Suppose we want to prove the a math statement is true. Merely put, us assume that the mathematics statement is false and also then present that this will bring about a contradiction. If it leader to a contradiction, then the statement have to be true




You are watching: Is the square root of 5 rational

If we are going come assume that √5is rational, climate we need to know what it way for a number to it is in rational.
Basically, if square root of 5 is rational, it have the right to be created as the proportion of 2 numbers as shown below:
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We get 5 × y2 = x2

In order to prove that square root of 5 is irrational, you should understand also this vital concept.

Another essential concept prior to we complete our proof: prime factorization Key question: is the variety of prime components for a number raised to the 2nd power an also or strange number?For example, 62, 122, and 15262 = 6 × 6 = 2 × 3 × 2 × 3 (4 element factors, so even number)122 = 12 × 12 = 4 × 3 × 4 × 3 = 2 × 2 × 3 × 2 × 2 × 3 (6 element factors, so even number)152 = 15 × 15 = 3 × 5 × 3 × 5 = (4 prime factors, so also number)There is a heavy pattern below to break up that any number squared will have an even variety of prime factors.In stimulate words, x2 has an even number of prime factors.Let\"s complete the proof then!5 × y2 = x2Since 5 × y2 is same to x2, 5 × y2 and also x2 must have the same number of prime factors.We just showed thatx2 has an even variety of prime factors.y2 has additionally an even number of prime factors.5 × y2 will then have an odd variety of prime factors.The number 5 counts as 1 element factor, therefore 1 + one even variety of prime factors is an odd variety of prime factors.5 × y2 is the same number as x2. However, 5 × y2 gives an odd number of prime element while x2 offers an even variety of prime factors.
This is a contradiction due to the fact that a number cannot have actually an odd number of prime factors and also an even variety of prime components at the exact same timeThe assumption that square root of 5 is rational is wrong. Therefore, square of 5 is irrational


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