what is an irrational number and a rational number?
An Irrational Number is a real number that cannot be written as a simple fraction
A rational number is a number that can be written as a simple fraction
any rational number can be written as a terminating decimal lor a repeating decimal
1/5 = 0.2000000
17/8 = 2.12500000
1/3 = 0.33333333...
9/7 = 1.28357142835714...
anything else is irrational
1.0203040506070809010011012... is irrational
pi = 3.14159265... is irrational
√2 = 1.414213562... is irrational
An irrational number is a number that cannot be expressed as a simple fraction (or ratio) of two integers. These numbers go on indefinitely without repeating any pattern. Examples of irrational numbers include √2, π (pi), and e (Euler's number).
On the other hand, a rational number is any number that can be expressed as a fraction (or ratio) of two integers, where the denominator is not zero. This includes integers, fractions, and terminating or repeating decimals. For example, 2, -5, 1/3, and 0.125 are all rational numbers.
To determine whether a number is rational or irrational, you can follow these steps:
1. Check if the number can be expressed as a simple fraction. If it can be written as a ratio of two integers, then it is rational. For example, 4 can be expressed as 4/1 and is therefore rational.
2. If the number cannot be expressed as a fraction, it could potentially be irrational. Irrational numbers are often identified through their decimal representation. If the decimal representation goes on forever without repeating or showing any pattern, then the number is irrational. For example, √2 ≈ 1.41421356... (it continues indefinitely) and π (pi) ≈ 3.14159265... (it never repeats or shows a pattern), so both are irrational numbers.
In summary, irrational numbers cannot be expressed as fractions and have non-repeating decimal representations, while rational numbers can be expressed as fractions and have terminating or repeating decimals.