Factorials and Arrangements of Unique Objects

Last weekend my dad asked me about my blog, and the last entry was about factorials of large numbers. Clicking on this link will take you there. On our way to classic car auto shop in Orange, CA; my dad and I talked about the factorials as I tried to come up with a way of finding applications using factorials. Last weekend became the inspiration for my blog entry. Love you, Dad.

Note: This blog entry will cover factorials of non-zero integers, that is n = 0, 1, 2, 3...

---

Factorials

The factorial of a non-negative integer, written with an exclamation mark after the number, is defined as:

n! = n × (n - 1) × (n - 2) × (n - 3) × ... × 3 × 2 × 1

Start by multiplying n by n less 1, then multiplying the product by n less 2, and repeat until you get to 1.

Examples
5! = 5 × 4 × 3 × 2 × 1 = 120
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320

---

What about 0! ?

By definition, 0! = 1. Dr. James Tanton, Ph. D (www.jamestaton.com) gives an explanation why mathematicians choose to in video: Link to Video

Simply put, 0! = 1 is just defined this way.

---

Arrangements

In this section, we consider three common arrangement problems. The task is to find the number of possible permutations a group of objects can be arranged. A permutation is an arrangement where the order of which the objects are placed is important.

In this section, we are going to arrange all the objects.

Arrangement of Unique Objects

Consider a bookshelf that has room for five books. For simplicity, let's call the books A, B, C, D, and E (and not "Combinatorics", "The Irrationals", "Euclid's Number", "Programming", and "Making Soup for Dummies" like I originally planned.)

The for the first slot there are five choices. The second slot provides four choices. Whatever is available for the second slot depends on what book was put in the first slot. For example, if I choose to put book A in the first slot, the books B, C, D, and E are available for the second slot. Instead, if I choose to put book C in the first slot, then A, B, D, and E are available.

For each choice I make on the first slot, I get four choices for the second. Considering the first two slots alone, this gives me a total of 5 × 4 = 20 arrangements.

Continuing in this way, there are three choices for the third slot, two choices for the fourth slot, and whatever is left over gets the fifth slot.

So, the total number of arrangements of five books is:

5 × 4 × 3 × 2 × 1 = 5! = 120

Yes, 120 different arrangements. Since order is important, the arrangement is considered a permutation.

In general, working with n objects and n slots:
There are n objects for the first slot,
there are n - 1 objects for the second slot,
there are n - 2 objects for the third slot,
and so on,
until we reach 1 slot left for the last object.

Hence, the number of arrangements are:

n × (n - 1) × (n - 2) × ... × 1 = n!

---

Number of Permutations of n Unique Objects = n!

---

Arrangement of Unique Slots, Limited Spaces Available

Let's go back to our problem of arranging five books (A, B, C, D, and E) but this time, we only have three slots available.

For the first slot, I have 5 books available to choose from. Depending on what I choose, I will have 4 books for the second slot. Each of those 4 books present a choice of the 3 books for the last slot. Whatever is left either goes somewhere else in the house or gets donated.

The number of arrangements that are available to me has decreased due to the fact I only have three slots available. Hence, 5 × 4 × 3 = 60.

Observe if we multiply 5 × 4 × 3 by (2 × 1)/(2 × 1) we get:

5 × 4 × 3
= (5 × 4 × 3 × 2 × 1) / (2 × 1)
= 5! / (2 × 1)
= 5! / (5 - 3)!

Basically we are taking all of the ways 5 books can be arranged, and then dividing that number by all the ways the 2 books that are not going to be used can be arranged.

Doing the above in the general case is how we arrive at the formula for permutations:

nPk = n! / (n - k)!

where we have n objects and k slots to fill.

---

Permutations: nPk = n! / (n - k)! where n ≥ k. Order is important.

---

Until next time, have a great day!

Eddie



This blog is property of Edward Shore, 2012.

Factorial of Large Numbers

Hi everyone! I am blogging from Coffee Klatch in San Dimas, CA! I was overdue for a latte. Their pumpkin brownies are to die for.

Factorial of Large Numbers

Calculators have an upper limit when working with numbers. Typically, the scientific calculators allow numbers up to 9.999999999 × 10^99. Anything over 10^100 is considered an overflow. Some of the higher graphing and CAS calculators allow for higher limits.

This means if you need to calculate anything over 69!, most calculators will give an "Overflow" error message.

---

Factorial:

The factorial of an integer n is the product:

n! = n × (n - 1) × (n - 2) × ... × 3 × 2 × 1

---

We can remedy this by using the laws of logarithms to calculate factorials of higher integers. Common logarithms (base 10) are used.

n! = n × (n - 1) × (n - 2) × ... × 2 × 1
log (n!) = log [ n × (n - 1) × (n - 2) × ... × 2 × 1 ]
log (n!) = log (n) + log (n - 1) + log (n - 2) + ... + log (2) + log (1)

log (n!) = ∑( log(k) from k = 2 to n)

Let T be this sum.

T = ∑(log (k) from k = 2 to n)

---

Note the following laws are used:

log (a × b) = log a + log b
log (1) = 0

---

If we take the anti-common logarithm of both sides, we get:

n! = 10^(∑( log(k) from k = 2 to n)
n! = 10^(T)
n! = 10^(frac(T) + int(T))
n! = 10^(frac(T)) × 10^(int(T))

Where:

(1) int(T) is the integer part of T. To find it, take the integer and ignore everything after the decimal point.

(2) frac(T) is the fractional part of T. To find this, take everything after the decimal point and ignore the integer part.

(3) int(T) + frac(T) = T

Example:
int(14.125) = 14
frac(14.125) = 0.125

A lot of programming and graphing scientific calculators have the int and frac functions. On most Hewlett Packard programming and scientific calculators, they are named IP and FP, respectively.

Example 1: 8!

Calculate 8! using the logarithm method.

T = ∑( log(k) for k = 2 to 8) = log 2 + log 3 + log 4 + log 5 + log 6 + log 7 + log 8 ≈ 4.605520523

In this case, int(T) = 4 and frac(T) = 0.605520523.

Then 8! ≈ 10^(0.65520523) × 10^4 ≈ 4.031999996 × 10^4

Because the accuracy of calculators and computer programs, I recommend that you round 10^(frac(T)) to a set amount of decimal places. I find that 5 decimal places works well. Then:

8! ≈ 4.03200 × 10^4

In fact, 8! = 40,320

Example 2: 111!

1. Calculate the sum ∑( log(k) from k=2 to 111). We find the answer to be 180.2462406.

2. Split the integer and fractional portions. Take the anti-common logarithm of the fractional portion. Hence, 10^0.2462406 ≈ 1.762952551. I round this answer to five decimal places to get 1.76295.

3. Answer: 111! ≈ 1.76295 × 10^180

In fact, using an HP 50g, 111! =
1,762,952,255,109,024,466,387,216,104,710,707,578,876,140,953,602,656,551,604,157,406,334,734,695,508,724,831,643,655,557,459,846,231,577,319,604,766,283,797,891,314,584,749,719,987,162,332,009,625,414,533,120,000,000,000,000,000,000,000,000

And yes, typing out this long number is not as easy it sounds.

Programming

Now we come to the programming portion of calculating n! using the logarithmic method.

This program was done on a HP 39gii, but can easily adapted on most graphing/programming calculators.

Program LFACT

EXPORT LFACT(N)
BEGIN
LOCAL T,M;
Σ(LOG(K),K,2,N)→T;
ROUND(10^FRAC(T),5)→M;
MSGBOX(M + "10^" + INT(T));
RETURN {M, INT(T)};
END;


Here is a program in BASIC language. I used the App techBASIC from ByteWorks.
Everything after explanation points signify comments.

