Is the square root of 12 a rational number? This question often arises in mathematics, particularly when exploring the properties of square roots and rational numbers. To answer this question, we need to delve into the definitions of rational and irrational numbers, as well as the nature 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.
The square root of 12, denoted as √12, is an interesting case. To determine whether it is rational or irrational, we can try to express it as a fraction. Let’s assume that √12 is rational and can be written as p/q, where p and q are integers and q is not equal to zero.
By squaring both sides of the equation, we get:
(√12)^2 = (p/q)^2
12 = p^2/q^2
Now, we can multiply both sides of the equation by q^2 to eliminate the fraction:
12q^2 = p^2
At this point, we can observe that p^2 must be a multiple of 12, since it is equal to 12q^2. This implies that p must also be a multiple of 12, since the square of an integer is a multiple of that integer. Let’s denote p as 12k, where k is an integer.
Substituting p with 12k in the equation, we get:
12q^2 = (12k)^2
12q^2 = 144k^2
Now, we can divide both sides of the equation by 12 to simplify:
q^2 = 12k^2
This shows that q^2 is also a multiple of 12, which means that q must be a multiple of 12 as well. Let’s denote q as 12m, where m is an integer.
Substituting q with 12m in the equation, we get:
(12m)^2 = 12k^2
144m^2 = 12k^2
Dividing both sides of the equation by 12, we get:
12m^2 = k^2
This implies that k^2 is a multiple of 12, which means that k must also be a multiple of 12. However, this contradicts our initial assumption that p and q are integers with no common factors other than 1.
Since our assumption led to a contradiction, we can conclude that the square root of 12 is not a rational number. Instead, it is an irrational number, as it cannot be expressed as a fraction of two integers. This means that the decimal expansion of √12 will never terminate or repeat, making it an infinite, non-repeating decimal.
