Rational Numbers: Definition and Examples

Rational numbers are a subset of the real numbers that can be expressed as the ratio of two integers, where the denominator is not zero. In other words, any number that can be written in the form of p/q, where p and q are integers and q is not zero, is a rational number. The word "rational" comes from the Latin word "ratio," which means a comparison or division.

Examples of Rational Numbers:
1. 2/3: The fraction 2/3 is a rational number since it is the ratio of two integers (2 and 3) and the denominator is not zero.
2. -5: Whole numbers can also be expressed as fractions, so -5 can be written as -5/1. Thus, -5 is a rational number as well.
3. 0.8: Decimal numbers can be represented as fractions too. 0.8 is equivalent to 8/10, which can be further simplified to 4/5. Hence, 0.8 is a rational number.

Standard Form of Rational Numbers:

The standard form of a rational number is the representation of a fraction in its simplest or most reduced form. It means that the numerator and denominator have no common factors other than 1. This form is also known as the simplest form or the lowest terms form.

To express a rational number in standard form, follow these steps:

1. Find the Greatest Common Divisor (GCD): Determine the largest number that divides both the numerator and denominator evenly. This is the GCD.
2. Divide Both Numerator and Denominator by the GCD: Divide both the numerator and denominator by the GCD obtained in step 1.
3. Simplify the Fraction: Write down the resulting quotient of the division as the new numerator and denominator. If the fraction can be simplified further, repeat steps 1 and 2 until it is in its simplest form.

For example, let's simplify the fraction 24/36 to its standard form:

1. Find the GCD: The GCD of 24 and 36 is 12.
2. Divide both numerator and denominator by 12: 24/12 = 2 and 36/12 = 3.
3. Simplify the fraction: The simplified fraction is 2/3, which is in standard form.

By expressing rational numbers in standard form, it becomes easier to compare and perform arithmetic operations with them.

In conclusion, rational numbers are numbers that can be represented as the ratio of two integers, where the denominator is not zero. They can be expressed in standard form, which is the simplest or most reduced form of the fraction. Simplifying fractions to their standard form allows for easier calculations and comparisons.