! Approximate Factorial (5 decimal mantissa plus exponent)
! Eddie Shore
! 10/20/2012

! To improve accuracy, set all variables as DOUBLE
01 DIM N as DOUBLE
02 DIM T as DOUBLE
03 DIM F as DOUBLE
04 DIM M as DOUBLE
05 DIM K as DOUBLE


10 PRINT "Factorial Program"
11 INPUT "N = "; N
12 T = 0

! Sum Loop
14 FOR K = 2 TO N
16 T = T + LOG10(K)
18 NEXT K

! Separation of T into its fractional and integer parts, we know T>0
20 M = INT(T)
22 F = T - M

! Round Fraction to five places - we have no ROUND function to work with
! 10^x when x<1 10="" a="" br="" has="" less="" than="" value="">! Use string functions to round the answer to 5 decimal places (7 characters: the integer, decimal point, and five decimal places)
30 F = 10^F
32 F = VAL(LEFT(STR(F),7))

! Final results
40 PRINT "N! = "; F ; " x 10^"; M


Test Data:
8! = 4.0320 × 10^4
66! = 5.4434 × 10^92
122! = 9.87504 × 10^202
425! = 1.59775 × 10^934

Thank you as always, I appreciate the comments and compliments. Hope you enjoy the weekend.

Until next time,

Eddie


This blog is property of Edward Shore, 2012.

What I am reading...

Julian Havil. "The Irrationals: A Story of the Numbers You Can't Count On" Princeton University Press, 2012

Clifford A. Pickover "The Mαth βook". Sterling Publishing, 2009

John P. Flynt and Danny Kodicek "Mathematics and Physics for Programmers" 2nd Ed. Course Technology, 2012


Talk to you all soon, Eddie. Positive vibes and blessings.

Calculator Tricks - Part 5

Welcome to Part 5 of Calculator Tricks

Part 5 will cover:
* Solving 2 by 2 equations
* Finding roots of the quadratic equations

The above tasks will be accomplished by using a simple calculator. That means we have only the arithmetic functions, square roots, percents, and memory to work with.


Solving 2 by 2 Equations

Suppose we are assigned the task solving for x and y:

Ax + By = E
Cx + Dy = F

This is a system of two simultaneous equations with two unknowns, x and y. We can use matrices to visualize this problem:

Take the inverse of the coefficient matrix and left multiply both sides to get:

We can find the inverse of the coefficient matrix by:

The product A × D - B × C is called the determinant. Finally, we solve for x and y:

Now that we have our general solutions, how do we tackle this on a simple calculator?

2 × 2 System of Equations

We only have one memory in the register to work with. Here is a strategy:

1. Calculate the determinant. Keystrokes: A × D - B × C =. Note this number down on paper. The calculator's memory will be required to find x and y.

2. Solve for x. Use the sequence:
MC
D × E = M+
B × F = M-
MR ÷ determinant =

3. Solve for y. Use the sequence:
MC
A × F = M+
C × E = M-
MR ÷ determinant =

---------------
Example: Solve the System

2x + 3y = -1
x - 3y = 2

We have A = 2, B = 3, C = 1, D = -3, E = -1, F = 2

Step 1:
Keystrokes:
MC
2 × 3 +/- = M+
3 × 1 = M-
MR

The determinant is -9. Write this result down.

Step 2:
Keystrokes:
MC
3 +/- × 1 +/- = M+
3 × 2 = M-
MR ÷ 9 +/- =

x ≈ 0.3333333

Step 3:
Keystrokes:
MC
2 × 2 = M+
1 × 1 +/- = M-
MR ÷ 9 +/- =

y ≈ -0.5555555

----------------

Solving Quadratic Equations

Let's tackle on how we find the roots of the equation:

Ax^2 + Bx + C = 0

Our focus for this blog entry will be finding real roots.

Referring to the quadratic formula:

We can derive our algorithm for finding both roots:

1. Calculate the discriminant. Store this result in the calculator's memory.

Keystrokes:
MC
B × = M+
4 × A × C = M-

2. Find the first root:

Keystrokes:
MR √ - B ÷ 2 ÷ A =

3. Find the second root:

Keystrokes:
MR √ +/- - B ÷ 2 ÷ A =
---------------
Example: Find the roots of

x^2 + 5x - 6 = 0

Let A = 1, B = 5, and C = -6

Step 1: Discriminant
MC
5 × = M+
4 × 1 × 6 +/- = M-

Discriminant = 49

Step 2: First Root
MR √ - 5 ÷ 2 ÷ 1 =

One of the roots is x = 1.

Step 3: Second Root
MR √ +/- - 5 ÷ 2 ÷ 1 =

The other root is x = -6


I hope this series is helpful. This series has showed how to make various calculations with a simple calculator: arithmetic procedures including fractions, area of a circle, distance between two points, calculating the total shopping bill, solving 2 x 2 simultaneous equations, and solving quadratic equations.

Until next time,

Eddie


This blog is property of Edward Shore, 2012.

Calculator Tricks - Part 4

Welcome to Part 4 of Calculator Tricks. This is series is where accomplish mathematical tasks with a use of a regular, simple calculator. Part 4 will cover:

* Dealing with Percents
* Shopping
* Simple Interest

Dealing with Percents

With most calculators, if you have to add tax and subtract discounts, you can just execute the operation directly.

However, on some calculators, like the Casio SL-300VC, require a different sequence:
× n % + (to add) and × n % - (to subtract).

Here is how I prefer to work with percents, and it avoids the percent key altogether.

---------------
Add Percent

Let A be the number you want to add N% percent to it. In a shopping application, N represents the sales tax. In construction, N can be thought be allowance for waste.

A + N%
= A + (A × N/100)
= A × (1 + N/100)

Keystrokes: N ÷ 100 + 1 × A
---------------
Example: Add 10% to 19.95.

Keystrokes: 10 ÷ 100 + 1 × 19.95

Result: 21.945

(On most calculators, 19.95 + 10 % works too)
---------------
Subtract Percent

Let A be the number you want to subtract N% from. In terms of shopping, N represents a discount.

A - N%
= A - (A × N/100)
= A × (1 - N/100)

Keystrokes: N ÷ 100 +/- + 1 × A
---------------
Example: Subtract 10% from 19.95.

Keystrokes: 10 ÷ 100 +/- + 1 × 19.95

Result: 17.955

(On most calculators, 19.95 - 10 % works too)
---------------

Shopping

Ever though what your bill would be as you shop? This section will show you what the potential bill will be and hopefully will lead you to make smart shopping decisions, and keep in budget.

Approach:
1. Clear Memory. We will use the memory register to keep track of our purchases.
2. Determine whether the item is subject to sales tax. You can press MR at any time to get a subtotal.
3. Add the total.
4. *If all of your items are subject to sales tax, add sales tax to the total.

If your purchases are "mixed", buying both non-taxable and taxable items (grocery store comes mind):

If the item or service is not subject to sales tax, use the keystroke sequence price M+ .

If the item or service is subject to sales tax, use the keystroke sequence tax factor × price M+ .

where tax factor = (1 + tax%/100)


In California, where I live, generally food, grocery items, and most services are generally not subject to sales tax, while sales of tangible goods are subject to sales tax.


Let's illustrate this strategy by a few examples.
---------------
Example 1: We are at a hardware store and purchasing the following items:

Hammer: $19.99
Nails: $3.95
Staple Gun: $14.95
Measuring Tape: $3.99

All items are subject to 8.5% sales tax.

Well, since we are dealing with just taxable purchases, we can total everything and add sales tax at the end.

