Other math breakthroughs of the 12th century

Although Fibonacci is said to have contributed the most to the math realm during the 12th century, there were some other additions provided by different mathematicians. One was scholar Abraham bar Hiyya, a Jewish mathematician who contributed in the areas of algebra and geometry. Hiyya is famous for his equation for area of a circle.
A=.5*L*R where L is the circumference and R is the radius.

Another important contributor was Omar Khayyum, a Persian mathematician. He is most well-known for solving cubic equations, and realizing they can have more than one solution. He did this using conic sections. The use of algebra would not be applied to these equations until later. He also wrote commentary on some of Euclid's postulates. He was able to prove that Euclid's definition of equality of ratios, and the earlier definition proposed by Islamic Mathematicians were equal to each other.

Lastly, another scholar was Nasir al-Din al-Tusi, also a Persian mathematician. He is credited with being the first person to separate the study of trigonometry from astronomy, and give it its own branch of mathematics. One famous law he is credited with is The Law of Sines for Plane Triangles.
a/sinA=b/sinB=c/sinC where a,b,c are the side lengths and A,B,C are the corresponding angle measurements.

 

http://www.mathwarehouse.com/trigonometry/law-of-sines/formula-and-practice-problems.php

Sources:
O'Connor, J. J. "Area of a Circle by Rabbi Abraham Bar Hiyya Hanasi." Area of a Circle by Rabbi      Abraham Bar Hiyya Hanasi. N.p., n.d. Web. 21 Mar. 2014.

O'Connor, J. J. "Nasir Al-Din Al-Tusi." Al-Tusi_Nasir Biography. N.p., n.d. Web. 21 Mar. 2014.

"Omar Khayyam." Khayyam Biography. N.p., n.d. Web. 21 Mar. 2014.

Liber Quadratorum

Liber Quadratorum is recognized as Fibonacci's most impressive collection of knowledge. It was written in 1225 and dealt with number theory. One interesting thing contained in his book is the equation he came up with for finding square numbers (1, 4, 9, 16, 25, etc). He realized that each square number was the sum of the previous square and some odd number:

For some n in the whole numbers: n²+(2n+1)=(n+1)² 

Let n=1,  1+(3)=4. 

Let n=2, 4+(5)=9. 

Let n=3, 9+(7)=16.

Another formula Fibonacci found dealt with finding Pythagorean Triples. He would first find an odd, square number. Then he would add up all the odd numbers from 1 to the chosen number, including one, but excluding the latter. Taking the square of that sum would produce the second square. Adding the two square numbers together would give him the square number whose square would produce the final part of the Pythagorean Triple. I included an example below.
Choose 5 as one of the first squares.

5²=25...5 is the first part of the Triple.

The odd numbers from 1 up to but not including 25 are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23.

The sum of these numbers is 144.
sqrt(144)=12<--- second part

144+25=169

sqrt(169)=13<--- third part

Therefore the Pythagorean triple is 5, 12, 13.

http://www.bing.com/images/search?q=pythagorean+triple&FORM=HDRSC2#view=detail&id=E8B31E2011B18D1FD0CDABE2BF67A9A48BCE68A4&selectedIndex=9

Sources:
O'Connor, J. J. "Leonardo Pisano Fibonacci." Fibonacci Biography. N.p., n.d. Web. 02 Mar. 2014.

"Liber Abaci"

Upon his return to Italy, Fibonacci wrote Liber Abaci in 1202. It was in this work that he introduced the Latin World (Europe) to the Hindu-Arabic number system (0, 1, 2,..., 9). Previously they had used the Roman Numeral numbering system. Liber Abaci  was written on a practical level that included important examples and strategies for merchant and bankers to help them do their mathematical calculations in a more efficient and easier way. Although Fibonacci is more so remembered for the "Fibonacci Sequence" (discussed below), the work he did in spreading the decimal numbers can be argued as his greatest and most influential achievement in regards to the impact it had on future mathematics.

Fibonacci Sequence: 1 1 2 3 5 8 13 21 34 55 89 144 233....
The next number in each sequence is the sum of the previous two numbers before it.
Example: 1+0=1, 1+1=2, 2+3=5, 5+3=8, 8+5=13, etc.

 http://www.bing.com/images/search?q=liber+abaci&qpvt=liber+abaci&FORM=IGRE#view=detail&id=6DF09779E77D0BEC280B8FB022EA68991E13AEAD&selectedIndex=4

Sources
Gies, Frances Carney. "Leonardo Pisano (Italian Mathematician)." Encyclopedia Britannica Online.  Encyclopedia Britannica, n.d. Web. 27 Feb. 2014.

Knott, R. "The Life and Numbers of Fibonacci." Plus.maths.org. N.p., n.d. Web. 27 Feb. 2014.