Suppose we want to prove the a math statement is true. Simply put, us assume the the math statement is false and also then show that this will result in a contradiction. If it leader to a contradiction, climate the statement need to be true
You are watching: Is square root of 5 a rational number
If we space going come assume that √5is rational, climate we require to recognize what it method for a number to be rational.
Basically, if square root of 5 is rational, it deserve to be composed as the proportion of 2 numbers as presented below:
We get 5 × y2 = x2
In order to prove the square source of 5 is irrational, you have to understand also this important concept.Another crucial concept before we finish our proof: prime factorization Key question: is the variety of prime components for a number increased to the second power an even or strange number?For example, 62, 122, and 15262 = 6 × 6 = 2 × 3 × 2 × 3 (4 element factors, so also number)122 = 12 × 12 = 4 × 3 × 4 × 3 = 2 × 2 × 3 × 2 × 2 × 3 (6 element factors, so also number)152 = 15 × 15 = 3 × 5 × 3 × 5 = (4 prime factors, so also number)There is a hard pattern right here to break up that any number squared will have actually an even variety of prime factors.In order words, x2 has an even number of prime factors.Let"s end up the evidence then!5 × y2 = x2Since 5 × y2 is equal to x2, 5 × y2 and also x2 must have actually the same number of prime factors.We just showed thatx2 has actually an even number of prime factors.y2 has also an even number of prime factors.5 × y2 will then have an odd number of prime factors.The number 5 counts together 1 element factor, for this reason 1 + an even variety of prime factors is an odd number of prime factors.5 × y2 is the same number as x2. However, 5 × y2 gives an odd variety of prime aspect while x2 gives an even variety of prime factors.
This is a contradiction due to the fact that a number cannot have an odd variety of prime factors and also an even variety of prime determinants at the same timeThe assumption that square source of 5 is reasonable is wrong. Therefore, square the 5 is irrational
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