Keystrokes:
MC
19.99 M+
3.95 M+
14.95 M+
3.99 M+
8.5 ÷ 100 + 1 = × MR =

Total: $46.52
---------------
Example 2: From an office store, we are purchasing:

Ream of 500 sheets of paper: 3 at $8.99 each
Pack of pencils: $2.99
Pack of pens: $3.95
Calculator: 2 at $9.95 each

All items are subject to 8.5% sales tax. In this case, we can add the sales tax at the end.

Keystrokes:
MC
3 × 8.99 = M+
2.99 M+
3.95 M+
2 × 9.95 = M+
8.5 ÷ 100 + 1 = × MR =

Subtotal: $53.81
With Sales Tax: $58.38
---------------
Example 3: At a grocery store, we are purchasing:

Bag of Grapes: $2.45
Apples: 5 at 99¢ each
Bananas: 6 at 24¢ each
Bread: $4.09 with a 5% discount coupon
Water: 2 gallons at $1.09 each
Box of Ziploc Bags: $3.99 with a 50¢ coupon

Only the Ziploc bag is subject to 8.5% sales tax. Assume coupons take affect immediately (before sales tax). Here we must use the "mixed purchases" strategy. Enter $3.49 for the Ziploc bags.

MC
2.45 M+ (grapes)
5 × 0.99 = M+ (apples)
6 × 0.24 = M+ (bananas)
5 ÷ 100 +/- + 1 = × 4.09 = M+ (bread)
2 × 1.09 = M+ (water)
8.5 ÷ 100 = + 1 = × 3.49 = M+ (Ziploc bags)
MR (final total)

Total bill: $18.69
---------------


Simple Interest

The simple interest formula is:

I = P × R% × T

I = amount of interest
P = amount of the principal
R% = annual interest rate (as expressed as a decimal)
T = time, in years

The total amount paid is principal and interest, in other words, P + I.

---------------
Example 1: A bank makes a short term loan to Fred and Suzy of $1,000. The bank charges 9.6% interest on short term loans. Fred and Suzy have to pay the loan in two months. If Fred and Suzy wait for the two months, how much interest have they paid?

We are looking for I. We have the following:
P = principal = 1000
R% = annual interest rate = 9.6/100
T = time = 2 months * 1 year/12 months

Note have to convert months to years. So the interest paid is:

Keystrokes:
1000 × 9.6 ÷ 100 × 2 ÷ 12 =

The total interest paid is $16.
---------------
Example 2: Terrell is looking over his credit card bill. The balance is $1,540.29. His credit card charges an annual rate of 15.99%. Terrell is planning to make a $300.00 payment. Assuming Terrell does not use his credit card for the next month, what will be his balance?

Variables:
P = $1,540.29 - $300.00 = $1,240.29
R% = 15.99%
T = 1/12 (1/12 of a year)

Keystrokes:
MC (Clear memory)
1540.29 - 300 = M+ (Subtract payment, interest will accrue on $1,240.29.)
MR × 15.99 ÷ 100 ÷ 12 = (Months interest: $16.53)
M+ MR (Add interest to memory)

Terrell's new balance next month would be $1,256.82.
---------------
Example 3: Lita deposited $500 in a Double Your Money CD. The bank will pay her $1,000 when the CD doubles in value. The bank pays 7.5% interest on these deposits. How long will Lita wait?

This time we are looking for T.

Variables:
P = $500
R% = 7.5%
I = $500

Why is I = $500? Lita deposits $500 and will wait for her account to grow $500, to earn $500 in interest. Solving for T, time:

T = I / (P × R%/100) = I × (P × R%/100)^-1 = (P × R%/100)^-1 × I

Keystrokes:
500 × 7.5 ÷ 100 =
÷ ÷ = (37.5^-1)
× 500 = (Display: 13.333333)

So it takes 13 1/3 years to double Lita's CD. She may want to rethink this investment.
---------------

So this wraps up Part 4 of our Calculator Tricks series. Coming up in Part 5 we will tackle two common algebra problems.

Eddie

This blog is property of Edward Shore, 2012.

Calculator Tricks - Part 3

Welcome to Part 3 of the Calculator Trick series using the Simple Calculator. This part covers:

* Squaring Numbers
* Reciprocals
* Geometric Applications

Just a reminder, a simple calculator has the following functions: arithmetic (+, -, ×, ÷), square root (√), percent (%), and memory (M+, M-, MR, MC). If your calculator has a MRC clear, this means press MRC once to recall memory, twice to clear it.

Squaring Numbers

---------------
To square a number, use the following key sequence:

× =
--------------
Example 1: 7^2

Keystrokes:
7 × =
---------------
Example 2: √(3^2 + 4^2) = 5

Use the memory keys to solve this problem. Squaring a number is multiplying a number by itself. Apply the square root last.

Keystrokes: MC (Again, always a good idea to clear memory to start)
3 × = M+ (3^2 = 9)
4 × = M+ (4^2 = 16)
MR (3^2 + 4^2 = 25)
√ (√25 = 5)
----------------

Reciprocals

---------------
To find the reciprocal of a number, use the following key sequence:

÷ ÷ =
(Yes, press the division key twice)
---------------
Example 1: 1/7 ≈ 0.142857

Keystrokes: 7 ÷ ÷ =
---------------
Example 2: 1/(1/5 + 1/3.2) ≈ 1.9512195

Strategy: Tackle the denominator first. Of course, anytime you have mixed operations, chances are you will be using the memory register.

Keystrokes:
MC
5 ÷ ÷ = M+ (1/5 = 0.2)
3.2 ÷ ÷ = M+ (adding 1/3.2 = 0.3125)
MR (1/5 + 1/3.2 = 0.5125 )
÷ ÷ = ( 1/(1.5 + 1/3.2) ≈ 1.9512195)
---------------

Geometric Applications

Area of the a Circle:
A = π r^2

where r is the radius and π is the constant pi. In this series, I am working with an 8 digit calculator, I will use the approximation π ≈ 3.1415927. I can use less digits, but I want as much accuracy as possible.

---------------
Area of a Circle: A = π r^2

Keystrokes: radius × = × 3.1415927 =
---------------
Example: Find an area of a circle with a radius of 14.5 inches.

Keystrokes: 14.5 × = × 3.1415927 =

The area is approximately 660.51986 square inches.
---------------


Distance between two Cartesian points (x1, y1) and (x2, y2):
√((x2 - x1)^2 + (y2 - y1)^2)

---------------
Distance between (x1, y1) and (x2, y2):

Keystrokes: MC
x2 - x1 = × = M+
y2 - y1 = × = M+
MR √
---------------
Example: Find the distance between the points (5,6) and (1,9).

Note: (x1, y1) = (5, 6) and (x2, y2) = (1, 9)

Keystrokes: MC
1 - 5 = × = M+ (Display: 16 M)
9 - 6 = × = M+ (Display: 9 M)
MR √ (Display: 5 M)

The distance between the points (5, 6) and (1, 9) is 5 units.
---------------

Right Triangles

Area = 1/2 × A × B

Pythagorean Theorem: H^2 = A^2 + B^2

---------------
With A = 3.9 inches, B = 2.4 inches. Find the area and the hypotenuse (H).

Area:
Keystrokes: 3.9 × 2.4 ÷ 2 =

The area is 4.68 square inches.

Finding the Hypotenuse:
Keystrokes: MC 3.9 × = M+
2.4 × = M+
MR √

The hypotenuse is approximately 4.5793012 inches.
---------------

Next time in Part 4, we will cover shopping and short-term loans with simple interest.

Eddie


This blog is property of Edward Shore, 2012.

