Rational Numbers

 Rational Number :-

A rational number is a number that can be expressed as the ratio of two integers, where the denominator is not zero. In other words, a rational number is any number that can be written in the form of p/q, where p and q are integers and q ≠ 0.

Examples of rational numbers include:

• 1/2
• 3/4
• 7/9
• -4/5
• 0 (which can be expressed as 0/1)

Rational numbers can be either positive, negative, or zero. They can also be expressed as decimals, either terminating (when the decimal digits end after a finite number of terms) or repeating (when the same sequence of digits repeats indefinitely).

For example, 1/3 can be expressed as the repeating decimal 0.333..., and 7/12 can be expressed as the terminating decimal 0.583333....

• Rational numbers are closed under addition, subtraction, multiplication, and division. This means that if you add, subtract, multiply, or divide two rational numbers, the result will always be another rational number.
• Any integer can be expressed as a rational number by setting the denominator to 1. For example, the integer 5 can be written as 5/1.
• Any terminating decimal can be expressed as a rational number by converting the decimal to a fraction. For example, the decimal 0.75 can be expressed as the fraction 3/4.
• Any repeating decimal can also be expressed as a rational number. To do this, you can use the fact that a repeating decimal can be represented as an infinite series of digits. For example, the repeating decimal 0.333... can be expressed as the fraction 1/3, since 0.333... = 3/10 + 3/100 + 3/1000 + ... = 1/3.
• Rational numbers have a well-defined order relation. That is, given any two rational numbers, you can always determine which one is greater than the other, or if they are equal. This property is useful in many applications, such as comparing prices, measuring quantities, or ranking data.
• The set of rational numbers is countable, which means that it can be put into a one-to-one correspondence with the set of natural numbers (i.e., the positive integers). This is because you can list all the rational numbers in a systematic way, such as by arranging them in order of increasing magnitude.

There are several formulas used in mathematics that involve rational numbers. Here are some of the most common ones:

1. Addition and subtraction: The formula for adding or subtracting two rational numbers is simply to combine the numerators over a common denominator. For example:

   (a/b) + (c/d) = (ad + bc) / bd

   (a/b) - (c/d) = (ad - bc) / bd

1. Multiplication: The formula for multiplying two rational numbers is to multiply the numerators and denominators separately, and then simplify the result by reducing the fraction. For example:

   (a/b) x (c/d) = (ac) / (bd)

1. Division: The formula for dividing two rational numbers is to invert the second fraction and then multiply the first fraction by the inverse. For example:

   (a/b) ÷ (c/d) = (a/b) x (d/c) = (ad) / (bc)

1. GCD and LCM: The greatest common divisor (GCD) and least common multiple (LCM) of two rational numbers can be found by first finding the GCD and LCM of the two denominators, and then simplifying the fractions accordingly.

1. Converting between decimals and fractions: To convert a decimal to a fraction, write the decimal as a ratio with a power of 10 in the denominator. Then simplify the fraction. For example:

   0.75 = 75/100 = 3/4

   To convert a fraction to a decimal, divide the numerator by the denominator. If the result is a terminating decimal, write it as a decimal. If the result is a repeating decimal, use the formula for converting repeating decimals to fractions.