Mathematical Knowledge Prime Numbers, Factoring Numbers, Inequality Signs


A prime number is an integer that cannot be separated by two numbers multiplied together.

1, 2, 3,  5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 are prime numbers in the range 1 to 50

Numbers ending with 0, 2, 4, 6, 8 can be divided by the number 2 .

Numbers 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 44, 48 can be divided by the number 3.

Numbers ending with a 5 or 0 can be divided by 5 and other numbers depending upon the number.

Number 49 can be divided by the number 7.  7 X 7 = 49



These numbers are an integer followed by the exclamation point sign , such as 9!

9!  is 9 X 8 X 7  X 6 X 5 X 4 X 3 X 2 X 1     504 X 120 X 6 = 362,880

4! is 4 X 3 X 2 X 1  = 24



<  is used to represent two numbers or variables one less than the other  a  <  b  , a < 8

> is used to represent two numbers or variables one greater than the other  a > b , a > 8

These are used to represent situations where the variable is to be less than or greater than a number

Less than or equal to, or greater than or equal to

The < and > signs are used, but a bar appears under the character.

<_        and   >_     (I am not sure how to create these symbols in this Word Press system, but it is probably a typing multiple keys simultaneously.

Another mathematical symbol is for “not equal to” . It is an equal sign  =  with a slash through it.

/=     n /= 2    means that the variable n is not equal to 2.


Mathematical Knowledge Equations with two variables

These equations have two letters for variables, such as a and b,  p and q, x and y.

An equation might be 5x + y +9  or a formula such as 5a[superscript 3, a X a X a ]  + 5xy

Arithmetic uses a system with the acronym PEMDAS for the order of calculations in a series of addition, subtraction, multiplication, division, and exponent power actions.  A frequently used phrase to recall the PEMDAS order is

1. Please Excuse My Dear Aunt Sally

2.  A phonetic alphabet phrase is Papa Echo Mike Delta Alpha Sierra

Number 1 is most frequently used in textbooks and study guides.

P  Parentheses  (  )  Perform the mathematical actions with the parentheses.

E Exponents   with a number raised one half line above the letter.  The number may be positive or       negative.

M  Multiplication

D  Division

A  Addition

S  Subtraction

What is the value of 5a(x + y) when a = 4,  x = 2,  and y = 7

Parentheses first (x + y) = 2 + 7 = 9  .  Add the two numbers together.  If there had been three variables between the parentheses,  add the numbers on either side of the + sign, and then subtract that number from the number on the other side of the  –   sign.

Exponents   there are no exponents in this equation, such as a X a  or ” a squared “.

Multiplication  Multiply 5 X a  to get 5 X 4 = 20

Division   there are no division actions for this equation.

Addition/Substraction  There are no additional actions to be done within the equation.

