Rational numbers are numbers that include the entire set of numbers, excluding the imaginary numbers. Negative numbers are also rational numbers. In general, rational numbers are in the fractional form of m/n, where n is never equal to zero. Note that all rational numbers are not fractions, but all fractions are real numbers. This is because fractions require both the numerator and denominator to be positive.
Irrational numbers are the counterpart of rational numbers. They cannot be expressed in rational or fractional forms. A decimal is used to represent an irrational number since it cannot be expressed as a simple fraction. After the decimal point, it has an infinite number of non-repeating digits. The Pi (π) = 3.142857…, the Euler’s Number (e) = 2.7182818284590452……. are some examples of irrational numbers.
Because a rational number is a subset of the real number, it will abide by all of the real number system's characteristics. The following are some of the most important features of rational numbers:
The closure property of rational numbers states that if two rational numbers are summed, subtracted, or multiplied, the result yields a rational number. Hence, we can deduce that rational numbers are bound or closed by addition, multiplication, and subtraction. However, division does not fall in this category as the rational numbers on dividing with zero, results in unimaginable numbers. Therefore, we can tell that except zero, all the other rational numbers are enclosed by division.
Commutative law or property states that for any two numbers, the sum of the first and second number is equal to the sum of the second and first numbers in case of addition. The multiplicative law states that the product of the first and second numbers is equal to the product of the second and first numbers. Commutative law of addition: p + q = q + p Commutative law of multiplication: p × q = q × p The commutative law is not true for subtraction and division, as with the change in orientation of numbers, the result changes.
The associative property for rational numbers is also true for only addition and multiplication. For example, let us assume that there are three rational numbers k, l, and m. Therefore, the associative property will be: Addition: k + (l + m) = (k + l) + m Multiplication: k (lm)=(kl)m
The distributive property for three rational numbers k, l, and m is given as follows: k x (l + m) = (k x l) + (k x m)
Identity Property: Identity property is the addition or multiplication of numbers to rational numbers, which does not alter the value and results in the original number. The additive identity is 0, and the multiplicative identity is 1 for all rational numbers. Examples:
Inverse Property: For any rational number m/n, the inverse while the addition is -m/n and multiplication is the reciprocal of the number n/m. Examples: The additive inverse of 7/8 is -7/8. Therefore, 7/8 + (-7/8) = 0 The multiplicative inverse of 7/8 is 8/7. Therefore, 7/8 x 8/7 = 1