159 = 2^7+31
158 = ∫dx/x [e<x<e^159]
157 = Im(z) - Re(z) [z = (8+9i)(4+9i)]
156 = (10-2i)(15+3i)
155 = gcd(775, 465)
154 = Perimeter of a rectangle with side-lengths 50 and 27
153 = (27/3)/((8/2)/68)
152 = ½(18!/16!)
151 = Smallest palindromic number greater than 145
150 = perimeter of a right triangle with leg lengths 25 and 60
More tomorrow. It's time for bed.
(PS, be sure to check all of these before using them. It's very late, so I might have made some mistakes).
Can you expand on this please?
Of course. I think "intelligence" is an emergent phenomenon. I don't think a program (or organism) is truly intelligent unless it can learn and I think emotions are a side effect to learning. To the best of my knowledge (which is very, very limited) there are two approaches to AI: the "top down" approach and the "bottom up" approach.
The top down approach is simply programming a computer to be knowledgeable. A program knows something because it is programmed to know something. An example of this I made is here: http://spamtheweb.com/ul/upload/140410/77502_face.php. This is a very simple program that will tell you if a mouse drawn face is smiling or frowning. Try it out by drawing a "smiley face" and then pressing space. How did it know? Simple, when I made it, I told it what a happy face looks like (And what a sad face looks like). This program may give the impression of intelligence, but (as far as I'm concerned) it's not.
The bottom up approach is to program a computer with very simple rules and have the program "learn" over time. Here the program is born stupid but it not stagnant. (I don't have a good example to post, sorry). My belief is that any being (organic or artificial) that can learn will develop emotions. To me, thinking and feeling are the same thing. As far as I'm concerned, the brain is simply a computer and any computer built to be as intelligent as a human brain is probably going to have many of the tendencies. Sufficiently intelligent computers will make mistakes, feel emotions and even behave irrationally.
Two books I recommend: Artificial Life by Steven Levy (non-fiction) and Galatea 2.2 by Richard Powers (fiction). Both of these books explore this topic extensively.
Interesting side issue: for a computer to pass the Turing test it would also need to be deceptive. Ask it for the square root of pi and it needs to go "Umm...let me think... about ummm.... 1 point ahhh..."
...or be genuinely bad at math. All of our computers today do computation well, but computers of the future might not work that way. It seems to me, if we want to have truly intelligent and creative computers, we need to make them dumb at some things.
As for why we would want to give computers emotions: I don't think we have a choice. I think emotions are a side effect of true intelligence. I believe that emotions are an unavoidable consequence of intelligence/creativity.
Is artificial intelligence an achievable gain? If not, why? Will we be able to simulate emotions (or will they be actual, not simulated, emotions)?
What advances do we have to make in hardware first? Software?
I'll share my opinions soon; I'm curious to know what everyone else thinks.
When I use Python, I use IDLE or PyScripter. I don't compile my code, so I guess I'm using a scripted language.
I look forward to someday learning Java, but I simply don't have the time right now. But if I don't have to learn it to use Jython, I can jump right into no problem.
What are typical "needs" of a compiled/scripted language?
Calccrypto: I don't understand your question. I use IDLE to write my programs (and more recently PyScriptor). Make no mistake, I have no idea what I'm doing, I'm doing this for fun.
Ricky: Wow, that's a whole bunch of techno-lingo, but at least I have a starting place. Am I going to have to learn Java to use Jython? How do I figure out which implementation of Python I'm using? IF I'm using the wrong implementation of Python, how do I use the right one? And how will that change what I'm doing?