Show that each of the following numbers is a rational number...

Show that each of the following numbers is a rational number by expressing it as a fraction with an integer numerator and a nonzero integer denominator: $6,-9,-\frac{7}{8}, 3 \frac{1}{2},-0.3,2.83.$

Key Concepts

Conversion of Decimals to Fractions

Decimals can be converted to fractions by using the place value system. The process involves writing the decimal as a fraction whose denominator is a power of ten, and then simplifying the fraction if possible. This conversion technique further illustrates the idea that decimals represent rational numbers when they either terminate or repeat.

Rational Numbers

A rational number is any number that can be expressed in the form a/b, where a and b are integers and b is not zero. This concept is fundamental in number theory and arithmetic, distinguishing numbers that can be written as a fraction from those that cannot, such as irrational numbers.

Fraction Representation

Expressing a number as a fraction involves writing it as a ratio of two integers. This method is a standard technique for demonstrating that a number is rational, as it clearly shows the numerator and denominator which satisfy the definition of a rational number.

Integer Representation as Fractions

Any integer can be expressed as a fraction by placing it over one. This is an important concept when proving that integers are rational because writing an integer n as n/1 maintains its value while fitting the required form of a rational number.

Conversion of Mixed Numbers to Improper Fractions

A mixed number, which combines an integer with a fraction, can be converted to an improper fraction by multiplying the whole number by the denominator of the fractional part and then adding the numerator. The result is a single fraction that clearly falls into the rational number form.

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