- B1 - Writing a googolplex
- B2 - Fraction woes
- N1 - Factor question
- A1 - I need your
help fast
- A2 - The 'two trains' problem
- A3 - Miminizing a weird function
- A2 - The 'two trains' problem
- T1 - Sine, Cosine, Tangent
- C1 - Graphs of functions and derivatives
- M1 - Profit curve
for econ?
- M2 - Too big for calculator!
- M3 - Scarecrow was not a wizard
- M4 - Good grapefruit prediction
- M5 - The Speed Of Dark?
- M2 - Too big for calculator!
- Dear Dan - Our math professor
at U of Louisville posed this question:
- If one million zeroes can be written on the front and back of a sheet of
- paper, how many sheets of paper are necessary for a googol of zeros?
We have to come up with answer (which was not a problem - it's 10 to- the 94th power sheets of paper) and we have to be able to explain the
- problem and solution to a 12-year old. How do you explain such a
- large number to someone - what can you relate it to? - Karen
- Hi Karen -- Thanks for writing -- interesting question. A googol is 10^100 and a 1 with
- a googol zeroes after it is called a 'googolplex.'
- I thought maybe you could relate the thickness of the paper (a ream of paper, 500 sheets, is
around 2" so each sheet is about .004" thick) and a large distance, say from the earth to the
moon (around 240,000 miles). I'll let you figure out how the number of sheets of paper
from here to the moon compares with 10^94 (still nowhere close?), so talk about how many
trillions of stacks of paper like that you'd need?- Another aspect is time: if you wrote 1 number per second (or a printer spit out one sheet
per min), how many centuries would it take if all 6 billion people wrote (or had printers)?
By the way, 1 million seconds = 12 days and 1 billion sec = 31 years, approx!- Ask the 12-yr-old how far away they've traveled (say New York, 3000 miles), then maybe
relate the earth-moon distance to that (80 times as far). -- Hope this helps!- Meanwhile don't waste paper writing out a googolplex in base 10. ;-}
- -- Dan the Math Man -- www.dansmath.com
- If one million zeroes can be written on the front and back of a sheet of
- Hello Dave, I am a 7th
grader and having a hard time with fractions.
- Could you point me in the direction of a helpful site? Thanks, Will
(Coming to you from Colorful Colorado, home of the Awesome Avalanche!!!)- Hi Will... (My name's Dan but I'll answer to Dave...)
- I was going to point you to my site, www.dansmath.com , but you're right, I don't
have much currently there on fractions. I will give you some pointers but I would
suggest www.google.com -- type in 'Fraction math help' and I swear it will give
you some excellent active choices. Look for some on-line quizzes as well.- Here's one I just found: www.mathleague.com/help/fractions/fractions.htm
- You can also go to 'expert sites' such as www.askme.com and look for message
items or ask questions there. I'll be happy to answer specific questions or problems
for you too.- As for fractions themselves: here's a quick review: The top is the 'numerator', and the
- bottom is the 'denominator'; in 5/7 the 5 is the numerator and the 7 is the denominator.
- Fractions with the same denominator can easily be added or subtracted: keep the same
denominator and add or subtract the numerators. 5/7 + 3/7 = (5 + 3)/7 = 8/7 or 1 1/7.- Fractions can be converted or reduced: 4/6 = 2/3 (divide top & bottom by 2),
or 5/7 = 15/21 (multiply top & bottom by 3).- Fractions with different denoms have to be converted to a common denom (LCM)
2/5 + 1/3 = 6/15 + 5/15 = 11/15 , etc.- Multiplying two fractions: multiply tops and bottoms separately:
2/3 * 5/8 = (2*5)/(3*8) = 10/24 = 5/12.- Dividing two fractions: iinvert the second fraction and then multiply:
(4/9) / (3/7) = (4/9) * (7/3) = 28/27 or 1 1/27.- Then there's the whole connection with fractions and decimals and percents...
- --Dan the Math Man
- Could you point me in the direction of a helpful site? Thanks, Will
-
- Question N1 - A Factor Question [ top of page ]
- Dan, Hello! My name is Brandon, I am a college student in KY. I was
- asked a question the other day, that has stumped me. The question was...
- what is the smallest number possible with exactly 100 factors? This
- question may seem easy to you, but it was kicking my butt. If you could
- help and email me back I would really appreciate it. Thanks again..
- Hey Brandon. You definitely askedd the right person! That's not a simple question so
- give your kicked butt a rest. The idea is to use the prime factoriz'n and work with the
exponents alone to find the number of divisors.- See my lessons page on supercomposites, also called highly composite or versatile numbers.
- For example 48 = 2^4 * 3^1 and if we make a list of all divisors of 48:
- multiples of 3 in 2nd row; others in top row:
1 2 4 8 16
3 6 12 24 48
there are 5 columns of 2, making 10 divisors, and notice the exponents are 4 and 1 ;- you add 1 to each and multiply : d(48) = (4+1)(1+1) = 5*2 = 10.
- The link for my page is: http://www.dansmath.com/lessons/suprcomp.html
read about using small primes and with larger exponents.- Now if you want 100 divisors, figure out how to produce100 as a product of (1+exps) :
way: expons prime fac'z'n number (you calc it)
10*10 9 , 9 2^9 * 3^9
10*5*2 9, 4, 1 2^9 * 3^4 * 5^1
5*5*4 4, 4, 3 2^4 * 3^4 * 5^3
5*5*2*2 4, 4, 1, 1 2^4 3^4 5^1 7^1- one of these (the smallest, duh) is the butt-kicker you seek. Hint: it's got 5 digits!
- Hope you enjoy my page; this one type of question has held my interest for years!
