Mytutorfriend- Your friendly tutor

September 7, 2008

Generating Pythagorean Triples

There is a simple formula that gives all the Pythagorean triples.

Suppose that m and n are two positive integers, with m < n. Then n2m2, 2mn, and n2 + m2 is a Pythagorean triple.

It’s easy to check algebraically that the sum of the squares of the first two is the same as the square of the last one.

Here are the first few triples for m and n between 1 and 10. You can also extend for numbers greater than 10.

   | m= 1          2           3           4            5           6              7            8            9
---+------------------------------------------------------------------------------------------------------------------
n= |
 2 | [3,4,5]
   |
 3 | [8,6,10]    [5,12,13]
   |
 4 | [15,8,17]   [12,16,20]  [7,24,25]
   |
 5 | [24,10,26]  [21,20,29]  [16,30,34]  [9,40,41]
   |
 6 | [35,12,37]  [32,24,40]  [27,36,45]  [20,48,52]  [11,60,61]
   |
 7 | [48,14,50]  [45,28,53]  [40,42,58]  [33,56,65]  [24,70,74]   [13,84,85]
   |
 8 | [63,16,65]  [60,32,68]  [55,48,73]  [48,64,80]  [39,80,89]   [28,96,100]  [15,112,113]
   |
 9 | [80,18,82]  [77,36,85]  [72,54,90]  [65,72,97]  [56,90,106]  [45,108,117] [32,126,130] [17,144,145]
   |
10 | [99,20,101] [96,40,104] [91,60,109] [84,80,116] [75,100,125] [64,120,136] [51,140,149] [36,160,164] [19,180,181]
Advertisements

August 30, 2008

Fibonacci Numbers and the Golden Ratio


Suppose you have a pair of cats given to you as babies. This pair of cats will begin to reproduce when they are two years old. They will produce another pair of cats every year. Each of these pairs of cats will also reproduce–one cat pair per year starting when they are two years old. How many cat pairs will you have after n years?

The answer to this question is a sequence of numbers known as the ‘Fibonacci sequence’:

1,1,2,3,5,8,13,…

These numbers can be described mathematically as follows:

F(n+2) = F(n+1) + F(n)

In other words, each Fibonacci number is the sum of the two preceding Fibonacci numbers. Another way of saying that is, If you know two successive Fibonacci numbers, you can find the next one by adding them together.

There are many reasons why Fibonacci numbers are interesting to mathemeticians. One of the reasons is that Fibonacci numbers are very closely related to the Golden Ratio, a number which the ancient Greeks  find very interesting.

August 13, 2008

Time and Distance-Short cuts

Filed under: Algebra — mytutorfriend @ 11:05 pm
Tags: , , ,

Time and Distance

60 mph = 1 mile per minute

  • Going 60 mph and the exit is in 10 miles? That’s 10 minutes.
  • Been driving a half hour? That’s about 30 miles at highway speeds.

Feet Per Second = MPH * 1.5
MPH = Feet Per Second * 2/3 (derivation)

  • 60 mph is about 90 feet per second (88 exactly), so just multiply by 1.5. Or, just add half to itself (60 + 30 = 90).
  • Going 100 mph? That’s 150 fps.
  • Going 10 fps? That’s about 7 mph (10 * 2/3 is 6.666). Or, just take away 1/3 (10 – 3 = 7).

speed of light = 1 foot per nanosecond (derivation)

  • The US is about 3000 miles long. There’s about 5000 feet/mile, so that’s about 3000 × 5000 or 15 million feet. 15 million feet takes 15 million nanoseconds, or 15/1000, or 15 milliseconds. That’s the minimum time for a signal to go across the country.
  • Inside a microchip, if you have a clock cycle every nanosecond (1 GHz), your signal can only travel 1 foot at most (or less, depending on the material). Even light takes 30ns to cross a 30 foot room.

1 year = 250 work days = 2000 work hours (derivation)

  • Project takes 1000 man hours? That’s 6 months for 1 person.
  • Daily commute of 1/2 hour? That’s .5 * 250 = 125 hours in the car each year.

Money and Finance

$1/hour = $2000/year (derivation)

  • Earn $25/hour? That’s about 50k/year.
  • Make 200k/year? That’s about $100/hour. This assumes a 40-hour work week.

$20/week = $1000/year (derivation)

  • Spend $20/week at Starbucks? That’s a cool grand a year.

