# General History

Number Systems
Babylonian had a place value system which used base 60, like our seconds and minutes. The numbers are written smallest to the right, just like what we normally do, but with a space between the place values.
Egyptian had a symbolic system, it had a symbol for each place value and number, so for 30 they would write their 10 3 times.
The numbers are written smallest to the left, the opposite of the Hindu-Arabic and Babylonian.
A good book on the history of mathematics of the Ancient Egyptians is Mathematics in the Time of the Pharaohs by Richard J. Gillings.
Hindu-Arabic is they number system most commonly used in the world today. It is a place value system using base 10, also called a decimal system. The symbols that can go in each place value are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Mayan had a vertical place value system which used base 20. The smallest value on the bottom.
Other Bases are commonly used today, such as base 12 when working with inches and feet. However, more common are the ones used with computers, binary (base 2), octal (base 8), and hexidecimal (base 16).
Any number can be a base, we just have to tell what base we are working in, if different than base 10. For example, if we write the number 576, we assume it is base 10, but it could be base 8. When there is a chance for confusion, we write the base as a subscript, that is 5768.

Place Value for most modern base systems we write our numbers with the largest place value to the left and smallest to the right.
Decimal System, we use the powers of 10 for our place values. We can write them easily in numbers. However to get used to other bases, the more formal mathematical notation using exponents is helpful.
Decimal Place Values
3
2
1
0
-1
-2
-3
words
thousands
hundreds
tens
ones
tenths
hundredths
thousandths
numbers
1000
100
10
1
0.1
0.01
0.001
Powers of 10
103
102
101
100
10-1
10-2
10-3

Other Bases, we use powers of our other bases for the place values, but we don't have nice words for them. The numbers in the base would be the same. Just like we do not have a single symbol for 10, we don't have one for our other bases, so if working in base 5, we would use 0, 1, 2, 3, 4. 5 would be written at 10 since that would be 51.
Other Bases Place Values
3
2
1
0
-1
-2
-3
numbers
1000
100
10
1
0.1
0.01
0.001
Base 2
23 (in base 10, 23 = 8)
22
21
20
2-1
2-2
2-3
Base 8
83 (in base 10, 83 = 512)
82
81
80
8-1
8-2
8-3

Arithmetic in Other Bases
This is very similar to clock arithmetic, or modulo arithmetic. For example when we start at 9AM, and say we are meeting someone in 4 hours, we don't say 13AM, we say 1PM.
Adding & Subtracting: When we reach 'our base' we put a 1 in the next higher power column, and keep going.
Multiplication & Division: Having a 'times table' to reference is best when starting out.

Symbols used in Math & their meaning (PDF ~ partial list)
Common Symbols for constants (To be added)
Symbols from a particular topic are with that topic
Illustrated Mathematics Dictionary https://www.mathsisfun.com/definitions/
Mathwords http://www.mathwords.com/
Number Words and Number Symbols by Karl Menninger

Some good book on the History of Mathematics
History of Mathematics by Victor Katz
The Historical Roots of Elementary Mathematics by Bunt, Jones, & Bedient
Math through the Ages by Berlinghoff & Gouvêa
History of Mathematics by Boyer
Sherlock Holmes in Babylon by Anderson, Katz, & Wilson
Mathematics of the Incas by Ascher & Ascher

Additional information on these topics can be found on the following sites, as well as many others:
Canadian Society for the History and Philosophy of Mathematics http://www.cshpm.org
History of Mathematics http://archives.math.utk.edu/topics/history.html
HOMSIGMAA - History of Mathematics SIGMAA http://homsigmaa.org/
Math History http://web.stcloudstate.edu/wbbranson/MathHistory.html
Mathematics History http://library.thinkquest.org/22584/
Teaching with Original Historical Sources in Mathematics http://www.math.nmsu.edu/~history/

The following sites have information about Binary, Hexidecimal, and other helpful information when working with computers. There are many other good ones.
http://www.w3schools.com
http://www.w3schools.com/charsets/default.asp
http://www.ascii-code.com/
https://mothereff.in/binary-ascii
http://www.colorhexa.com