We know that for two integers m and n, we get the sum m + n, product of m x n, and difference of m - n which are always integers. However it may not always be possible for a given integer to exactly divide another given integer. In other words, the result of division of an integer by a non-zero integer may or may not be an integer. In fact, 9/4 is a fraction. Thus, there is a need to extend the system of integers so that it may also be possible to divide any integer by any other given integer different from zero (because division by zero is not possible).
So to cover this flaw in the number system, rational numbers were invented.
A rational (ratio) number is a number that can be written as p/q, where p and q are integers and q is not equal to 0.
Some examples or rational numbers are -5=-5/1, 3=3/1, 0.73=73/100, and 0.333...=1/3.
Rational numbers include decimals that can either be terminating such as 0.73 or repeating 0.333... since they can be written as 73/100 and 1/3, respectively.
Examples:
-2 1/5=-11/5=-2.2 -- (terminating decimal)
3/7=0.4285714285714... -- (repeating decimal)
Note:Since the pattern from digits 4 to 1 is repeating, we place a bar to those digits that are repeated or sometimes dots above the first and last digits of the repeating pattern.
MATH FOCUS
|
To convert pure repeating decimals into rational numbers, we can follow these steps:
Examples: 0.1...=1/9; 0.17...=17/99; 0.492...=492/999 |
MATH FOCUS
|
Rule to express mixed repeating decimals into rational numbers: Step 1: Write the (mixed repeating decimal) - (nonrepeating number) as numerator. Step 2: Put in the denominator as many nines as in the number of nonrepeating digits. Examples: 0.2|1...=(21-2)/90=19/90 |
Just like integers, rational numbers can also be illustrated on a number line. The negative rational numbers are represented on the left side of zero and the positive rational numbers on the right side of zero.
THe multiplicative inversse or reciprocal numbers is obtained by interchanging the numerator and the denominator of the rational number.
Therefore, the multiplicative inverse of p/q is q/p where p is not equal to 0, q is not equal to 0.
Not all numbers on a number line can be written in the form p/q. For example, √2, √3, √5, etc. These numbers are called irrationals or irrational numbers.
The sets of rational and irrational numbers make up the set of real numbers. These can be represented on a number line, that is, every point on teh number line corresponding to a eal number, we are plotting the real number.
One can see that there are other numbers in between any two integers which can be represented as the quotient of two integers.
On the number line, the length between two integers is divided into 5 equal parts.