Calculator Tricks - Part 2

Fractions

Welcome to Part 2 of the Calculator Tricks series. We are using a simple ("four banger" calculator) to tackle common mathematical problems. If you missed Part 1,
you can check it right here by clicking on this link.

Simple Fractions

Simple calculators do not have the ability to display numbers as fractions, just their decimal equivalents.

Here are the decimal equivalent of some fractions that are handy to remember:

1/8 = 0.125
1/4 = 0.25
1/3 ≈ 0.3333333
3/8 = 0.375
1/2 = 0.5
5/8 = 0.625
2/3 ≈ 0.6666667
3/4 = 0.75
7/8 = 0.875

Remember the order you press the keys is critical, since the simple calculator operates in Chain Mode.

This blog assumes that you are working with an 8-digit calculator.

1. Adding and Subtracting Fractions

Adding and subtracting fractions will require the use of memory. Remember to always clear memory before beginning a calculation.

---------------
Adding Fractions: A/B + C/D

Keystrokes:
MC (clear memory because we will need it)
A ÷ B = M+
C ÷ D = M+
MR
---------------
Subtracting Fractions: A/B - C/D

Keystrokes:
MC
A ÷ B = M+
C ÷ D = M-
MR
---------------
Example 1: 4/11 - 3/99 = 1/3 ≈ 0.3333333

The decimal equivalent (0.3333333 on a 8-digit calculator) is what we are after in this example.

Keystrokes:
MC (Yes, I can't emphasize clearing the memory at the start enough!)
4 ÷ 11 = M+
3 ÷ 99 = M-
MR (Display: 0.3333333 M)
--------------
Example 2: 1/7 + 3/8 - 4/9 = 37/504 ≈ 0.073412698

We can use this technique to add and subtract more than two fractions.

Keystrokes:
MC
1 ÷ 7 = M+ (adding 0.1428571)
3 ÷ 8 = M+ (adding 0.375)
4 ÷ 9 = M- (subtracting 0.4444444)
MR (Display: 0.0734127 M)

Depending on the calculator the last digit may be rounded or not. This result is correct to eight decimal places.

2. Multiplying and Dividing Fractions

Let's multiply the fractions A/B × C/D. Using some algebra, we can simplify this expression.

A/B × C/D
= (A × C)/(B × D)
= (A × C)/B × 1/D
= (A × C) ÷ B ÷ D

---------------
Multiplying Fractions: A/B × C/D

Keystrokes: A × C ÷ B ÷ D =

Multiply the numerators, divide the denominators.
---------------
Example: 4/7 × 2/3 = 8/21 ≈ 0.3809524

Keystrokes: 4 × 2 ÷ 7 ÷ 3 =
---------------

Let's divide the fraction A/B by C/D. (Calculate A/B ÷ C/D) To divide number by a fraction, multiply it by the fraction's reciprocal.

A/B ÷ C/D
= A/B × D/C
= (A × D)/(B × C)
= (A × D)/B × 1/C
= (A × D) ÷ B ÷ C

----------------
Dividing Fractions: A/B ÷ C/D

Keystrokes: A × D ÷ B ÷ C =
---------------
Example: 7/9 ÷ 2/5 = 7/9 × 5/2 = 35/18 ≈ 1.9444444

Keystrokes: 7 × 5 ÷ 9 ÷ 2 =
----------------

Mixed Fractions

A strategy is to to covert mixed fractions to simple (improper fractions) first.

Adding Mixed Fractions

For the mixed fraction A B/C is converted into the simple fraction (A × C + B)/C. Adding two mixed fractions will require the use of the memory register.

---------------
Adding Mixed Fractions:

A B/C + D E/F = (A × C + B)/C + (D × F + E)/F

Keystrokes: MC
A × C + B ÷ C = M+
D × F + E ÷ F = M+
MR
---------------
Example: 8 1/9 + 7 3/5 = 15 32/45 ≈ 15.711111

Keystrokes: MC
8 × 9 + 1 ÷ 9 = M+
7 × 5 + 3 ÷ 5 = M+
MR
---------------

Multiplying Mixed Fractions

Unlike multiplying simple fractions, multiplying mixed fractions will require the use of memory. We can get a form for an algorithm by simplifying.

A B/C × D E/F
= (A × C + B)/C × (D × F + E)/F
= ((A × C + B) × (D × F + E))/(B × C)
= ((A × C + B) × (D × F + E)) ÷ B ÷ C

---------------
Multiplying Mixed Fractions

A B/C × D E/F = ((A × C + B) × (D × F + E)) ÷ C ÷ F

Keystrokes:
MC
A × C + B M+
D × F + E =
× MR =
÷ B ÷ F
---------------
Example 1: 4 1/2 × 8 1/3 = ((4 × 2 + 1) × (8 × 3 + 1)) ÷ 2 ÷ 3 = 37.5

Keystrokes:
MC (Start with the memory cleared!)
4 × 2 + 1 = M+ (Store 4 × 1 + 2 = 9 in memory)
8 × 3 + 1 = (Display: 25 M)
× MR = (numerator is done, Display: 225)
÷ 2 ÷ 3 = (no divide by the denominators)

The display should be 37.5 (with the memory indicator). This is a complex algorithm, so it will take some practice. Let's do another example.
---------------
Example 2: 5 5/9 × 16 2/3 = ((5 × 9 + 5) × (16 × 3 + 2)) ÷ 9 ÷ 3 ≈ 92.59259

Keystrokes:
MC (Start with the memory cleared!)
5 × 9 + 5 = M+ (Store 4 × 1 + 2 = 50 in memory)
16 × 3 + 2 = (Display: 50 M)
× MR = (numerator is done, Display: 2500 M)
÷ 9 ÷ 3 = (no divide by the denominators)

The display should have 92.59259.
---------------

In the next example we will add and multiply mixed fractions. Be willing to write immediate results on a note pad with complex problems.

---------------
Example 3:
4 1/8 + 3 3/7 × 6 2/3 = 1511/56 ≈ 26.982143

The Order of Operations says we must multiply the mixed fractions first. However, we only have one memory register, and we don't have the ability to "switch" whatever is in the display with memory. Tackling this problem requires a plan. Here is one way:

1. Change 4 1/8 into a fraction and write it down.
2. Multiply the mixed fractions 3 3/7 × 6 2/3.
3. Add the resulting decimal equivalent of 4 1/8 to the result obtained from step 2.

Step 1 Keystrokes:
4 × 8 + 1 ÷ 8 =
Note the result, which is 4.125. Don't store this number in memory, but on a notepad.

Step 2 Keystrokes:
MC
3 × 7 + 3 = M+ (Display: 24 M)
6 × 3 + 2 = (Display: 20 M)
× MR = (Display: 480 M)
÷ 7 ÷ 3 = (Display: 22.857142 M)

Step 3 Keystrokes:
Add 4.125 obtained from step 1 to 22.857142 obtained from step 2.
+ 4.125 = (Display: 26.982142 M)

We have arrived at our result.

The lesson here is to plan your calculation.

Another strategy to calculate 4 1/8 + 3 3/7 × 6 2/3 is to change all fractions to their decimal equivalents first, noting the decimal equivalents on paper. Let's see how this works out:

1. Change each mixed fraction to their decimal equivalent.
2. Complete the calculation.

Step 1 Keystrokes:
4 × 8 + 1 ÷ 8 = (4 1/8 = 4.125)
3 × 7 + 3 ÷ 7 = (3 3/7 ≈ 3.4285714)
6 × 3 + 2 ÷ 3 = (6 2/3 ≈ 6.6666666)

Now we have 4.125 + 3.4285714 × 6.6666666

Step 2 Keystrokes:
3.4285714 × 6.6666666 + 4.125 =
Result: 26.982142


The next time we meet, we will work with square numbers, reciprocals, and some geometry.

