Mathematics Odds and Ends





The Earth rotates once per day on its axis towards the east and orbits around the Sun once in about 365 days.

The Moon rotates once in about 28 days on its axis towards the west. The Moon orbits around the Earth once in about 28 days.

 If the Moon didn't rotate on its axis at all, then eventually it would show its far side to the Earth while moving around our planet in orbit. As the rotational period is exactly the same as the orbital period, the same portion of the Moon's sphere is always facing the Earth.  

Assume there is only one house on the Moon and you can see it on day one from your house on Earth.  After 7 days, the Moon has orbited one quarter of the way around the Earth.  The Moon  has turned one quarter revolution on its axis in those 7 days, so you and anyone else on the Earth are still looking at that same house. After 14 days, the Moon has orbited one half of the way around the Earth to the opposite side.  The Moon  has turned one half revolution on its axis in those 14 days, so you and anyone else on the Earth are still looking at that same house.  This means you always are looking at the same side of the moon, whether you are at day 3, 9, 22, etc. in the 28 day orbit of the Moon around the planet Earth.

On earth, what day of the year is noted as having the most collect phone calls?  Father's Day.

Q: "Why does the Moon orbit the Earth?"
A: "To get to the other side.

Q: Why didn't the moon eat?
A: It was full.

Q: What holds the moon up?
A: Moonbeams.

Q: What was the problem with the restaurant on the moon?
A: The food is excellent, but it had no atmosphere.


Some facts:  Trivia:  More trivia: The inclusion of the motto, "In God We Trust", on all currency was required by law in 1955, and first appeared on paper money in 1957.

The first one-dollar bill was issued in 1862 as a Legal Tender Note (United States Note) with a portrait of Salmon P. Chase, the Secretary of the Treasury under President Abraham Lincoln. The $1 United States Note was redesigned in 1869 with a portrait of George Washington in the center.  It has undergone further design changes over the years.  The back of the one dollar bill is printed in green (this is why the dollar bill is sometimes called a greenback). It pictures the word "ONE" flanked by two circles, picturing the front and back of the Great Seal of the United States of America. The circle on the left of the bill pictures an unfinished pyramid with 13 steps. There is an eye within a triangle above the pyramid; light radiates from the eye. The circle on the right pictures the front of the Great Seal of the United States of America. It shows a bald eagle holding olive branches and arrows in its talons. There is a banner in the eagle's bill reading, "E PLURIBUS UNUM" (which means, "Out of many, one," referring to the union of the states). 13 stars are above the eagle and a shield with 13 stripes is in front of the eagle.

The $2 bill has had many designs.  In March 1862, the first $2 bill was issued as a Legal Tender Note (United States Note) with a portrait of Alexander Hamilton. By 1869 the $2 United States Note was redesigned with the now familiar portrait of Thomas Jefferson to the left and a vignette of the United States Capitol in the center of the obverse. In 1886, the first $2 Silver Certificate with a portrait of United States Civil War General Winfield Scott Hancock on the left of the obverse was issued. This design went on until 1891 when a new $2 Silver Certificate was issued with a portrait of U.S. Treasury Secretary William Windom in the center of the obverse. In 1896 The "Educational Series" Silver Certificate was issued. The entire obverse of the note was covered in artwork with an allegorical figure of science presenting steam and electricity to commerce and manufacture. The reverse of the note featured portraits of Robert Fulton and Samuel F. B. Morse surrounded by an ornate design that occupied almost the entire note. By 1899 however, The $2 Silver Certificate was redesigned with a small portrait of George Washington surrounded by allegorical figures representing agriculture and mechanics. The only large-sized, Federal Reserve Note–like $2 bill was issued in 1918 as a Federal Reserve Bank Note. Each note was an obligation of the issuing Federal Reserve Bank and could only be redeemed at the corresponding bank. The obverse of the note featured a border-less portrait of Thomas Jefferson to the left and wording in the entire center. The reverse featured a World War I battleship.  In 1929, when all U.S. currency was changed to its current size, the $2 bill was issued only as a United States Note. The obverse featured a cropped version of Thomas Jefferson's portrait that had been on previous $2 bills. The reverse featured Jefferson's home, Monticello.

