Mytutorfriend- Your friendly tutor

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

Easy Summation

Filed under: Algebra — mytutorfriend @ 11:29 pm
Tags: , ,

A formula that would sum all the numbers between two numbers, inclusive.

Formula:

x is the smaller number of the two. y is the larger one.

This formula has the advantage that it can be used quickly for huge numbers of integers, such as databases and arrays.  -www.mytutorfriend.com

Create a free website or blog at WordPress.com.