Eddie


This blog is property of Edward Shore, 2012.

Calculator Tricks - Part 1

First of all, I am home taking care of my dad today. He's doing fine, recovering from surgery.

I also want to thank everyone who reads this blog and leaves comments. I appreciate the input and conversation.

Eddie

---------------

Introduction

Don't have a scientific calculator and only have a simple calculator with you? Interested in maximizing the abilities of a simple calculator? Want to impress your friends and co-workers? This series is for you.

What do I mean by a simple calculator? it is the calculator that you find everywhere, not just produced by the big calculator manufacturers Hewlett Packard, Texas Instruments, Sharp, and Casio, but as a novelty item from companies with almost any color or design you like. If you prefer, there are thousands of calculator apps on almost any cell phone, tablet, or iPod Touch. I refer to this type of calculator as a "four-banger" because it's primary functions are the arithmetic functions (+, -, ×, ÷).

Part 1 will cover:

* Chain Mode
* The Memory Keys
* Arithmetic Operations

Chain Mode

I estimate that 99% of simple calculators operate on chain mode. That is calculations take place as you enter them, without regard to the Order of Operations. You may remember the expression "My Dear Aunt Sally" or it's expanded version "Please Excuse My Dear Aunt Sally" as a mnemonic for the Order of Operations.

Order of Operations:
Please (everything in Parenthesis gets priority)
Excuse (Exponents, roots, [sine, cosine, tangent, logarithm])
My Dear (Multiplication, Division)
Aunt Sally (Addition, Subtraction)

Here is how to know whether your calculator is operating in Chain Mode:

Type, in order: 4 + 2 × 3 =

Now using the proper order of operations, 4 + 2 × 3 = 10. However, calculators in chain mode don't "know" the order of operations. In that case, that calculator completes 4 + 2 first before multiplying the result by 3, giving a result of 18.

So if you type 4 + 2 × 3 = , in that order, and get 18, your calculator operates in Chain Mode. This means we have to manually take the Order of Operations into account to ensure we get the correct answer of 10.

One way to do this is rearrange the expression to 2 × 3 + 4. Typing the expression in this order will give us the correct answer, 10.

For this series, we will work with calculators operating in Chain Mode, which covers about 95% of simple calculators.

The Memory Keys

The simple calculator has four memory operations:
M+: Add whatever is in the display to Memory.
M-: Subtract whatever is in the display from Memory.
MR: Recall the contents of Memory.
MC: Clears the contents of Memory.

Often, you will see the key MRC. Press this key once to recall the contents of memory, twice to clear it.

For this series, I will keep MR and MC separate. Just remember if you are working with the MRC key, pressing MRC twice will clear memory.

The memory register is a key to advanced calculations on a simple calculator. One, it can help keep our calculations in proper operation. Second, it can store a number for later use.

Your calculator will give an indicator ("M" or "MEMORY") whenever a number other than zero is stored in memory. The memory is consider cleared when 0 is stored and the calculator does not display a memory indicator.

On to the Arithmetic Section.

Arithmetic Section

Let's tackle some common arithmetic calculations with the simple calculator to get the correct answers. Remember the order you press the keys is critical, since the simple calculator operates in Chain Mode.

For each section, I will give a proper keystroke to tackle the problem. Then I will give an example. Each capital letter (A, B, C, etc.) represents a variable.

1. A × B + C

This is a fairly simple expression. Just keep "Please Excuse My Dear Aunt Sally" in mind and you're gold.

---------------
A × B + C

Keystrokes: A × B + C =
---------------
Example: 8 × 6 + 3 = 51

Keystrokes: 8 × 6 + 3 =

The result is 51, which the correct result with the Order of Operations.

2. A × B + C × D

Now we have two multiplications to do before the addition. This is where the memory keys (M+, M-, MR, MC) come in handy. Before you start any operation involving memory, clear it first! Remember if your calculator has a MRC key, press it twice to clear memory.

---------------
A × B + C × D

Keystrokes:
MC
A × B = M+
C × D = M+
MR
---------------
Example: 4 × 1.95 + 3 × 0.99 = 10.77

Key:
MC
4 × 1.95 = M+ (Display: 7.8 M)
3 × .99 = M+ (Display: 2.97 M)
MR (Display: 10.77 M)

3. A × B - C × D

This is similar to problem 2.

---------------
A × B - C × D

Keystrokes:
MC
A × B = M+
C × D = M-
MR
---------------
Example: 4 × 8.25 - 3 × 1.95 = 27.15

Key:
MC (always want to clear memory before beginning)
4 × 8.25 = M+ (Display: 33 M)
3 × 1.95 = M- (Display: 5.85 M)
MR (Display: 27.15 M)

The answer is 27.15.

4. A × (B + C)

Note the parenthesis around the addition of B and C. This time we work the addition first.

---------------
A × (B + C)

Keystrokes: B + C × A
---------------
Example: 8 × (6 + 3) = 72

Keystrokes: 6 + 3 (Display: 9)
× 8 = (Display: 72)

5. (A + B) ÷ (C + D)

A strategy is to work the denominator first, store the result in memory. Then work left to right.

---------------
(A + B) ÷ (C + D)

Keystrokes: MC
C + D = M+ (store denominator in memory)
A + B
÷ MR =
---------------
Example: (48 - 16) ÷ (3 + 1) = 8

Keystrokes: MC
3 + 1 = M+ (4 is stored in memory)
48 - 16 (working the numerator)
÷ MR = (divide by memory)

Display: 8 M



This ends Part 1 of our series. In part 2, we will work with fractions. Thanks,

Eddie



This blog is property of Edward Shore, 2012.

Distribution of Digits for π, e, Euler's Constant, and √2

Here are four famous numerical constants, extended to 50 decimal places: π, e, γ, and √2. For each constant, the approximation will be listed and a histogram of each of the 50 decimal places and the integer part (51 digits in total) will be presented.

Pi
π ≈ 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

Digit Distribution: (digit: number of occurrences)
0: 2
1: 5
2: 5
3: 9
4: 4
5: 5
6: 4
7: 4
8: 5
9: 8

e

e ≈ 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995

Digit Distribution:
0: 3
1: 3
2: 8
3: 4
4: 5
5: 6
6: 4
7: 7
8: 5
9: 6

γ (Euler's Constant)

γ ≈ 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992

Digit Distribution:
0: 9
1: 5
2: 7
3: 5
4: 4
5: 6
6: 5
7: 2
8: 2
9: 6

√2

√2 ≈ 1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694

Digit Distribution:
0: 5
1: 5
2: 4
3: 4
4: 5
5: 4
6: 6
7: 6
8: 7
9: 5

Of the four constants presented, √2 has the most even distribution, at far as the first 50 decimal points are concerned.

Fun Fact: 99/70 gives √2 accurate to four decimal places.
99/70 = 1.41428 571428...

Resources

π, e, γ

Zwillinger, Daniel. CRC - Standard Mathematical Tables and Formulae 32nd Edition. CRC Press, Boca Raton, FL. 2012

√2

√2 Wikipedia: Retrieved October 2, 2012.


Enjoy! This is something I wanted to do for a while, as I am fascinated by numerical constants.

Eddie


This blog is property of Edward Shore, 2012.

Honnold-Mudd Library and The Odds of Winning the World Series

Greetings everyone!

