A rational number expressed in the form 
(in-line notation) or 
(traditional "display" notation), where a is called the numerator and b is called the denominator. When written in-line, the slash "/" between numerator and denominator is called a solidus.
A proper fraction is a fraction such that 
,
The Egyptians expressed their fractions as sums (and differences) of unit fractions. Conway and Guy (1996) give a table of Roman notation for fractions, in which multiples of 1/12 (the uncia) were given separate names.
The rules for the algebraic combination of fractions are given by
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(1) |
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(2) |
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(3) |
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(4) |
Note however that the above results will not necessarily be in lowest terms.
Adjacent Fraction, Anomalous Cancellation, Common Fraction, Complex Fraction, Continued Fraction, Denominator, Egyptian Fraction, Farey Sequence, Golden Rule, Half, Improper Fraction, Lowest Terms Fraction, Matrix Fraction, Mediant, Mixed Fraction, Numerator, Pandigital Fraction, Proper Fraction, Pythagorean Fraction, Quarter, Rational Number, Solidus, Unit Fraction
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 22-23, 1996.
Courant, R. and Robbins, H. "Decimal Fractions. Infinite Decimals." §2.2.2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 61-63, 1996.