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... ely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers. Leonardo died sometime during the 1240 s, but his contributions to mathematics are still in use today. Now I would like to take a closer look at some of Leonardo's contributions along with some examples. II Fibonacci's Contributions to Math Decimal Number System vs. Roman Numeral System Algorithm Root Finding Fibonacci Sequence Decimal Number System vs.
Roman Numeral System As previously mentioned Leonardo was the first person to introduce the decimal number system or also known as the Hindu-Arabic number system into Europe. This is the same system that we use today, we call it the positional system and we use base ten. This simply means we use ten digits and a decimal point. In his book, Liber abbacy, Leonardo described and illustrated how to use this system. Following are some examples of the methods Leonardo used to illustrate how to use this new system: 174 174 174 174 i 28 = 6 remainder 6 + 28 - 28 x 28 202 146 3480 + 1392 4872 It is important to remember that until Leonardo introduced this system the Europeans were using the Roman Numeral system for mathematics, which was not easy to do. To understand the difficulty of the Roman Numeral System I would like to take a closer look at it.
In Roman Numerals the following letters are equivalent to the corresponding numbers: I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000 In using Roman Numerals the order of the letters was important. If a smaller value came before the next larger value it was subtracted, if it came after the larger value it was added. For example: XI = 11 but IX = 9 This system as you can imagine was quite cumbersome and could be confusing when attempting to do arithmetic. Here are some examples using roman numerals in arithmetic: CLXXIM + XXVIII = CCII (174) (28) (202) Or CLXXIV - XXVIII = CXLVI (174) (28) (146) The order of the numbers in the decimal system is very important, like in the Roman Numeral System. For example 23 is very different from 32. One of the most important factors of the decimal system was the introduction of the digit zero.
This is crucial to the decimal system because each digit holds a place value. The zero is necessary to get the digits into their correct places in numbers such as 2003, which has no tens and no hundreds. The Roman Numeral System had no need for zero. They would write 2003 as MMIII, omitting the values not used. Algorithm Leonardo's Elements, commentary to Euclid's Book X, is full of algorithms for geometry. The following information regarding Algorithm was obtained from a report by Dr.
Ron Knott titled Fibonacci's Mathematical Contributions: An algorithm is defined as any precise set of instructions for performing a computation. An algorithm can be as simple as a cooking recipe, a knitting pattern, or travel instructions on the other hand an algorithm can be as complicated as a medical procedure or a calculation by computers. An algorithm can be represented mechanically by machines, such as placing chips and components at correct places on a circuit board. Algorithms can be represented automatically by electronic computers, which store the instructions as well as data to work on. (page 4) An example of utilizing algorithm principles would be to calculate the value of pi to 205 decimal places. Root Finding Leonardo amazingly calculated the answer to the following challenge posed by Holy Roman Emperor Fredrick II: What causes this to be an amazing accomplishment is that Leonardo calculated the answer to this mathematical problem utilizing the Babylonian system of mathematics, which uses base 60. His answer to the problem above was: 1, 22, 7, 42, 33, 4, 40 is equivalent to: Three hundred years passed before anyone else was able to obtain the same accurate results.
Fibonacci Sequence As discussed earlier, the Fibonacci sequence is what Leonardo is famous for today. In the Fibonacci sequence each number is equal to the sum of the two previous numbers. For example 13 K) O 13 Leonardo used his sequence method to answer the previously mentioned rabbit problem. I will restate the rabbit problem: A certain man put a pair of rabbits in a place surrounded on all sides by a wall.
How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? I will now give the answer to the problem, which I discovered in the Mathematics Encyclopedia. It is easy to see that 1 pair will be produced the first month, and 1 pair also in the second month (since the new pair produced in the first month is not yet mature), and in the third month 2 pairs will be produced, one by the original pair and one by the pair which was produced in the first month. In the fourth month 3 pairs will be produced, and in the fifth month 5 pairs. After this things expand rapidly, and we get the following sequence of numbers 13, 21, 34, 55, 89, 144, 235, K This is an example of recursive sequence, obeying the simple rule that two calculate the next term one simply sums the preceding two. Thus 1 and 1 are 2, 1 and 2 are 3, 2 and 3 are 5, and so on. (page 1) III Conclusion Conclusion Leonardo Fibonacci was a mathematical genius of his time.
His findings have contributed to the methods of mathematics that are still in use today. His mathematical influence continues to be evident by such mediums as the Fibonacci Quarterly and the numerous internet sites discussing his contributions. Many colleges offer classes that are devoted to the Fibonacci methods. Leonardo's dedication to his love of mathematics rightfully earned him a respectable place in world history. A statue of him stands today in Pisa, Italy near the famous Leaning Tower.
It is a commemorative symbol that signifies the respect and gratitude that Italy endures toward him. Many of Leonardo's methods will continue to be taught for generations to come. Works Cited Dr. Ron Knott Fibonacci's Mathematical Contributions March 6, 1998 web (Feb. 10, 1999) Mathematics Encyclopedia web realms / encyclop /articles / fibonacci . html Works Cited Dr. Ron Knott Fibonacci's Mathematical Contributions March 6, 1998 web (Feb. 10, 1999) Mathematics Encyclopedia web realms / encyclop /articles / fibonacci .
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