Fractional Numbers
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A fractional number is a decimal number that is expressed as a ratio of two Integers, where the numerator is the dividend and the denominator is the divisor.
Definitions
Definition 1: Algebraic Representation
A fraction can be algebraically represented as \(\frac{p}{q}\), where \(p\) and \(q\) are Integers and \(q\) is non-zero. The value of the fraction is then calculated by dividing the numerator \(p\) by the denominator \(q\). For example:
\(\frac{6}{8} = \frac{3}{4}\)
Definition 2: Decimal Representation
A fractional number can also be represented as a decimal, which is a rational number that has an infinite number of digits. The Decimal Representation of a fraction can be obtained by dividing the numerator by the denominator.
Characteristics of Fractional Numbers
- Ratios: Fractional numbers are Ratios of two Integers.
- Decimals: Fractional numbers can also be represented as Decimals.
- Negative and Positive Divisors: A negative fractional number can be expressed as a positive integer divided by its absolute value, while a positive fractional number can be expressed as a negative integer divided by its absolute value.
Types of Fractions
1. Positive Fractions
A positive fraction is a ratio of two Integers where both the numerator and denominator are positive.
\(\frac{3}{4}\)
2. Negative Fractions
A negative fraction is a ratio of two Integers where both the numerator and denominator are negative.
\(-\frac{1}{2}\)
Operations on Fractional Numbers
Addition: To add two Fractions, the denominators must be the same. \(\frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1\)
Subtraction: To subtract two Fractions, the denominators must be the same. \(\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\)
Multiplication: To multiply two Fractions, the denominators must be the same and the numerators must be multiplied together. \(\frac{3}{4} \times \frac{2}{2} = \frac{6}{8}\)
Example Use Case: Mixed Fractions in Decimals
Let’s consider a mixed fraction of two different decimal numbers:
\(0.\overline{5}\) and \(1.25\)
To convert the mixed Fractions to improper Fractions, we can use Algebraic Representation: \(\overline{5} = \frac{5}{9}\) \(1.25 = \frac{5}{4}\)
Real-World Applications of Fractional Numbers
- Mathematics: Fractional numbers are used extensively in mathematics to solve equations and problems involving Fractions.
- Science: Fractional numbers are used to describe quantities such as velocities, speeds, and distances.
- Engineering: Fractional numbers are used in engineering to design and calculate structures, machines, and mechanisms.
Converting Between Fractions and Decimals
To convert a fraction to a decimal, divide the numerator by the denominator:
\(\frac{3}{4} = 0.75\)
Alternatively, you can use long division or a calculator to convert Fractions to Decimals.
Similarly, to convert a decimal to a fraction, multiply it by an appropriate power of 10 and then express the result as a fraction in simplest form.
\(\begin{aligned} 2.\overline{5} &= \frac{2.55}{10}\\ 1.25 &= \frac{5}{4}\\ \end{aligned}\)