complex fraction
(noun)
A ratio in which the numerator, denominator, or both are themselves fractions.
Examples of complex fraction in the following topics:
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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:
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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:
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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.
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Fractions
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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.
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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:$\displaystyle \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = {\frac{10\cdot\sqrt{3}}{{\sqrt{3}}^2}} = {\frac{10\sqrt{3}}{3}}$
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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.
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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.
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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:
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Roots of Complex Numbers