Abraham Lincoln is currently on the front of the $5 bill and the Lincoln Memorial is on the back. The $5 bill is sometimes called a "fin".  There has been many designs. The first $5 bill was issued as a Demand Note in 1861 with a small portrait of Alexander Hamilton.  A new $5 United States Note was issued in 1869 with a small portrait of Andrew Jackson on the left and a vignette of a pioneer family in the middle. In 1870 National Gold Bank Notes were issued specifically for payment in gold coin by participating banks. The obverse featured vignettes of Christopher Columbus sighting land and Columbus with an Indian Princess; the reverse featured US gold coins.   In 1886, the first $5 Silver Certificate was issued with a portrait of Ulysses S. Grant on the obverse and five Morgan silver dollars on the reverse. In 1896 the famous "Educational Series" Silver Certificate was issued. The entire obverse was covered with artwork representing electricity and the reverse featured portraits of Ulysses Grant and Phillip Sheridan.  In 1899, there was a new $5 silver certificate with a portrait of Running Antelope on the face was issued. In 1914, the first $5 Federal Reserve Note was issued with a portrait of Abraham Lincoln on the obverse and vignettes of Columbus sighting land and the Pilgrims' landing on the reverse.

Only two non presidents are currently on our paper currency Hamilton and Franklin).   Alexander Hamilton is on the $10 bill and the US Treasury is on the back as Hamilton was the first US Secretary of the Treasury.  See He is the only person on our currency not born in the US.  Hamilton was born in the West Indies.

Andrew Jackson, or "Old Hickory" is on the front of the $20 bill and the White House is on the back.  Jackson was the first President to invite the public to attend the White House ball honoring his first inauguration. Many poor people came to the inaugural ball in their homemade clothes. The crowd became so large that Jackson's guards could not keep them out of the White House, which became so crowded with people that dishes and decorative pieces inside were eventually broken. Some people stood on good chairs in muddied boots just to get a look at the President. The crowd had become so wild that the attendants poured punch in tubs and put it on the White House lawn to lure people outside. Jackson's raucous populism earned him the nickname "King Mob".

There have actually been nine images to bless the front of the $50 dollar bill, now referred to as a "Grant".  The US Capitol building is now on the back.

  1. 1861: Bald Eagle
  2. 1862: Alexander Hamilton
  3. 1869: Henry Clay
  4. 1870: George Washington
  5. 1874: Benjamin Franklin
  6. 1878: Edward Everett
  7. 1882: Silas Wright
  8. 1891: William H. Seward
  9. 1913: Ulysses Grant

Benjamin Franklin is on the $100 dollar bill which are called "C Notes" or "Franklins" and Independence Hall is on the obverse side as Franklin was a leader in the cause to be independent and actually wrote much of the Declaration of Independence.  Unlike Thomas Jefferson, who is credited with writing the Constitution, Franklin was against slavery.  See

The $100,000 bill was the largest bill ever printed. It was only used between banks and contained Woodrow Wilson's picture.

See  The use of different types of check digits in the world of business and commerce is so widespread that it’s almost easier to list where they’re not used. The Wikipedia page about check digits gives lots of examples of their use, including ways to verify bank routing codes, credit card numbers, ISBN numbers on books, magnetic stripe cards, Postnet and Intelligent Mail barcodes used by the United States Postal Service, vehicle identification numbers, and on and on.  Check digits play a major role in our daily lives. Information below contains methods used in a number of areas, with differing algorithms used to calculate the check digit. 

"Check Digit" in Credit Cards, using the Luhn algorithm

The Luhn algorithm or Luhn formula, also known as the "modulus 10" or "mod 10" algorithm, is a simple checksum formula used to validate a variety of identification numbers, such as credit card numbers, IMEI numbers, National Provider Identifier numbers in US and Canadian Social Insurance Numbers. It was created by IBM scientist Hans Peter Luhn and described in U.S. Patent No. 2,950,048, filed on January 6, 1954, and granted on August 23, 1960.

The algorithm is in the public domain and is in wide use today. It is not intended to be a cryptographically secure hash function; it was designed to protect against accidental errors, not malicious attacks. Most credit cards and many government identification numbers use the algorithm as a simple method of distinguishing valid numbers from collections of random digits.

The formula verifies a number against its included check digit, which is usually appended to a partial account number to generate the full account number.

LUHN Formula (Mod 10) for Validation of Primary Account Number

The following steps are required to validate the primary account number:

Step 1: Double the value of alternate digits of the primary account number beginning with the second digit from the right (the first right--hand digit is the check digit.)

Step 2: Add the individual digits comprising the products obtained in Step 1 to each of the unaffected digits in the original number.

Step 3: The total obtained in Step 2 must be a number ending in zero (30, 40, 50, etc.) for the account number to be validated.

