complex fraction

(noun)

A ratio in which the numerator, denominator, or both are themselves fractions.

Examples of complex fraction in the following topics:

  • Complex Fractions

    • A complex fraction is one in which the numerator, denominator, or both are fractions, which can contain variables, constants, or both.
    • A complex fraction, also called a complex rational expression, is one in which the numerator, denominator, or both are fractions.
    • When dealing with equations that involve complex fractions, it is useful to simplify the complex fraction before solving the equation.
    • The process of simplifying complex fractions, known as the "combine-divide method," is as follows:
    • Let's apply this method to the first complex fraction presented above:

  • Partial Fractions

    • In algebra, partial fraction decomposition (sometimes called partial fraction expansion) is a procedure used to reduce the degree of either the numerator or the denominator of a rational function.
    • Reducing complex mathematical problems via partial fraction decomposition allows us to focus on computing each single element of the decomposition rather than the more complex rational function.
    • We can then write $R(x)$ as the sum of partial fractions:
    • We have rewritten the initial rational function in terms of partial fractions.
    • We have solved for each constant and have our partial fraction expansion:

  • The Quadratic Formula

    • The quadratic formula can always be used to find the roots of a quadratic equation, regardless of whether the roots are real or complex, whole numbers or fractions, and so on.

  • Fractions

  • Fractions

    • A fraction represents a part of a whole.
    • Find a common denominator, and change each fraction to an equivalent fraction using that common denominator.
    • To subtract a fraction from a whole number or to subtract a whole number from a fraction, rewrite the whole number as a fraction and then follow the above process for subtracting fractions.
    • To multiply a fraction by a whole number, simply multiply that number by the numerator of the fraction:
    • The process for dividing a number by a fraction entails multiplying the number by the fraction's reciprocal.

  • Fractions Involving Radicals

    • Root rationalization is a process by which any roots in the denominator of an irrational fraction are eliminated.
    • In mathematics, we are often given terms in the form of fractions with radicals in the numerator and/or denominator.
    • This same principal can be applied to fractions: whatever we do to the numerator, we must also do to the denominator, and vice versa.
    • You are given the fraction $\frac{10}{\sqrt{3}}$, and you want to simplify it by eliminating the radical from the denominator.
    • Therefore, multiply the top and bottom of the fraction by $\frac{\sqrt{3}}{\sqrt{3}}$, and watch how the radical expression disappears from the denominator:$\displaystyle \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = {\frac{10\cdot\sqrt{3}}{{\sqrt{3}}^2}} = {\frac{10\sqrt{3}}{3}}$

  • Rational Algebraic Expressions

    • Then you rewrite the two fractions with this denominator.
    • The denominator in the second fraction can not be factored.
    • The first fraction has two factors: $y$ and $(x^2+2)$.
    • The second fraction has one factor: $(x^2 + 2)$.
    • We then rewrite both fractions with the common denominator.

  • Simplifying, Multiplying, and Dividing

    • A rational expression can be treated like a fraction, and can be manipulated via multiplication and division.
    • A rational expression is a fraction involving polynomials, where the polynomial in the denominator is not zero.
    • Just like a fraction involving numbers, a rational expression can be simplified, multiplied, and divided.
    • Rational expressions can be multiplied and divided in a similar manner to fractions.
    • Dividing rational expressions follows the same rules as dividing fractions.

  • Negative Exponents

    • Numbers with negative exponents are treated normally in arithmetic operations and can be rewritten as fractions.
    • There is an additional rule that allows us to change the negative exponent to a positive one in the denominator of a fraction, and it holds true for any real numbers $n$ and $b$, where $b \neq 0$:
    • To understand how this rule is derived, consider the following fraction:

  • Roots of Complex Numbers