Properties of Rational Numbers

Key Questions

• They can be written as a result of a division between two whole numbers, however large.

Example: 1/7 is a rational number. It gives the ratio between 1 and 7. It could be price for one kiwi-fruit if you buy 7 for $1. In decimal notation, rational numbers are often recognised because their decimals repeat. 1/3 comes back as 0.333333.... and 1/7 as 0.142857... ever repeating. Even 553/311 is a rational number (the repeating cylce is a bit longer) There are also IRrational numbers that cannot be written as a division. Their decimals follow no regular pattern. Pi is the best-known example, but even the square root of 2 is irrational. • Answer: Different types of fractions are: 1. Proper Fractions 2. Improper Fractions 3.. Mixed Fractions Explanation: 1. Proper Fractions are the one in which there is a smaller no. in the numerator and a larger no. in the denominator. 2. Improper Fractions are fractions in which there is a larger no. in the numerator and a smaller no. in the denominator. 3. Mixed Fractions are fractions in which there is whole no. followed by a fraction which can either be proper or improper. • Let's assume that we know what integer numbers are and what is an operation of their multiplication, so we know how to multiply any integer number by any other. Considerations of symmetry and harmony lead us to a desire to reverse the operation of multiplication, that is to be able to divide any integer number by any other. Obviously, it's not always possible within the realm of integer numbers. Operation of multiplication is not really complete in the space of only integer numbers since its reverse, division, is not possible for some integer numbers. For instance, we can multiply $3$by $7$getting $21$and we can divide $21$by $7$getting $3$, but we cannot divide $21$by $6$within a set of integer numbers. Rational numbers are completely new entities that allow us to divide any integer number by any other (not equal to $0$). So, by definition, a rational number is a set of two integer numbers, the first, usually called numerator, and the second (not equal to $0$), usually called denominator, that has one important property: if multiplied by a denominator, result is a numerator. Traditionally, if a numerator is $M$and a denominator is $N$, the rational number formed by them is written as $\frac{M}{N}$with property defined for it: $\frac{M}{N} \cdot N = M$. Introduction of rational numbers completes the operation of multiplication by enabling its reverse in a broader set of numbers. Now we can divide $21$by $6$using rational numbers and the result, by definition of rational numbers , is $\frac{21}{6}\$.

The harmony has been restored by expanding the concept of numbers from integer to rational.

Obviously, we have to prove the correctness of our definition, that certain properties of operations of addition and multiplications of integer numbers are preserved within a set of rational numbers, but this is a different topic.