Is root 13 a rational number? This question often puzzles many students of mathematics. To understand the answer, we must delve into the concepts of rational and irrational numbers, as well as the properties of square roots.
Rational numbers are those that can be expressed as a fraction of two integers, where the denominator is not zero. In other words, a rational number can be written in the form of p/q, where p and q are integers and q is not equal to zero. On the other hand, irrational numbers cannot be expressed as fractions and have decimal expansions that never terminate or repeat.
Now, let’s examine the square root of 13. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. However, not all numbers have a rational square root.
To determine whether the square root of 13 is rational, we can try to express it as a fraction of two integers. Suppose we have a fraction p/q, where p and q are integers and q is not equal to zero. Squaring both sides of the equation, we get:
(p/q)^2 = 13
This simplifies to:
p^2 = 13q^2
Now, since p^2 and 13q^2 are both integers, we can conclude that 13 must be a perfect square. However, 13 is not a perfect square, as it cannot be expressed as the product of two integers. Therefore, the square root of 13 cannot be expressed as a fraction of two integers, and it is an irrational number.
In conclusion, the square root of 13 is not a rational number. This is because it cannot be expressed as a fraction of two integers, and its decimal expansion is non-terminating and non-repeating. Understanding the properties of rational and irrational numbers is crucial in mathematics, as they play a significant role in various fields, such as algebra, geometry, and calculus.