Rule of 72: Years To Double = 72/Interest Rate (derivation)

  • Have an investment growing at 10% interest? It will double in 7.2 years.
  • Want your investment to double in 5 years? You need 72/5 or about 15% interest.
  • Growing at 2% a week? You’ll double in 72/2 or 36 weeks. You can use this rule for any duration of time, not just years.
  • Inflation at 4%? It will halve your money in 72/4 or 18 years.

Mental Arithmetic

Numbers

10,000 = hundred hundred
million = thousand thousand
billion = thousand million
trillion = million million

  • 1% of 10k is 100. The Dow is roughly 10k (it’s about 12k now). So if the dow drops 100, it’s about a 1% loss.
  • What’s 5k x 50k? That’s 250 * thousand * thousand or 250 million.

Visualizing numbers

  • 12 days = 1 million seconds
  • 1 year = 31 million seconds (about pi * 10 million)
  • 30 years = 1 billion seconds
  • 30,000 years = 1 trillion seconds
  • One “part per million” means an accuracy of 1 second every 12 days. One “part per trillion” means an accuracy of 1 second every 30,000 years.

Powers of 2

2^6 = 64 (the sixes match: six and sixty-four)
2^10 ~ thousand (1 kb)
2^20 ~ million (1 mb)
2^30 ~ billion (1 gb)

  • Sure, 2 to the tenth = 1024, but 1000 is good enough for government work. (Read on about KB vs KiB).
  • Have 32-bit color? That’s 2 + 30 bits, aka 2^2 billion, or 4 billion (4gb exactly).
  • Have a 16-bit number? That’s 6 + 10 bits, or 2^6 thousand, or 64 thousand (64 kb).

a% of b = b% of a

  • It’s not immediately clear, but it’s true: a% of b = .01 * a * b, which is the same as b% of a (.01 * b * a).
  • What’s 16% of 25? The same as 25% of 16: 4
  • What’s 43% of 200? Same as 200% of 43: 86.

http://www.mytutorfriend.com

July 8, 2008

Magic Square

Filed under: Math tricks — mytutorfriend @ 11:29 pm
Tags: ,

What is a Magic Square?

A magic square is an arrangement of the numbers from 1 to n^2 (n-squared) in an nxn matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same. It is not hard to show that this sum must be n(n^2+1)/2.

The simplest magic square is the 1×1 magic square whose only entry is the number 1.

The next simplest is the 3×3 magic square

and those derived from it by symmetries of the square. This 3×3 square is definitely magic and satisfies the definition given above.

The 4×4 Dürer magic square (or, what is essentially the same, the 4×4 magic square I use) has many interesting special properties that are not shared by magic squares in general. They are so interesting that they are often pointed out when this square is presented. That is good, but can sometimes lead to misunderstandings as to which is the meat and which is the gravy. The meat is the definition I gave above. The gravy (or some of it), suggested by Jerome S. Meyer in his book, Fun with Mathematics , follows.

In the case of this 4×4 magic square:

in addition to having the sum 34 (= 4(4^2+1)/2) in each row, column and main diagonal,

  1. The four corners add to 34.
  2. The four numbers in the center add to 34.
  3. The 15 and 14 in the top row and the 3 and 2 facing them in the bottom row add to 34.
  4. The 12 and 8 in the first column and the 9 and 5 facing them in the last column add to 34.
  5. The four squares in the corners add to 34.
  6. If you go clockwise around the square and choose the first squares away from the corners (15,9,2,8), they add to 34. The same holds if you go counterclockwise.
  7. If you replace each entry by its square, you get the following square:This square is not magic, but it has some noteworthy properties:
    1. The first and last column have the same sum; likewise the 2nd and 3rd columns have the same sum;
    2. The same holds with rows instead of columns (although the sums one gets are different from the sums for the columns);
    3. From (1) and (2), it follows that the left half of the square has the same sum as the right half and that this is one-half the sum of all of the numbers in the square. The same holds for the top half and the bottom half; hence the left half equals the right half equals the top half equals the bottom half. Furthermore, this also equals the sum of the 8 numbers on the diagonals and the sum of the 8 numbers off the diagonals.
  8. If instead you replace each entry by its cube, you get the following square:This square has the property that the sum of the 8 numbers on the diagonals equals the sum of the 8 numbers off the diagonals.       – http://www.mytutorfriend.com

Blog at WordPress.com.