Honnold Mudd Library

Yesterday I visited the Honnold Mudd Library, which is part of the libraries of the Claremont Colleges. The library has open for most of the day and has room for the public. Amazingly, this is the first collegiate library that I have ever been to where the mathematics section is located on the first floor. The bookshelves for this section are on rollers - which allowed one aisle to be open at a time.

I spent the better part of four hours there, reading about game theory, probability, and the Zeta function.

Here is the link to the Claremont College Libraries' web page.

The Odds of Winning the World Series

Every October in the United States (and Canada should the Toronto Blue Jays be one of the contenders), Major League Baseball (a.k.a. MLB) has an annual tournament of 10 teams to determine the World Series winner. The 10 teams consists of six division winners and four "wild-card" teams. MLB started the current format this year.

Out of the 10 teams, 2 final teams, the National League Champion and American League Champion meet for the World Series. The series is a best of seven series, where the first team to win four games wins the Series and is the Championship team for that year.

We can calculate the odds of a team winning that series by using the Binomial distribution, with the following formula:

nCr × p^r × (1-p)^(n-r)

Where:
n = number of trials
r = number of successes
p = probability of success of one trail
1 - p = q = probability of failure of one trail. Note that p + q = 1.
nCr = combination of n objects taken r objects at a time = n! ÷ ((n - r)! r!)

Even Odds

Let's say two teams are evenly matched. That is, p = 0.50, and q = 1 - p = 0.50. What will be the odds whether one team wins the World Series in four games? Five games? Six games? Seven games?

Win the Series in Four Games

If a team wins a game in four games, it is said that the victorious team swept the losing team. To achieve the probability of a sweep, find the combination of winning all four games and multiply by the odds. That is:

4C4 × 0.50^4 × 0.50^0 = 1 × 0.50^4 = 0.0625

Winning the Series in Five Games

In order to win the series in five games, you must win three games and lose one in the first four games. Obviously, you have to win the fifth game. In calculating probability for this situation, it really does not matter which order you win the first three in four, but that it happens. The "lack of order" requirement allows us to use combinations.

Multiply the result of winning 3 out of 4 games by the odds of winning the fifth game to get the total probability. The total probability of winning the World Series in five games is:

P(winning 3 games out of 4) × P(winning the fifth game)
(4C3 × 0.50^3 × 0.50^1) × (0.50)
= 0.25 × 0.50
= 0.125

Winning the Series in Six Games

In order to win the series in six games, you must win three games and lose two in the first five games. Using a similar approach to the 5 game situation, the total probability of winning the World Series in six games is:

P(winning 3 games out of 5) × P(winning the sixth game)
(5C3 × 0.50^3 × 0.50^2) × 0.50
= 0.3125 × 0.50
= 0.15625

Winning the Series in Seven Games

Similarly, in order to win the series in seven games, you and your opponent win three games apiece in the first six games. Using a similar approach to the 6. game situation, the total probability of winning the World Series in seven games is:

P(winning 3 games out of 6) × P(winning the seventh game)
(6C3 × 0.50^3 × 0.50^3) × 0.50
= 0.3125 × 0.50
= 0.15625

Not surprisingly, the odds of your team winning the World Series in an evenly matched series is 50%. (0.0625 + 0.125 + 0.15625 + 0.15625 = 0.50.)

Underdogs and Favorites

What if the teams are not evenly matched? Instead, we have an underdog, whose odds of winning each game is less than 50%; and a favorite, whose odds of winning each game is greater than 50%. We can use the analysis presented above to calculate the odds of winning the World Series.

For example, say the National League Champion San Francisco Giants, the underdog, plays against the American League Champion New York Yankees. Expert odds makers estimate that the Yankees win about 55% of their games against the Giants. What are the odds of each team winning the World Series?

From the Giants' Point of View, the odds of winning a game is 45%, or p = 0.45. (q = 0.55) Remember each team can win the World Series in one of four ways: sweep (in four games), in 5 games, in 6 games, and in 7 games.

Sweep:
P(win 4 games of 4) = 4C4 × 0.45^4 × 0.55^0 ≈ 0.0410063

Win in 5 Games:
P(win 3 games of 4) × P(winning the 5th game)
= (4C3 × 0.45^3 × 0.55^1) × 0.45 ≈ 0.0902138

Win in 6 Games:
P(win 3 of 5 games) × P(winning the 6th game)
= (5C3 × 0.45^3 × 0.55^2) × 0.45 ≈ 0.1240439

Win in 7 Games:
P(win 3 of 6 games) × P(winning the 7th game)
= (6C3 × 0.45^3 × 0.55^3) × 0.45 ≈ 0.1364483

Total Chance of the Giants winning the World Series (with p = 0.45):
0.0410063 + 0.0902138 + 0.1240439 + 0.1364483 = 0.3917122 (approximately 39.17%)

Going to the Yankees, the favorites in this example. Now the probabilities are reversed, with p = 0.55 (q = 0.45).

Sweep:
P(win 4 games of 4) = 4C4 × 0.55^4 × 0.45^0 ≈ 0.0915063

Win in 5 Games:
P(win 3 games of 4) × P(winning the 5th game)
= (4C3 × 0.55^3 × 0.45^1) × 0.55 ≈ 0.1647113

Win in 6 Games:
P(win 3 of 5 games) × P(winning the 6th game)
= (5C3 × 0.55^3 × 0.45^2) × 0.55 ≈ 0.1853002

Win in 7 Games:
P(win 3 of 6 games) × P(winning the 7th game)
= (6C3 × 0.55^3 × 0.45^3) × 0.55 ≈ 0.1667701

Total Chance of the Yankees winning the World Series, with p = 0.55,
0.0915063 + 0.1647113 + 0.1853002 + 0.1667701 = 0.6082878 (approximately 60.83%)

Note: 0.3917122 + 0.6082878 = 1. As a check, see if the total probabilities add to 1. Being a favorite can significantly increase your chance of winning the World Series!

A RPN Program (HP 15C)

Here is a RPN (Reverse Polish Notation) keystroke program to calculate

nCr × p^r × (1-p)^(n-r)

I used a HP 15C. On the 15C (and 32S, 35S), recall arithmetic can be used to save space. Adjust accordingly.


(Key: Key Code)
LBL A: 42 21 11
RCL 3: 45 3
RCL 2: 45 2
y^x: 14
1: 1
RCL- 3: 45 30 3
RCL 1: 45 1
RCL- 2: 45 30 2
y^x: 14
×: 20
RCL 1: 45 1
RCL 2: 45 2
Cy,x: 43 40
×: 20
RTN: 43 32
(15 steps)


Input:
Preload the following memory registers: R1 = n, R2 = r, and R3 = p

Output:
Probability on the x stack level (display), with the previous contents of the x stack level on the y, z, and t stack levels.


Source of the formula: Packel, Edward. "The Mathematics of Games and Gambling". MAA 1981. New York

Before I go for today, I leave you with several fun facts:

Let z be any complex number, where z = a + bi, with a the real component and b the imaginary part

The Riemann Zeta Function has no zeros (roots) when a is greater than 1.

The Gamma Function has no zeros when a is greater than 0.

Source: Patterson, S.J. "An Introduction to the Theory of the Riemann Zeta Function" Cambridge University Press 1988.


This blog is property of Edward Shore, 2012.

iOS App Review: PowerOne FE - 9/24/2012

As we move closer to using our tablets, smartphones, and other "one-stop" machines for literally everything, I want to start reviewing calculator apps on occasion. Today, it is the PowerOne App. All the screen shots are from the iPad app.


