What Are Consecutive Fractions?

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A consecutive fraction is a number written as a series of alternating multiplicative inverses and integer addition operators. Consecutive fractions are studied in the number theory branch of mathematics. Consecutive fractions are also known as continued fractions and extended fractions.

Consecutive Fractions

Consecutive fractions are any number written in the form a(0) + 1/(a(1) + 1/(a(2) + ...))) where a(0), a(1), a(2) and so on are integer constants. The consecutive fraction can continue indefinitely or finitely. Any real number can be written as a finite or infinite consecutive fraction.

Rational Numbers

Rational numbers can be written in the form p/q where p and q are both integers. Rational numbers are one of the two categories of real numbers. Any rational number can be written as a finite consecutive fraction in the form a(0) + 1/(a(1) + 1/(a(2) + ... 1/a(n))) where a(0), a(1) ... a(n) are integer constants as well.

Irrational Numbers

Irrational numbers cannot be written in the form p/q where "p" and "q" are two integers. Common irrational numbers include the √2, pi and e. Irrational numbers cannot be written as finite consecutive fractions, but they can be written as infinite consecutive fractions.

Calculating Finite Consecutive Fractions

To calculate the value of a finite consecutive fraction in the form a(0) + 1/(a(1) + 1/(a(2) + ...1/a(n))), where a(0), a(1) ... a(n) are integers, start from the bottom of the fraction. Solve 1/a(n), add a(n-1), divide 1 by this number and repeat until you solve the fraction. For example, consider 1 + 1/(2 + 1/(3 + 1/4)) = 1 + 1/(2 + 1/(13/4)) = 1 + 1/(2 + 4/13) = 1 + 1/(30/13) = 1 + (13/30) = 43/30.

References

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