5a(x + y) when a = 4, x = 2, and y = 7  is  5(4)(2 + 7) = 5(4)(9) = 180




ASVAVB and similar tests study help websites

Internet Sources that I think are useful for preparation to take the Arithmetic and Mathematics Knowledge Parts of the Armed  Services Vocational Aptitude Battery
It is my understanding that these portions of the ASVAB are the ones with the highest failure rate.
I have been looking for internet accessible study help.  I have found two groups of sources
Internet web pages and web sites.
Youtube videos.
I have looked at sample GED Tests and sample ASVAB Tests.  I think that success in completing GED type questions will help a  a person succeed in completing the Arithmetic and Mathematics segments of the ASVAB.
The ASVAB  has many questions that are based on algebra or geometry.  The following lists of website addresses (URLs) are some  I think will help a person.  There are many other sites with good instruction.  I chose to limit my list to websites  that usually have a domain ending in:  edu  , k12, or org.
There are many study guides available.  I found books by 7 different publishers at a bookstore.  Public libraries may have  “Test Book Section” with a variety of study guides for civil service, military service, undergraduate, and professional  education programs.  Used book stores may have copies of older editions of study guides.  A major change in the GED test  system was made in 2014, so only GED guides with copyright dates of 2014 or later should be purchased or borrowed.
     Professor Perez  .
This site was started by Professor Perez of Saddleback Community College, Mission Viejo, California.
The segments begin with pre-algebra mathematics, but continue up through algebra, and perhaps even into calculus.
Web pages are available in English AND in Spanish.   The file formats are in several different programs,  including .wmv, and Youtube videos.
     Khan Academy
The Khan Academy was started by by Salman Khan in 2006 to provide free, high-quality instruction to anyone with access  to a computer.  Broad area of education include Mathematics, Science and Engineering, Arts and Humanities, Computing, and  Test Preparation.
The segments with videos to help those preparing for the ASVAB include Arithmetic, Algebra, Geometry, and Probability and Statistics.
A typical URL begins with     and one of  these words  algebra , arithmetic , geometry , probability   .
There are subtopics for each of the words.  I recommend getting an ASVAB study guide, using it, and then looking at the Kahanacademy video listings to find    the video or videos that are most likely to match the study guide topic.  (New study guides may have a CD disc or password card included within which will provide links to the publisher’s own help site.  Take advantage of this material if it is in the guide.)
GEOMETRY    The sections that show up on the left hand side of the web page under Courses,  Math 351 down through Intermediate Algebra may be useful to prepare for examinations.  The publishers of the “Dummies” book series have web pages about the concepts of geometry and use of geometry.
 Some of the websites that I found are:
 While scanning Youtube,  I came upon a video titled  “Most missed GED Math Test Problem”  .
I have not checked my ASVAB Test Guides to see if this problem type is also used in the ASVAB test, but I have a hunch that this problem type does appear in the ASVAB.   (I hope that I have the letters and numbers correct in capitalization.) has videos offering instruction in geometry.  I think that a person studying for an exam with a pre-examination information sheet, or a common study guide, should look at the sheet or guide and the list of Khan Academy videos to see which ones will help them prepare for that specific examination.
 There is a free website  which may be helpful in studying the types of problems on that exam that are probably similar to those found on the ASVAB test.   The author of this site offers a larger, pay for use, site for the total GED.
You may not need to subscribe to it unless you do not have a high school diploma.  GED certificate holders must receive a higher score on the AFQT and ASVAB test to be considered enlistment.
If you have a chance to get a high school diploma OR go for a GED, get the diploma./
Studying  for the GED test in SPANISH.
I found three sites that offer preparation material to take the Spanish language GED examination.  The ASVAB is an English language test.  If you feel more
comfortable learning the arithmetic and mathematics in Spanish, you might try these sites.
Spanish GED
Khanlatinoamericano  (Khan Academy)
I located a set of 11 lessons for geometry in Khanlatinoamericano  when I searched  Geometrica 1   ….. Geometrica 11
There may be a set of lessons for algebra    The English and spanish spellings of algebra are the same except for the accent over the a.
The master webpage for Khanlatinoamericano may have a table of contents list similar to the ones of the English language web site.

Mathematical Knowledge One Variable Equation

A one variable equation may look like this, with informnation on either side of the equal = sign.

12a + 8 = 53 – 3a

The variables can be on the left side, and the numbers on the right side.  The numbers could be put on the left side and the variables on the right side.

Variables on the left:  12a + 3a = 53 -8

Numbers on the left:  8 – 53 = -3a – 12a    ,  or  -45 = -15a

It is easier to work with positive values on both sides of the = sign when it possible.

Let’s use  15a  = 45  .  Divide both side of the equation by the number 15 .  15a/15 = 45/15

a = 3      Mathematics check  12(3) + 8 = 44.   53 – 3(3) = 53  – 9 = 44 .

Linear equations where one or more sides have a fraction.

2x + 7 =  x/4      .    The first step is to remove the fraction.

4 X (2x+ 7) = 4 X  (x/4)       8x + 28 = x   .  Move the number or the 8x to one side.

8x – x  = – 28   7x  = -28    7x/7 = -28/7     x = -4

Mathematical Knowledge Question Monomial Binomial Polynomial

Algebra expressions are composed of terms.  A term is a number, a variable, or numbers multiplied by variables.  A number such as 3, a variable can be something like 3a or 3a[superscript2]  [3 times a times a], a combination of letters representing variables such as ab .

Monomial:  an algebra expression of one term.  An example is (4 + 2z

Binomial:  an algebra expression of two terms.  An example is (4+2z + (5-2y)

Polynomial:  an algebra expression of three or more terms.  An example is (4+2z) + (5+6y) + (4-a)

Terms can be added or subtracted IF they have they same variables and the variables have the same exponents.  (4+2z and (5+6z can be added together.

(4+2z) + (5+6z)= 9 + 8z , but (4+2x) + (5+6y) cannot be added together.

Unlike terms such as a + a [superscript2]  [a to the second power], or a + b , cannot be added together.


Numbers and variables can be multiplied together and divided.

(4 + x) can be multiplied by a variable such as 6y.   (4 + x)(6y) = 24y + 6xy

Two variable can be multiplied together.  (4d)(6y) = 24dy  .  It may be easier to rearrange and separate the components of each variable, combining the numbers first and then combining the letters.     (4)(6)(d)(y)  (4)(6) = 24  .  (d)(y) = dy .   24 times dy  or using the mathematical symbol for multiplication  24 x dy , equals =  24dy .

