Quotients {fraction} {simple fraction} {common fraction} {ordinary fraction} {ratio of terms} have top expression {numerator} {antecedent, ratio} divided by bottom expression {denominator, fraction} {consequent, ratio}. Term can divide another term: a/b, x/y, and a*x^b / c*y^d.
simplification
Simplify fraction by factoring same number, variable, coefficient, or polynomial from both numerator and denominator. Then cancel factor. After all common factors cancel, smallest possible denominator {least common denominator} remains, and numerator is simplest.
addition
To add or subtract two simple fractions, first make denominators equal by multiplying each-fraction top and bottom by other-fraction denominator. Then add resulting numerators. Simplify fraction by factoring out terms, to make smallest possible denominator.
multiplication
To multiply two fractions, multiply numerators and multiply denominators. Simplify resulting fraction.
division
To divide two fractions, multiply top-fraction numerator by bottom-fraction denominator to make numerator. Multiply top-fraction denominator by bottom-fraction numerator to make denominator. Simplify fraction by factoring.
rational numbers
Integers divided by integers make rational numbers.
Two fractions can have same denominator {common denominator}.
An oblique line {solidus, arithmetic} / denotes fraction.
Fractions {aliquot part} {unit fraction} can have the number one in numerators: 1/n.
Polynomials can divide into other polynomials {complex fraction}: (a*x + b) / (b*y + d). Alternatively, fractions can have numerator and/or denominator fractions: (3/4) / (7/8).
A fraction {continued fraction}| can have an integer plus a numerator-1 fraction, and that fraction can have a denominator with an integer plus a numerator-1 fraction, and so on: a + 1/(b + 1/(c + 1/(...))), where a, b, and c are integers. a + (b + (c + (...)^-1)^-1)^-1. Rational numbers can be terminating continued fractions. Quadratic irrational numbers can be non-terminating periodic continued fractions. Real numbers can be non-terminating non-periodic continued fractions.
Two fractions, a1/b1 and a2/b2, can make a fraction {mediant fraction} whose value is between the original fractions: (a1 + a2) / (b1 + b2).
Fractions {partial fraction} can have form A / (a*x + b)^n or (A*x + B) / (a*x^2 + b*x + c)^2, and so on. Proper fractions can be sums of partial fractions whose denominators are proper-fraction denominator factors: 16/12 = 1/2 + 1/3 + 1/2.
Numerator can be less than denominator {proper fraction}. Numerator can be greater than denominator {improper fraction}.
Fractions {similar fraction} can have same denominator.
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Date Modified: 2022.0225