Root 4 is irrational!

Let us prove that 4\sqrt 4 is irrational:

Let us assume that 4\sqrt 4 is rational, then it can be expressed in the form of pq\frac pq where p,qp,q have no factors in common, thus pq=4\frac pq = \sqrt 4. or p2/q2=4\color{#D61F06}{p^2 /q^2 = 4} then p2=4q2p^2 = 4q^2 , with p has a factor of 4\color{#3D99F6} {\text{with } p \text{ has a factor of 4}} . Let p=4mp = 4m , then (4m)2=4q216m2=4q2,4m2=q2\color{#20A900}{(4m)^2 = 4q^2 } \rightarrow 16m^2 = 4q^2, 4m^2 = q^2 , then q also has a factor of 4 \color{#EC7300}{q \text{ also has a factor of 4}} . But this contradicts the fact that p,qp,q are coprimes. Thus the assumption that 4\sqrt 4 is rational is wrong. Hence, 4\sqrt 4 is also irrational.

This concludes that 4=2\sqrt 4 = 2 is an irrational number.

Which of these above colored equations are not necessarily correct?

"1" represents red\color{#D61F06}{red}, "2" represents blue\color{#3D99F6}{blue}, "3" represents green\color{#20A900}{green}, "4" represents orange\color{#EC7300}{orange}.


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