- Thanks for letting me tell you about this; no extra charge! -- Dan the Divisor Man
- Question N1 - A Factor Question [ top of page ]
- Hi Dan, I live in Cleveland,
Ohio, and I need your HELP.
- Can you teach me algebra? I need to learn, and learn fast,
- I am applying for an apprenticeship program and the
- requirements are algebra. I really need this opportunity,
- so can you please teach me? -- Brian
- Hi Brian - Algebra is a very important subject ; I call it the 'gateway to the
technical world'. If this apprentice program is asking you to know algebra, it- must mean it's for a job that uses it and depends on it. This means you can't
- learn it hastily or all at once. If I wanted to be fluent in speaking Russian, I
- wouldn't be able to do it in a few days or weeks.
- But I do have a lot of lessons on my website, in all levels; arithmetic, prealgebra,
beginning alg. and on up. Check them out at www.dansmath.com -->- math lessons --> Basic Skills. There are also lots of links on my site or go to
- www.google.com and type 'algebra help.' It's a great way to find stuff.
- I also have a great textbook that could be used for self-study, check it out at my
site at 'meet dan' then click textbook. If you want to order the book online, get- the details ready (Bach/Leitner Prealgebra, Houghton Mifflin) and then go to
- bookcenter.dvc.edu and order with a credit card, they ship! -- DaMathMan
- Can you teach me algebra? I need to learn, and learn fast,
- Dear Dan - two trains
leave stations 100 miles apart.
- "A" train leaves at 12:00 and "B" train leaves at 1:00.
- when, and at what mile, will they collide? there will be
- no loss of life. > mark
i would appreciate any help. big test on sat am. thank you.
- Hi Mark - I mean mark.
- If we knew the speed of each train then we could set up distances:
- t = hours past 12:00 ; and the trains have gone dA and dB:
dA = (speed of A)*(t hours) ; dB = (speed of B)*(t - 1 hours)
because B starts at 1:00, 1 hr later, so 1 hr less time. (both are d = r t.)- Now our equation is dA + dB = 100 miles, but we can't solve for t unless
we have some real numbers for those speeds. If you forgot to include those,
that's where you'd use them; if the speeds weren't given in the problem
then it wasn't a well-phrased question.- One thing we do know is if A is going more than 100 mph, the collision
- will occur at B's home station, maybe with loss of life! - Dan the Math Man
- "A" train leaves at 12:00 and "B" train leaves at 1:00.
- Dear Dan, How would you
find the minimum of
- y = square root of (x^2 + 4) + square root of (x^2 - 6 +10).
- Do you have to use calculus? - Tara
- Hi Tara.
- You can use calculus, to minimize y = sqrt(x^2 + 4) + sqrt(x^2 - 6x + 10)
by taking the derivative dy/dx and setting it equal to zero. That looked icky,
but do-able. But you'd prefer a calculus-free approach...- Or you can type the function into your graphing calculator or some
computer algebra system like Mathematica (see my website), and try
to spot a min. This one looks like x = 2 gives a smallest value of about
y = 4.25 or so. Then you could go back to the formula and see if you
could complete the square to see the min at x = 2:
y - sqrt[x^2 + 4] = sqrt[(x-3)^2 + 1] ; square both sides, simplify,
isolate the square root term, square both sides again, and the result should
have some sort of factor of (x - 2) if my intuition is right.- I'll let you have fun with the algebra. Hope this helps! - Dan the Man
- y = square root of (x^2 + 4) + square root of (x^2 - 6 +10).
- Dan, please help!! I have
been trying to figure out sine, cosine, and tangent
- for about 2 months, and I can't quite get it. I just don't understand how to
- get to the answer from: sin/cos/tan(angle)=(side length over side length)
I also do not know how to do sin/cos/tan when you only have one side length.- Please, please, please help!!! ~ Michelle ~
- Hi Michelle - Have you heard of 'SOHCAHTOA' ?
- Draw a right triangle with right angle at the lower right; the angle T at the left;
- the Opposite (vertical) and the Adjacent (base) and the Hypotenuse (rising diagonal).
- It means Sine = Opp / Hyp , Cosine = Adj / Hyp , Tangent = Opp / Adj .
You might like sin(T) = y / r , cos(T) = x / r , tan(T) = y / x .- Along with the Pythagorean Theorem x^2 + y^2 = r^2 , you can figure
out the whole story just by knowing any one side, and one other angle.- Please visit my (free!) web lessons at www.dansmath.com for more ; go to the
- Precalculus area and click Trig. There are pictures and even some animations
- there for you to enjoy / learn from. And write back any time with further or more
- specific questions.-- Dan the Math Man strikes again
- for about 2 months, and I can't quite get it. I just don't understand how to
Calculus Questions
- Dear Dan: Could you explain
the principles on graphs with
- relation to their derivative? - Shane
- Hi Shane - Happy to 'explane'; thanks for asking.
- The derivative of y = f(x) at a point (a, b) is the slope of the tangent line
to the graph at the point (a, b). This means that b = f(a) and that the- graph and the tangent line have the same slope at x = a.
- But the graph of the derivative is different; it records the slope of the graph
at each point, so if the function is increasing, the derivative is positive, and
if the graph of f(x) is decr, then f '(x) is neg.- I have a picture on my site of a graph, its deriv, and second deriv. The link is:
- http://home.earthlink.net/matica.html ; go to the 'options for graphics' section.
- You've inspired me to make a movie of a moving tangent line along a curve
y = f(x), and a second curve recording the slope f '(x) as it goes along.- Hope this helps a little; try graphing a few pairs {function , derivative}
on your graphing calculator, like {y = x^2 , y = 2x} to see the relation. -Dan - relation to their derivative? - Shane