Multiplication of bionomial and trinomial numbers (one number and two variables)

Multiplication of binomial numbers is shown first.  Use a method known as FOIL.

The capital letter X is used as the symbol for multiplication.

F is for First, O is Outside, I is for Inside, and L is for Last.

Multiplying (4 + 2x) times (5 + 6y) using FOIL  (4 + 2x) X (5 + 6y)

Multiply the FIRST part of the term inside each parentheses.  4 X 5 = 20  F for First

Multiply the OUTSIDE part of each term)  4 X 6 y  = 24y  O for Outside

Multiply the INSIDE or second part of each term  2x  X  5 = 10x

Multiply the Last part of each term  2x X 6y  = 12xy

Add the result of each multiplication   20 + 24y + 10x + 12xy

MULTIPLICATION of TRINOMIAL or POLYNOMIAL numbers.  Polynomial is many terms.

Multiply each term within the first variable by each term of the second variable.

(4 + 2n)  X  (2 + p + q)

Multiply the first part of the first term and the first part of the second term    4 X 2 = 8

4 X p = 4p

4 X q = 4q

2n X 2 = 4n

2n X p  = 2np

2n X q = 2nq

Add the result:  8 + 4p + 4q + 4n + 2np + 2nq

Here is an example of the result using n = 2 ,  p = 3 , q = 5

(4 X 2) + (4 X p) _ (4 X q) + (4 X n) + (2 X n X p) + (2 X n X q)

(4 X 2) + (4 X 3) + (4 X 5) + (4 X 2) + (2 X 2 X 3) + (2 X 2 X 5)

8 + 12 + 20 +8 + 12 + 20 = 80     (4 + (2  X 2)  X  (2 + 3 + 5) = 80

Three common quadratics

(x + y)(x-y)  =  x [squared] – y [squared]  the number 2 appears one half line above the x and y

(x + y) [squared] = x [squared] + 2xy + y [squared]

(x-y) [squared] = x [squared] -2xy + y [squared]

A perfect square number such as 4, 16, 25, 36, 49, 64, 81 in front of the x[squared] or y[squared] in a quadratic equation means that it is probably one of the three common quadratics.




Trying to answer mathematical knowledge questions

I have started helping to tutor students who are studying for arithmetic knowledge and mathematical knowledge tests.  These students need help in these areas for a variety of different tests.

Trying to find the right phraseology and wording to explain a concept is not easy.  This is the reason why there are so many different published study guides.  Each group of authors and editors is searching for the “magic wording” that will “turn on the light bulb” so to speak.    A sidebar note in an economics textbook I used in a course back in 1997 said that there were about 50 textbooks that were written to explain the basic principles of economics.

This is my attempt to explain some of the principles of Algebra and Geometry.  If anyone knows of a calculus textbook that explains calculus for to a person who got a D minus and a very low F, please contact me.   I am well beyond college, and am still looking.

ALGEBRA     Solving mathematical problems when one or more numbers are unknown.  These numbers are called variables.  Letters a, b, c, n, p, x, y, z are frequently used as letter symbols to represent the unknown number.  In some situations, there may be two numbers that will solve the mathematical formula.

Some word problems are like this:  Eddie is 4 years older than his cousin George.  The total of their ages is now 30.  In three years, how old will Eddie be?

A parellelogram has two lines that are the same distance apart and two lines connecting them.  The angle of the left side line is between 0 and 90 degree, and the angle of the right side line is between 90 and 180 degrees.  if the degree value of the bottom angle of the right hand line is X, and the angle at the top line is Y , and the degree  angle of  X is three times Y   What is the value of angle Y

Well, if I created the algebraic equations correctly,  Eddie is now 17 and George is 13.  Eddie will be 20 in three years.

The bottom angle X is 135 degrees, and the top angle Y is 45 degrees.  The lines from the bottom line to the top line  are like the   /       /     slash lines that we see in an internet Uniform Resource Locator.

My next step is to describe Algebra equations with one variable, in my next posting.




My research work on Daniel Shays (174?-1825) and the Daniel Shays Rebellion

My home town in Massachusetts is near the home town of Daniel Shays, who was one of the leaders of the insurgency in 1786-1787 to protest property taxation, foreclosures of farms, and the state government actions of the officials in Boston that affected the people of Central and Western Massachusetts.

