Assume Rational Number: Assume that 5 is a rational number. According to the definition of rational numbers, if 5 is rational, it can be expressed as a fractionba, where a and b are integers with no common factors other than 1, and b is not zero.
Square of 5: Since 5=ba, then 5=(ba)2=b2a2. This implies that 5b2=a2.
Express a as 5k: The equation 5b2=a2 indicates that a2 is a multiple of 5. Therefore, a must also be a multiple of 5, because the square of a non-multiple of 5 cannot be a multiple of 5. Let's write a as 5k, where 5k1 is an integer.
Substitute a in equation: Substitute a with 5k in the equation 5b2=a2 to get 5b2=(5k)2. This simplifies to 5b2=25k2, and further to b2=5k2.
b is a multiple of 5: The equation b2=5k2 shows that b2 is also a multiple of 5, and hence b must be a multiple of 5 as well.
Contradiction Found: Since both a and b are multiples of 5, this contradicts our initial assumption that a and b have no common factors other than 1. Therefore, our initial assumption that 5 is rational is incorrect.
Final Conclusion: We have reached a contradiction, which means our initial assumption is false. Hence, 5 cannot be expressed as a fraction of two integers with no common factors other than 1. Therefore, 5 is irrational.
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