For example, to validate the primary account number 49927398716:

Step 1:

        4 9 9 2 7 3 9 8 7 1 6

         x2  x2  x2  x2  x2 


         18   4   6  16   2

Step 2: 4 +(1+8)+ 9 + (4) + 7 + (6) + 9 +(1+6) + 7 + (2) + 6

Step 3: Sum = 70 : Card number is validated

Note: Card is valid because the 70/10 yields no remainder.

The account number 4992739871x  was determined by using the above steps which yields the sum (64 + x). Then x is determined by subtracting the units digit 4 from 10 and getting x = 6 (or multiplying the units digit 4 by 9 and using the units digit 6 of the 36 result).


You can amaze your friends by asking them to look at their  credit card number. If they will give you  all but one digit you will tell them what digit is  missing.  However  they must give you the missing odd positioned digits in no set order and the missing even positioned digits in no set order. 


If 49927398716 was a credit card number, but they did not give you the 8 (an even positioned digit from the right) you would add the odd positioned digits of 6, 7, 9, 7, 9, 4 and get a sum of 42.  Then you would calculate the doubles of the even positioned digits  (1+8), 4, 6, and 2 and get another sum of 21, which when added to 42 yields 63.  To make that divisible by 10, we need 7.  But 7 comes from an even positioned digit that  was doubled.  So the missing digit cannot be 3.5, but can be 8 which when doubled is 1+6 = 7.


CARD TYPE Prefix Length Check digit algorithm
MASTERCARD 51-55 16 mod 10
VISA 4 13, 16 mod 10
AMEX 34 37 15 mod 10
Diners Club/Carte Blanche 300-305  36  38 14 mod 10
Discover 6011 16 mod 10
enRoute 2014   2149 15 any
JCB 3 16 mod 10
JCB 2131   1800 15 mod 10

Check Digits in a Universal Product Code: UPC

Universal product codes are those ubiquitous sequences of bars and numbers—aka barcodes—found on just about everything sold in stores. The black bars of varying widths in a UPC are read by scanners and then converted into the 12-digit sequences of numbers you see written at the bottom of barcodes. 

UPC Check Digit Algorithm

  1. Start by adding up all the digits in the odd-numbered positions.

  2. Now multiply this sum by 3.

  3. Next, add all the digits in the even-numbered positions (except the final check digit) to this result.

  4. Then divide this number by 10 and find the remainder.

  5. If the remainder isn’t 0, subtract it from 10.

If you do this, the number you get will always be the same as the final check digit of the UPC! For example, here’s the UPC of a magazine: 0 7 4 4 7 0 2 4 0 9 3 8. See if this is a valid UPC by calculating the check digit:


  1. Add odd-positioned digits: 0 + 4 + 7 + 2 + 0 + 3 = 16

  2. Multiply this sum by 3: 16 x 3 = 48

  3. Add even-positioned digits to the total: 48 + 7 + 4 + 0 + 4 + 9 = 72

  4. Divide by 10 and find the remainder: 72 / 10 = 7 remainder 2

  5. Subtract the remainder from 10: 10 – 2 = 8

Which is exactly the same number as the check digit of the UPC…so this is indeed a valid barcode.


The UPC system was developed to give you a faster, more accurate, and automated way to buy things at stores by removing the need for every object to be tagged and rung up individually (a process that’s highly prone to errors). But with automation always comes a certain amount of “danger” since a human might not be present to verify things. To help ward off problems that could arise from this, the UPC system was designed to have numerous error checking safety measures built in—including the check digit. In looking for barcode errors, it’s much easier to detect any mistakes, misprints, or forgeries.


"Check Digit" in a book's  ISBN:  An ISBN (International Standard Book Number) is a controlled, 10- or 13-digit identification number allowing publishers, libraries, and book dealers to locate books. An International Standard Book Number consists of 4 or 5 parts. For a 13 digit ISBN:
  1. the GS1 prefix: 978 or 979 (indicating the industry (978 denotes book publishing)
  2. the group identifier (language-sharing country group)
  3. the publisher code
  4. the item number (title of the book)
  5. a check digit.

The calculation of an ISBN-13 check digit begins with the first 12 digits of the thirteen-digit ISBN (thus excluding the check digit itself). Each digit, from left to right, is alternately multiplied by 1 or 3, then those products are summed modulus 10 to give a value ranging from 0 to 9. Subtracted from 10, that leaves a result from 1 to 10. A zero (0) replaces a ten (10), so, in all cases, a single check digit results.

