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Chapter 4 Gas

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Powers of Pascal’s Triangle

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Pascal’s Triangle

One day a man’s boss needed help gambling.  The boss asked his employee how he could win in gambling.  This employee made Pascal’s Triangle.

Blaise Pascal (1623-1662) studied mathematics, philosophy, and physics.  His father collected taxes in Rouen, France.  Pascal always loved creating calculators.  After 3 years, Pascal made 20 of the first calculators.  He called these calculators Pascalines.

Nevertheless, of all of Pascal’s discoveries and inventions, humanity most commonly remembers him for making Pascal’s Triangle.

Pascal’s triangle consists of a one on the first line, then two ones on the second line, then a one, a two, and a one on the third line.  Each line is formed of numbers which are the sum of the two numbers above it, as shown below.

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As you can see, each row is formed of numbers which are the sum of the two numbers diagonally on top of the number.  Pascal’s triangle continues forever, always containing increasingly large numbers.  The border of Pascal’s triangle is always one.

Powers of 11

1 = 1                                                                  110
11 = 11                                                              111
11 * 11 = 121                                                    112
11 * 11 * 11 = 1331                                          113
11 * 11 * 11 * 11 = 14641                               114
11 * 11 * 11 * 11 * 11 = 161051                     115
11 * 11 * 11 * 11 * 11 * 11 = 1771561           116

The zeroth row of Pascal’s triangle is one.  11is one.
The first row of Pascal’s triangle is one one, or 11.  11is one.
The second row of Pascal’s triangle is one two one, or 121.  112 is 121.

This pattern progresses until we get to the fifth row of Pascal’s Triangle.  The fifth row of Pascal’s triangle is one five ten ten five one, or 15,101,051.  115 is 161,051.  So does that mean that Pascal’s triangle only tells us the powers of 11 up to 114?  Of course not! One just has to know how to read Pascal’s Triangle.

1.jpgWhenever you have a two digit number produced by adding together the two numbers above it in Pascal’s triangle, you keep the unit digit and take all the digits after it and add it to the next number’s one’s digit.  For example, in the fifth row of Pascal’s triangle, you have:

1   5   10   10   5   1
Take the one in the first ten and carry it over to the next ten, you are left with:
1 5 11 0 5 1
then you take the tens digit one in 11 and carry it over to the leftmost 5. You get
1 6 1 0 5 1
11is 161,051.

This pattern works for every power of 11 greater than 4.  Lets try 116. The sixth row of Pascal’s Triangle is 1 6 15 20 15 6 1.

1 6 15 20 15 6 1
Take the one from the leftmost 15 and carry it over to the twenty.  You get:
1 6 15 21 5 6 1
Then take the 2 from the 21 and bring it over to the 15. You should now have:
1 6 17 1 5 6 1
Finally, take the 1 from the 17 and move it over to the 6.  You end up with:
1 7 7 1 5 6 1
11is 1771561

Now you know how to read the powers of 11 from Pascal’s Triangle!

Prime Numbers

In pascal’s triangle, the uppermost row is called the 0th row.  The second highest row is called the first row.  The leftmost number in each row is called the 0th number.  The second number to the left is called the first number.

Now examine the first number of each row.  If it is a prime number, then try dividing that number into each of the other numbers in that row except for the last number and the 0th number.  You should get a whole number for each answer.

Whenever the first number in a row is prime, all the preceding numbers except for the last one are multiples of that first number.  pascals2.jpg

Powers of 2

1 = 1                                                       20
2 = 2                                                      21
2 * 2 = 4                                               22
2 * 2 * 2 = 8                                        23
2 * 2 * 2 * 2 = 16                               24
2 * 2 * 2 * 2 * 2 = 32                        25
2 * 2 * 2 * 2 * 2 * 2 = 64                 26

The zeroth row of Pascal’s triangle consists of the number one.   2is 1.
The first row of Pascal’s triangle consists of 2 ones.  1 + 1 = 2.  2is 2
The second row of Pascal’s triangle consists of 2 ones and a two.  1 + 2 + 1 = 4. 2is 4.
The third row of Pascal’s triangle consists of 2 ones and 2 threes.  1 + 3 + 3 + 1 = 8. 2is 8.

This pattern continues in Pascal’s triangle forever.

Hockey Sticks

Pick a number, any number.  That is, a number on Pascal’s triangle.  Then choose a diagonal direction going downwards.  Continue however many rows down you want.  On the last number, change the direction of the diagonal.  That number will be the sum of all the numbers in the first diagonal.  2.jpg

Diagonals

Triangular numbers are the number of units in an equilateral triangle.  In an equilateral triangle with a side length of 3, there are 6 units in the triangle.triang.dots.gif

The third diagonal going down on Pascal’s triangle contains all triangular numbers.

Counting numbers are positive integers such as 1, 2, 3, 4, &c.  The second row of Pascal’s triangle has every counting number.

Tetrahedral numbers are “numbers that can be represented that can be represented by a regular geometric arrangement of equally spaced points. Tetrahedral numbers correspond to placing discrete points in the configuration of a tetrahedron.”

 

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