There are Infinite Natural Number between two positive numbers. Rational Numbers means the Numbers which are in the form of P/q. for eg- 13.4,13.5..14.5,14.7..you can take any number so there will be **Infinite numbers**.

∴ **4-9and16-36 represent the same rational number**.

Solution : Rational number is **a number that can be represented as the quotient of a/ b of two integers such as b ≠ 0**. Rational numbers include all integers each of them can be written as numerator and 1 as denominator.

There are Infinite Natural Number between two positive numbers. Rational Numbers means the Numbers which are in the form of P/q. for eg- 13.4,13.5..14.5,14.7..you can take any number so there will be **Infinite numbers**.

**The denominator = the number of equal parts that make one whole unit.** **The numerator is the number of parts you are counting.**

rational number, in arithmetic, a number that can be represented as **the quotient p/q of two integers such that q ≠ 0**. In addition to all the fractions, the set of rational numbers includes all the integers, each of which can be written as a quotient with the integer as the numerator and 1 as the denominator.

A rational number is **a number that can be expressed as a ratio of p/q, where p and q are integers, and q does not equal to zero**. The numerator and the denominator of a rational number will be integers. This means that 0 can also be a rational number as a rational number can be represented as 0/5 or 0/100 etc.

EXAMPLE 1 Show that **the terminating decimal 0.625 is rational** by writing it as the quotient of two integers. The next example shows how to express a nonterminating repeating decimal as the quotient of two integers.

To find the rational numbers between two rational numbers with different denominators, **the denominators should be equated**. Equating the denominators can be done either by finding their LCM or by multiplying the denominators of one to both the numerator and denominator of the other.

Starting with an introduction to real numbers, properties of real numbers, Euclid's division lemma, fundamentals of arithmetic, Euclid's division algorithm, revisiting irrational numbers, revisiting rational numbers and their decimal expansions followed by a bunch of problems for a thorough and better understanding.

**Types of numbers**

- Natural Numbers (N), (also called positive integers, counting numbers, or natural numbers); They are the numbers {1, 2, 3, 4, 5, …}
- Whole Numbers (W).
- Integers (Z).
- Rational numbers (Q).
- Real numbers (R), (also called measuring numbers or measurement numbers).

Real numbers include whole numbers, integers, rational numbers, and irrational numbers. Ordering real numbers is **the act of listing the numbers from smallest to largest**, with numbers that are farther to the left on the number line being considered smaller than numbers that are farther to the right.

Real numbers are, in fact, pretty much any number that you can think of. This can include whole numbers or integers, fractions, rational numbers and irrational numbers. **Real numbers can be positive or negative, and include the number zero**.

**Rational numbers are the numbers that can be expressed in the form of a ratio (i.e., P/Q and Q≠0) and irrational numbers cannot be expressed as a fraction**. But both the numbers are real numbers and can be represented in a number line.

infinite number

Addition: Two rational numbers can be added in any order, i.e., commutativity holds for rational numbers under addition, i.e., for any two rational number a and b, **a + b = b + a**. Hence, subtraction is not associative for rational numbers. (iii) Multiplication: Multiplication is commutative for rational numbers.

A number is **a mathematical object used to count, measure, and label**. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words.

**Types of Numbers**

- Natural Numbers.
- Whole Numbers.
- Integers.
- Rational Numbers.
- Irrational Numbers.
- Real numbers.
- Complex numbers.

Hence, by the Second Principle of Mathematical Induction, we conclude that **P(n) is true for all n∈N with n≥2**, and this means that each natural number greater than 1 is either a prime number or is a product of prime numbers.

Real numbers are **numbers that include both rational and irrational numbers**. Rational numbers such as integers (-2, 0, 1), fractions(1/2, 2.5) and irrational numbers such as √3, π(22/7), etc., are all real numbers.

Rational numbers are needed **because there are many quantities or measures that integers alone will not adequately describe**. Measurement of quantities, whether length, mass, time, or other, is the most common use of rational numbers.

Dated : 23-May-2022

Category : Education