For example, the ISBN-13 check digit of 978-0-306-40615-? is calculated as follows:

s = 9x1 + 7x3 + 8x1 + 0x3 + 3x1 + 0x3 + 6x1 + 4x3 + 0x1 + 6x3 + 1x1 + 5x3
  =   9 +  21 +   8 +   0 +   3 +   0 +   6 +  12 +   0 +  18 +   1 +  15
  = 93       93 / 10 = 9 remainder 3   Then 10 - 3 = 7
Thus, the check digit is 7, and the complete sequence is ISBN 978-0-306-40615-7.

The ISBN-13 check digit calculation is:

x_{13} = 10 - \big(x_1 + 3x_2 + x_3 + 3x_4 + \cdots + x_{11} + 3x_{12}\big) \,\bmod\, 10.

This check system does not catch all errors of adjacent digit transposition. If the difference between two adjacent digits is 5, the check digit will not catch their transposition. For instance, the above example allows this situation with the 6 followed by a 1. The correct order contributes 3x6 +1x1 = 19 to the sum; while, if the digits are transposed (1 followed by a 6), the contribution of those two digits will be 3x1+1x6 = 9. However, 19 and 9 are congruent modulo 10, and so produce the same check digit.  The ISBN-10 formula uses the prime modulus 11 avoids this blind spot, but requires more than the digits 0-9 to express the check digit. A 10 digit ISBN develops a check digit computed so that multiplying each digit by its position in the number (counting from the right) and taking the sum of these products modulus 11 is 0. The furthest digit to the right (which is multiplied by 1) is the check digit, chosen to make the sum correct. It may need to have the value 10, which is represented as the letter X. For example, take the ISBN 0-201-53082-1. The sum of products is 0x10 + 2x9 + 0x8 + 1x7 + 5x6 + 3x5 + 0x4 + 8x3 + 2x2 + 1x1 = 99 ≡ 0 modulo 11. So the ISBN is valid.  While this may seem more complicated than the first scheme, it can be validated very simply by adding all the products together then dividing by 11.  More info on frequency of errors is at

Social Security Numbers

SSNs are nine digits long and consist of three different parts. The parts consist of three area numbers, two group numbers and four serial numbers.

Area numbers are the first three numbers on the left. In the past, area numbers represented the state in which you first applied for your Social Security card. The East Coast represented the lowest numbers, and the West Coast represented the highest. In 1972, changes were made to the determination technique of area numbers. These changes are still in effect today. Area numbers are now determined by the zip code that is found on your original application form. Many people receive their SSN at birth, and their area number reflects the state in which they were born.

Group numbers consist of two digits, 01 to 99, and are used to divide area numbers into smaller pieces. Within each area, the group numbers (middle two digits) range from 01 to 99 but are not assigned in consecutive order. For administrative reasons, group numbers issued first consist of the ODD numbers from 01 through 09 and then EVEN numbers from 10 through 98, within each area number allocated to a State. After all numbers in group 98 of a particular area have been issued, the EVEN Groups 02 through 08 are used, followed by ODD Groups 11 through 99.

Group numbers are assigned as follows:

First: ODD 01, 03, 05, 07, 09
Second: EVEN 10, 12, 14, 16,  to 98
Third: EVEN 02, 04, 06, 08
Fourth: ODD 11, 13, 15, 17,  to 99

Serial numbers are assigned consecutively. As they are the last four digits of your Social Security number, they range from 0001 through 9999.

On June 25, 2011, Social Security changed the SSN assignment process.  Since 1973, social security numbers have been issued by the central office. The first three (3) digits of a person's social security number are determined by the ZIP Code of the mailing address shown on the application for a social security number.

 Prior to 1973, social security numbers were assigned by the field offices. The number merely established that his/her card was issued by one of the offices in that State. The same area, when shown more than once, means that certain numbers have been transferred from one State to another, or that an area has been divided for use among certain geographic locations.  


 Area Number




New Hampshire












Rhode Island






New York



New Jersey















North Carolina



West Virginia



Not Issued



South Carolina



















































Area Number










North Dakota



South Dakota





















New Mexico



























District of Columbia



Virgin Islands



Puerto Rico






American Samoa



Philippine Islands



Not Issued



Not Issued



Not Issued



Not Issued



Railroad Board**



Enumeration at Entry



Not Issued

***700-728 Issuance of these numbers to railroad employees was discontinued July 1,1963.    Any number beginning with 000 will NEVER be a SSN.




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