Is sqrt 2 a rational number true or false? This question has intrigued mathematicians for centuries and remains a topic of debate even today. The concept of rational and irrational numbers is fundamental to mathematics, and the nature of sqrt 2 falls right at the heart of this debate.
Rational numbers are those that can be expressed as a fraction of two integers, where the denominator is not zero. For example, 1/2, 3/4, and -5 are all rational numbers. On the other hand, irrational numbers cannot be expressed as fractions and have non-terminating, non-repeating decimal expansions. Examples of irrational numbers include sqrt(2), pi, and the golden ratio.
The question of whether sqrt 2 is rational or irrational was first posed by the ancient Greek mathematician Hippasus of Metapontum. According to legend, Hippasus was sentenced to death by drowning for revealing this mathematical truth, which was considered heretical at the time. This story highlights the importance of the question and the significance of the answer.
To determine whether sqrt 2 is rational or irrational, we can use a proof by contradiction. Assume that sqrt 2 is rational, which means it can be expressed as a fraction of two integers, a/b, where a and b are coprime (i.e., they have no common factors other than 1). Squaring both sides of this equation, we get:
(sqrt 2)^2 = (a/b)^2
2 = a^2/b^2
This implies that a^2 is even, which means a must also be even. Let a = 2c, where c is an integer. Substituting this into the equation, we have:
2 = (2c)^2/b^2
2 = 4c^2/b^2
Dividing both sides by 2, we get:
1 = 2c^2/b^2
This shows that b^2 is also even, which means b must be even. However, this contradicts our initial assumption that a and b are coprime. Therefore, our assumption that sqrt 2 is rational must be false. Hence, sqrt 2 is an irrational number.
The proof that sqrt 2 is irrational is a classic example of a proof by contradiction and has been used to demonstrate the irrationality of other numbers as well. It is a testament to the power of mathematical reasoning and the beauty of the subject. So, to answer the question, “Is sqrt 2 a rational number true or false?” the answer is a resounding false.
