What is rational and irrational?
The terms "rational" and "irrational" are used to describe different types of numbers.
1. Rational Numbers: A rational number is any number that can be expressed as a fraction, where both the numerator and the denominator are integers (whole numbers) and the denominator is not zero. In other words, a rational number can be expressed in the form p/q, where p and q are integers, and q is not equal to zero. For example, 1/4, -3/5, and 7/1 are all rational numbers. Rational numbers can also be positive, negative, or zero.
2. Irrational Numbers: An irrational number is any number that cannot be expressed as a fraction, or it cannot be written as a terminating decimal or a repeating decimal. These numbers have decimal representations that go on forever without repeating any pattern. The decimal representation of irrational numbers is always non-terminating and non-repeating. Examples of irrational numbers include π (pi), √2 (square root of 2), and e (Euler's number).
Now, let's see how you can determine whether a number is rational or irrational:
1. Rational Numbers:
- If you are given a fraction, where both the numerator and denominator are integers, then it is a rational number.
- For example, 3/4 is a rational number because both 3 and 4 are integers.
2. Irrational Numbers:
- If you are given a non-repeating decimal that goes on forever, it is likely an irrational number.
- For example, if you have the decimal representation of a number like 1.73205080757..., it represents the square root of 3 (√3), which is an irrational number.
To determine whether a number is rational or irrational, you need to analyze its properties and its representation.
rational numbers can be represented as fractions of integers
3 is rational, 15/5 is a integer fraction that represents it.
irrational numbers cannot be written as a whole integer fraction, such as sqrt5
Another difference, as a decimal, rational numbers repeat, such as
1.666666666... is a rational number
2.456456456456... is rational
Irrational numbers in decimal form never repeat digits.