Daniel Shays led a force of about 1200 men to Springfield, Massachusetts to attempt to seize the federal arsenal (now known as the Springfield Armory) which had 7000 muskets and 1300 kegs of gunpowder.  The state militia led by General Shepard repulsed them, and a militia force from the Boston area led by General Lincoln went after the insurgents.  The insurgents first encamped in Pelham and tried to negotiate with General Lincoln who was encamped in Hadley.  After negotiations failed, the insurgents retreated NE and encamped in the town of Petersham.  They were probably trying to return to their base in Rutland, 12 miles NW of the city of Worcester.  General Lincoln marched his militia by night during a blizzard and surprised the insurgents at Petersham.  They had not posted guards because they thought that no one would ever attempt to march at night in a blizzard.  About 150 were captured, but Shays and others escaped.  Shays eventually went to a town in Quebec just over the border line  (I think that it had an English name at the time, but now has a name that includes the words Sainte  and Richelieu in it.)

Shays and some of his party settled in Sandgate, Vermont (Bennington County).  He moved to Sparta, New York, where he died in 1825.

There are some intriguing gaps in the story about the Rebellion that I have been trying to research and fill.  Why did Daniel Shays leave his home in Pelham and go 50 miles east (2 days by horseback) to join others in Rutland?  Who recruited him?  Who provided food and supplies for the insurgents?  The insurgents encampment was at the barracks built in 1778 after the Convention Army of British General Sir John Burgoyne was moved from Cambridge to Rutland.

The Convention Army of 4to 5 thousand was causing the local firewood supply to be depleted, and was a strain on the local economy of Cambridge and Boston.  It was also thought that a British fleet with marines and soldiers might try to land and rescue the prisoners, or return them into a fighting military force.   Rutland, 50 miles inland, was considered to be a better location,

There is a place in Rutland known as Barrack Hill, fronted by Barrack Hill Road.  The U.S. Geological Survey Geographic Names Information System Feature ID number for Barrack Hill is 617546  search Domestic Names  to get to the search engine.  This system also includes several mapping systems with topographic and or satellite imagery.    Using the Microsoft Virtual Earth System a few days ago, I saw that the home of Colonel Rufus Putnam is about 1 mile east of the site of the POW barracks that were used by the insurgents led by Shays.   The Putnam home is now a bed and breakfast inn the RufusPutnamInn.

Shays met with Putnam and told him that he was not the overall leader of the movement.  Putnam had been one of Shays’s regimental commanders during the Revolutionary War, as commander of the 4th or 5th Massachusetts Regiment of the Continental Army.    Putnam was also one of George Washington’s chief engineering officers during the war.    Putnam led a group that moved to the area that is now Marietta, Ohio and settled it when it was still a frontier area.

One of the leaders of the insurgent movement was Francis Stone (1740-1802) of North Brookfield, a nearby town.  He had been a captain in the army during the Revolutionary War.  His brother was one of the militia members who remained loyal to the state.

Shays fled Massachusetts after the failed attempt to take the arsenal.  Francis Stone was able to remain in Massachusetts.

Putnam was loyal to the Commonwealth of Massachusetts government led by Governor James Bowdoin.  But what might have he known about the insurgent force assembling in his home town of Rutland, and its leadership group and those who were financing and training the force.  Did he have any information about who the leaders were, and did he send any intelligence information to the Governor or other state officers?  Did he know which members of the militia in Worcester County could be counted on to support the state government, and which members were of dubious loyalty or active supporters of the movement?

Who owned the land that contained the Barrack and other cabins on its grounds?  Who permitted it to be used as an assembly point for the insurgent force?

I know the town of Rutland parcel code numbers for the land that is found on both sides of Barrack Hill Road.  Is it possible to trace the ownership of the parcels back to the 1780s by looking at Registry of Deeds microfilms?  Could a title search company do this kind of work, and what is it likely to cost to have a deed searched back into the 1780s?

Slavery was legal in Massachusetts until several cases in 1780 by the Supreme Judicial Court ruled that it was not permitted under the new constitution enacted in 1780.  The 1790 Federal Census tallied 5,403  All other free persons and 0 slaves in Massachusetts.  Some of the “all other free persons” may have been non-white free people (African-American descent, and some may have been Native American descent in 1780.  But looking at the Heads of Families schedules for the 1790 Census show that most of the AOFP were living in households headed by a white male or female.


How many of them were owned as slaves in the same household in 1780, and then emancipated by the Supreme Judicial Court decisions of 1780?  Were they staying in the same household because they did not have money of their own to move out and buy property of their own in town or to migrate to another town.  The name of any person who was in the AOFP group was not recorded unless he or she was the head of a household.