App: PowerOne
Software created by Infinity Softworks
Type of Calculator: Finance, Scientific, Construction, among possibly others
Are there Lite editions? Yes
Price of the Full Edition: $4.99 (Update 9/15/2012)
Available on: iPhone/iPod Touch, iPad (I do not know if an Android version is available)

Features:

* Full force scientific calculator with calculus, matrices, vectors, complex numbers, and basic logic computations
* Templates for every kind of application. They are easily accessed in with the Library app.
* You have the ability to create your own templates and share them via email. You can also share results from templates.
* The calculator works in either Algebraic or RPN mode.

Pros:

* Templates are available for most applications. You can share results and templates through email.
* You can mark templates as favorites and organize them.
* Often, the templates will self-calculate results as you enter data.
* The version you purchase will pre-load templates from appropriate subject. However, with any version, you can access all templates. I am running the Finance version.
* The factorial function (fact) works for all real arguments.
* Complex functionality extends to the trigonometric functions and exponential functions.

Four examples of templates are shown below:

Cons:

* The calculator is very cumbersome to use if you want to use advanced functions. First, you have to select different keyboards (trigonometry, number, etc) to get access to different commands. The Advance tab gets you access to these keyboards.
* The Help button gives only basics, and but syntax for each function. If you want syntax, you have to call up the Advanced Tab, pressing more, and scrolling down. Don't forget if you want to access the different keyboards, you have to scroll back up.
* The language is cumbersome. I have not tried to program a template in this version in the app, but the learning curve seems to be steep.
* If you want statistical functions, use the templates rather than the calculator, because the functions are confusing to get to run properly.
* Regarding the Function Graph template, you will need to type the entire function yourself. No access to the function calls. Worse, the keyboard does not stay on the alphabetic keyboard, and inconsistently switches back to the numeric keyboard. Almost impossible to type functions. Note: This complaint is limited to the iPad. Thankfully the iPhone/iPod Touch version does not have this problem.
* When entering arguments in templates, the numeric keypad shows up with no functionality. The arithmetic functions show up by the press of the Math key, but it blocks your number pad! Also, no way to enter π other than typing the value out: 3.14159265359... (See below)

Overall:

* If you are looking for solvers to common problems, this app is great for it. Many templates are available.
* For this version (I can't find the version number at this time), I would not recommend this app for regular calculator use, at least the iPad version. The iPhone version may be more useable as far as the calculator is concerned.

Syntax of some functions and numbers are presented here. Spaces between arguments are added for readability only; not required.

Any principal root: root(y; x) returns y^(1/x)

Complex Numbers - rectangular: (real ; imag)

Complex Numbers - polar: (radius ;@ angle). The Help screen got the syntax wrong, you need both the color and at sign.

Base Designations: _d, _b, _o, _h (but Hexadecimal numbers must start with a number) are entered by pressing the dec, bin, oct, and hex keys on the Programmer keyboard, respectively.

Matrices: [ [a_11 ; a_12 ; a_13 ; ... ] ; [a_21 ; ...] ... ]. Colons separate row entries as well as columns.

Numeric Derivative: nDeriv("f(x)" ; "x" ;point). The function and its variable are enclosed in quotes. The quotes are included on the Calculus keypad.

Numeric Integral: fnInt("f(x)" ; "x" ; lower limit ; upper limit). The function and variable are enclosed in quotes.

In the Statistics keyboard, the nCr and nPr keys calls up the comb and perm functions, respectively. Both take two arguments, n and r.

As always, comments and questions are welcome.

Until next time,
Eddie

This blog is property of Edward Shore. © 2012

HHC 2012 Day 2 Summary - 9/23/2012

Here is what went on during Day 2 of the HHC Conference:

Wlodek Mier-Jedzejowicz - "Is It a Really Pathetic Name." (My apologizes in advance if I misspell the names)
* RPN (Reverse Polish Notation) came to be by Polish mathematician and logician Jan Łukasiewicz. Łukasiewicz was also a philosopher, teacher, and worked with multi-value logic.
* Mier-Jedzejowicz's family has connections with Łukasiewicz: he was a professor to several of his relatives.
* At the 10th Anniversary of the HPCC (based in Great Britian), Dr. William Wickes wrote an article in "RCL 20: People, Dreams, & HP Calculators" about the naming of RPN. Dr. Wickes was not too happy with the name (see the title for the hint).
* Although several alternative names were given, including "natural operating system". Mier-Jedzejowicz defended the RPN name.

An example of notations: Say we want to calculate 7 + 11.
Infix or algebraic notation (TI, Casio, and most calculators): 7 + 11
Polish notation: + 7 11
Reverse Polish Notation (HP): 7 11 +

Namir Shammas - "Variable Sampling Integration"
* The program has first written on the HP 41.
* The main difference is that the change in x (often labeled "h" or "Δx") is not constant. Where the intervals are placed depends on the slope of f(x). More intervals are placed when f(x) is changing quickly.
* Smaller areas are calculated using the area under a quadratic or cubic curve and then aggregated.
* In general, variable sample integration is more accurate than Simpson's rule when using cubic polynomials to calculate areas of sub-intervals. However, Simpson's rule is better when f(x) contains trigonometric functions.

Rick Furr - "The Making of the HPV2 Poster"
* Furr is an avid collector of calculators: over 1,000 of them. The collection includes Curta mechanical calculators.
* The HP calculator poster is an update to 2005 poster. I purchased one and will be posting a picture of it.
* Furr produced several posters: one covering a Cuerta machine and its hundreds of parts, the history of telephones, and the history of money, specifically greenbacks.
* The process of the new poster in a nutshell:
1. Take a picture of each calculator. Furr commented that inspection lights work best. Also take the picture away from the calculator and use the camera's zoom features.
2. Furr used the desktop publishing software Serif PagePlus x5 to plan the poster. This preferable to Photoshop since images can be scaled while the resolution is preserved.
3. Find data for each calculator and peripheral. This proved to be a challenge.
4. Use a printer that uses a stochastic printing process. The random grouping of the dots (cyan, magenta, yellow, black) allows for sharper printing.

As soon as I can will post a picture of the HP poster, probably when I get back home.

Rick Furr's website: http://www.vcalc.net/

Jakie Waldering - Raspberry Pi
* Waldering showed the Raspberry Pi, a small computer that runs on Linux. How small is it? It can fit in your hand!
* Some specifications:
The computer boots from a 1G SD Card
The computer has an Ethernet port and two USB ports
It has an HDMI connection, so the computer can run on high-tech televisions
Graphics: 48 GPU Cores - screen refreshes at 40 to 50 screens per second
Does not get not after use - which was discovered after running a network of six Raspberry Pis ran a local network. The students were playing Quake 3.
This is a low cost device. Some assembly required.

More information about the Raspberry Pi can be found at:http://www.raspberrypi.org/faqs. This page contains links to suppliers if you want to purchase one. This is a nonprofit organization.

Final Thoughts
* I always have so such fun at the HHC conferences. If you get a chance, go. The HHC is an annual conference will be about 9/21/2013. Place to be determined.

Web Pages:
Gene Wright: http://www.rskey.org/gene/hpgene/
HPcalc.org, run by Eric Rechlin: http://www.hpcalc.org/. Rechlin also makes the keyboard stickers for the WP 34S.

I leave you with some pictures from the conference.

Eddie

HHC 2012 Day 1 Summary - 9/22/2012

Greetings from Nashville!

I am at the HHC 12 conference. It is an annual conference for Hewlett Packard Calculator enthusiasts, held every September. This has been my fourth conference I attended. Everyone that I meet at the conference is super. I love going to these conferences. Here is a summary of went on during the 13-plus hour day, which honestly, went by amazingly fast.

Before the conference started, I was privileged to purchase a HP 32s II calculator from Bob Patton. A picture of it is shown below. The HP 32s II is a highly sought RPN programmable calculator that was in production during most of the 1990s.

