Fourier division
Encyclopedia
Fourier division or cross division is a pencil-and-paper method of division
which helps to simplify the process when the divisor has more than two digits. It was invented by Joseph Fourier
.
Suppose that we wish to divide c by a, to obtain the result b. (So a × b = c.)
Note that a1 may not have a leading zero; it should stand alone as a two-digit number.
We can find the successive terms b1, b2, etc., using the following formulae:
Each time we add a term to the numerator until it has as many terms as a. From then on, the number of terms remains constant, so there is no increase in difficulty. Once we have as much precision as we need, we use an estimate to place the decimal point.
It will often be the case that one of the b terms will be negative. For example, 93,−12 denotes 9288, while −16,32 denotes −1600 + 32 or −1568. (Note: 45,−16,32 denotes 448432.) Care must be taken with the signs of the remainders also.
The general term is
should be multiplied by 100, and the next term added (or, if negative, subtracted). This result should be multiplied by 100, and the next term added or subtracted, etc., until all terms are exhausted. In other words, we construct partial sums of the b terms:
The last partial sum is the value for b.
≈ 3.14159.
The result is 32,-17,10 or 31,83,10 yielding 0.318310.
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
which helps to simplify the process when the divisor has more than two digits. It was invented by Joseph Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...
.
Method
The following exposition assumes that the numbers are broken into two-digit pieces, separated by commas: e.g. 3456 becomes 34,56. In general x,y denotes x·100 + y and x,y,z denotes x·10000 + y·100 + z, etc.Suppose that we wish to divide c by a, to obtain the result b. (So a × b = c.)
Note that a1 may not have a leading zero; it should stand alone as a two-digit number.
We can find the successive terms b1, b2, etc., using the following formulae:
Each time we add a term to the numerator until it has as many terms as a. From then on, the number of terms remains constant, so there is no increase in difficulty. Once we have as much precision as we need, we use an estimate to place the decimal point.
It will often be the case that one of the b terms will be negative. For example, 93,−12 denotes 9288, while −16,32 denotes −1600 + 32 or −1568. (Note: 45,−16,32 denotes 448432.) Care must be taken with the signs of the remainders also.
The general term is
Partial quotients with more than two digits
In cases where one or more of the b terms has more than two digits, the final quotient value b cannot be constructed simply by concatenating the digit pairs. Instead, each term, starting with should be multiplied by 100, and the next term added (or, if negative, subtracted). This result should be multiplied by 100, and the next term added or subtracted, etc., until all terms are exhausted. In other words, we construct partial sums of the b terms:The last partial sum is the value for b.
Example
Find the reciprocal of πPi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
≈ 3.14159.
The result is 32,-17,10 or 31,83,10 yielding 0.318310.
External links
- Alternative Division Algorithms: Double Division, Partial Quotients & Column Division, Partial Quotients Movie