Was resentment against the Supreme Judicial Court for emancipating slaves without compensation an issue at the time of the protests that led to shutting down sessions of the court?   I am assuming that that no compensation was made by the state government after the Supreme Judicial Court ruled that slavery was now illegal in Massachusetts.

I could go look up the Acts and Resolves of the Legislature, which are online, to see if there was any compensation.  I can also go look at copies of early newspapers which have been digitally scanned online to see if there are stories or commentaries about the effect of freeing the slaves in Massachusetts.

I am also trying to find the names of members of the state senate and the state house of representatives.  I think that these journals have been digitally scanned and put up for research on the internet.  I am making an assumption that the names of members of each chamber appear somewhere as in a list of those who have taken an oath of office.  This is another thing that I will be looking into in the future.

I know that a substantial number of members of the 1786 legislature were replaced by newly elected members in the 1787 legislature.  Who were the members of the 1780-1786 legislatures?  Did any of them win election to future legislatures?

There is a lot to research, and I have many email messages to send out to libraries and archives asking for help.

So long for now,  Tom Lindsey









My current government documents projects in my post U T Arlington career

I watch library offers lists for materials of interest to particular libraries, and notify specific libraries of the existence of this material.

I had noticed that some of the agency entries in a guidebook for government publications had a ? mark as to the last year of existence of the agency.  I am a person who likes to have database records checked and up to date, so I use information from other sources to try to find the date that the agency closed down, and the legal citation that caused its closure or relocation to another agency.     This can be easy some times, and difficult most of the time.

I was checking on one agency this afternoon, and discovered that it was now encompassed within a larger agency that had also taken another agency under its structure.  The new bigger agency has never published any publications, so most government documents librarians are probably unaware of its existence, unless they are avid readers of the Federal Register, the Code of Federal Regulations, and the annual appropriations bills and appropriation hearings of the Congress.

If any library out there is looking for someone to help prepare publication offers lists to send to their regional library or a national needs and offers lists, I can do a lot of telecommuting work for you without having to visit your library.

Contact me using my email addresses.  I am also found on LinkedIn and Facebook, but I do not check in regularly.



The “Invisible Federal Government Information” Collection is no longer in the library

Back in 2011 and 2012 I created some Libguides (TM) about what I called the “Invisible Federal Government Information ” collection at the University of Texas at Arlington Library.  The collection had about 900,000 publications in print, microfiche, and sheet map form.  More than half the print collection was not included in the online catalog.

I discovered that about 15,600 U.S. Geological Survey 7.5 minute series topographic maps were not listed in the catalog, but could be found only by checking a shelf list of cards arranged in latitude/longitude order and then looking for the A1 to A8 … H1 to H8 code for each of the 64 sheets in a 1 by 1 degree square.  There were other things like this

It was never going to be cataloged in its entirety.  So my solution was to come up with a multipage Superintendent of Documents classification number listing of the major stems such as A 13 for U.S. Forest Service, C 13 for National Bureau of Standards/National Institute of Standards and Technology, D 100 for Army, D 200 for Navy, D 214 for Marine Corps, and so on up through Y 10  Congressional Budget Office.  It was an attempt to let people know whatF agencies and their publications might be available.  Used with resources such as other library catalogs which included every federal government publication in the collection, the G.P.O. Catalog of Government Publications,  the Federal Digital System , and specialty printed indexes, it was my attempt to “open up the stacks”,

I visited the library on Saturday, June 25, 2016.  The entire collection is gone  from the library building; there may be remnants left in the Library Collection Depository building.  There is no government documents librarian.

What a shame to see 17 years of your working career to have disappeared in 3 1/2 years.

Harrison E. Salisbury on how our country should conduct itself

This quotation is from a lecture by Harrison E. Salisbury (1908-1993), “The Book Enchained”, sponsored by the Center for the Book in the Library of Congress and the Authors League of America.  It was presented at the Library of Congress on September 28, 1983.

A printed copy was distributed to federal depository libraries that had chosen to receive publications in item number 785-J or 0785-J.  It has Superintendent of Documents Classification number LC 1.38: 10 .

“Let us conduct ourselves that no one may ever ask whether the Statue of Liberty still stands in New York harbor.  If we follow this course not only will we protect and preserve our own freedom but the light of Liberty’s torch slowly but surely will penetrate all corners of the world.”

A good reminder to all who live in the United States, citizens or non-citizens, in the year 2016.

Mr. Salisbury had spend a considerable part of his reporting career in countries run by Communist governments, such as the Soviet Union, the People’s Republic of China, and North Vietnam.   As an editor of the New York Times, he was also aware about other countries where “the light of Liberty’s torch” was a faint vision.




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