Is root 7 a rational number? This question has intrigued mathematicians and enthusiasts alike for centuries. The concept of rational and irrational numbers is fundamental to mathematics, and understanding whether the square root of 7 falls into one of these categories can shed light on the nature of numbers and their properties.
Rational numbers are those that can be expressed as a fraction of two integers, where the denominator is not zero. They include all integers, fractions, and terminating or repeating decimals. On the other hand, irrational numbers cannot be expressed as a fraction of two integers and have non-terminating, non-repeating decimal expansions. Examples of irrational numbers include the square root of 2, pi, and the natural logarithm of 2.
To determine whether the square root of 7 is rational or irrational, we can use a proof by contradiction. Assume that the square root of 7 is a rational number. This means that we can express it as a fraction of two integers, say a/b, where a and b are coprime (i.e., they have no common factors other than 1). Therefore, we have:
√7 = a/b
Squaring both sides of the equation, we get:
7 = a^2/b^2
Multiplying both sides by b^2, we obtain:
7b^2 = a^2
Now, since 7 is a prime number, it must be a factor of either a or b. Without loss of generality, let’s assume that 7 is a factor of a. This means that a can be written as 7c, where c is an integer. Substituting this into the equation, we have:
7b^2 = (7c)^2
Simplifying, we get:
7b^2 = 49c^2
Dividing both sides by 7, we obtain:
b^2 = 7c^2
This implies that 7 is also a factor of b^2, and therefore, 7 is a factor of b. However, this contradicts our initial assumption that a and b are coprime. Hence, our assumption that the square root of 7 is a rational number must be false.
Therefore, we can conclude that the square root of 7 is an irrational number. This discovery is significant because it demonstrates that not all square roots of integers are rational, expanding the realm of irrational numbers and highlighting the complexity of numbers in mathematics.