Today's conference started at the bright and early time at 8:15 AM. Here is a short summary:

"WP-34S and the HP Calculator Way" - Jake Schwartz:
* In the 1970s, Hewlett Packard (from here on in will be referred to as HP), was wowing calculator owners by constantly exceeding expectations. This due to the fact that handheld electronic calculators were relatively new.
* Around the time that the HP 49g+ was introduced in 2003, HP and calculator users crossed a point. Now, we users are suggesting more features to HP than HP has produced. This resulted in some users making home made calculators by repurposing them. For example, the HP 20B and 30b are repurposed to become a scientific programming calculator, the WP 34S, despite working with limitations of memory, manual keyboard labeling, and limited screen.
* Today, there is less documentation. Also calculators have been aimed towards certain niche markets, primarily education.

"Tablets vs. HP Calculators" - Gene Wright
* Of a poll of 41 attendees: 18 have tablets. 8 of the tablet have Apple tablets, including me. In fact, I write my blog with my iPad, using the Blogger+ app. 24 of 41 attendees use smart phones.
* We are going towards a digital world. Textbooks are now being converts to PDF files for use on tablets. The university I work for, Cal Poly Pomona, has just participated in an e-book pilot program.
* The majority of young students now have access to tablets and smartphones. A concern among teachers is the issue of cheating with these devices. That has not stopped some schools from standardizing tablet use, such as the iPad.
* There is a consensus that the use of stand alone technology, such as cameras, physical calculators, and GPS devices, is shrinking in favor of do-everything tablets.
* Jakie Woldering, a computer science professor at Cleveland State University, commented that we bring skills that will help us transition from one technology to the other.

"Calculator Root Finding: Scan Range Method" - Namir Shammas
* This program finds root, extrema (function minimums and maximums), and inflection points of f(x), using a Laguerre method.
* The user gives a domain to find points of interest of f(x) using a step size and other criteria.
* More information can be found here. Check under the HHC 2012 section.


"Measuring HP Calculator Current" - Richard Nelson
* Richard Nelson demonstrates how to find current and voltage of calculators in four states: off, during boot up, on stand by, and running a calculation. This allows us a way to determine if the keyboard and microprocessor are working.

"Calculating Before Calculators" - Palmer Hanson
* Palmer Hanson brought machines pre-dating handheld calculators. Among them were abacuses, slide rules of every kind, a circular percentage wheel, and a three-arm protractor used for navigation.

"The HP-41 is Alive and Well" - Warren Farrow
* Warren Farrow recounted his story about how he purchased the HP-41. He runs a website, www.hp41.org, a rich resource for the HP 41 calculator.

"Where Can You Find the History of Calculator Applications?" - Felix Gross
* Felix Gross shows that we have so much information regarding calculators including books, hardware manuals, journals, and application journals. However, the last category lacks an organized system of archive. The applications range from finance, cattle herd composition, optimizing factors for heat transfer, map projection, and education.
* The peak for calculator documentation occurred around 1977-1984.
* Today's resources include NASA.gov and eric.ed.gov.
* Currently, he is looking for articles concerning finance, engineering, and documentation in foreign languages.

"Re-tooling the Calculator Tool" - Pavneet Arora
* Pavneet Arora makes a case that we are not merging towards one do-everything tablet. On the filed, there is still a need for specialized and durable tools.
* Tablets are not the best for rugged conditions. A problem also is present on relying on only one tool, with the risk of failure. Multiple tools that can handle a similar function still serves a purpose. In my opinion, it is good to have a backup device.
* Arora favors the use of watches and belts that can be used on the arm. In addition, he states that a smaller Kindle Touch, with its e-ink screen, can serve as a multi-use rugged device due to its operating software. E-ink is preferred over back-lit devices.
* He stresses that tool design must be appropriate for working conditions.

Katie Wessman quilted a lovely calculator quilt that will be given as a door prize.

Gene Wright brought and show some of his non-HP (and non-TI) collection, which was really neat. Among the collection is Royal 99T Scientific Calculator (1976), Royal 94F Business Calculator (1976), Unitrex SC30, Unitrex 90SC RPN Calculator (had a 12 digit display which is unheard of in the 1970s), Canon Palmtronic F-7, and several Commodore models including the N-60 (navigation based), S-61 (geared towards statistics), and M-55, which he let me borrow for the night. The weirdest one was the Sanyo CZ-8008, which had only five scientific functions: sine, cosine, tangent, π, and square root. Yep, no inverse trig functions. Gene, who had the Commodore calculators, acquired most of the others calculators from Scott Reynolds.

"Converting HP-41C Programs to PDF" - Jakie Woldering
* Jakie Woldering described he was able to convert HP 41C programs from the calculator and classic journals to a modern format (PDF). This was done by using wands to read magnetic storage cards, a lot of typing, and conversion software.

"Even Larger Scaled Reptiles of the Nordic Countries" - Eric Smith
* Eric Smith updated progress on a calculator he and his group is making. The most significant update is a bigger capacity RAM chip which allows a more complex operating system.

"Curve Fitting with Least Squares Relative Error" - Namir Shammas
* Namir discusses a curve fitting algorithm that minimizes relative error instead of absolute error.
* A result: when the range of data is small, the least-squares method with relative error is more accurate than using absolute error.

About the Commodore M-55 (1977)

Advanced Features Include:
* Three dimensional vectors arithmetic, dot product, cross product
* Logarithmic Gamma Function (note the factorial function still only works for integers)
* Integration by discrete points
* Linear Regression
* Complex Number Arithmetic
* 2 × 2 matrices including arithmetic, inverses, and determinants
* Error function
* Bessel Function (a very rare preprogrammed function!)
* Poisson Distribution for distinct points
* Quadratic Formula including complex roots
* Gaussian Distribution (Normal Distribution) - lower tail area

The battery is rechargeable, and lasted up to three hours on a single charge. Amazingly, the manual can be found online.


Eddie

Day 2 to come!

Volume of a Parabolic Vase and Casio fx-78

Volume of a Parabolic Vase

Introduction

The following procedure is how to find the volume of a parabolic vase. This is a vase where the "sides" (for the lack of a better term) form a parabolic curve. Each side of a vase has a smooth bump, away from the center, with one peak. Imagine if that you can turn a vase on its side. The result is like the illustration above.

Note: The following procedure works with vases where the "sides" curved towards the center, rather than away.

Steps

Using calculus, we can approximate the the volume by taking the following steps:

1. Measure the length of the vase. Let x be this length.
2. Find the "center line" of the vase. From the center line, find the distance from the center line to the end of the case. Do this for both ends of the vase, and at half of the length. Get the best measurement you can. From one end to the other, these are lengths y_0, y_1, and y_2, respectively.
3. Use the data to form the parabolic curve. Find a, b, and c.

4. With a, b, and c determined, plug in a, b, c, and x and calculate the following integral.

Most graphing calculators can handle this task easily.

Note: If there is an official term for these type of vases, I would like to know. Many thanks in advance.

Example

Find the volume of a parabolic vase 4 inches long with the following: (see the illustration "Parabolic Vases with measurements").

So we have these points to form the parabolic equation with:
(0,4), (2,6), (4,4)

Then:

which leads to:

c = 4
b = 2
a = -0.5

The integral is:

Calculating the integral gives an approximate area of 134.041287 in^2.

Found at the Swap Meet: Casio fx-78

Several Sundays ago I purchased this calculator at the Azusa Swap Meet. This calculator, made in 1982, is not only basic, but quite small. It still works.

Have a great day, Eddie


This blog is property of Edward Shore. © 2012