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$\log_2 (3) \approx 1.58496$ as you can easily verify The compendium is very brief and doesn't explain what this means. $ (\log_2 (3))^2 \approx (1.58496)^2 \approx 2.51211$

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$2 \log_2 (3) \approx 2 \cdot 1.58496 \approx 3.16992$ I just came across this annotation in my school's maths compendium $2^ {\log_2 (3)} = 3$

Do any of those appear to be equal

(whenever you are wondering whether some general algebraic relationship holds, it's a good idea to first try some simple numerical. So, when you square both sides of an equation, you can get extraneous answers because you are losing the negative sign That is, you don't know which one of the two square roots of the right hand side was there before you squared it. We can square both side like this

$ x^2= 2$ but i don't understand why that it's okay to square both sides What i learned is that adding, subtracting, multiplying, or dividing both sides by the same thing is okay But how come squaring both. We can't simply square both sides because that's exactly what we're trying to prove

$$0 < a < b \implies a^2 < b^2$$ more somewhat related details

I think it may be a common misconception that simply squaring both sides of an inequality is ok because we can do it indiscriminately with equalities. The mean of the square is greater than or equal to the square of the mean, from jensen's inequality. What is the appropriate parametric equation of the boundary of a square For example, the unit circle has a parametric equation $x(t)=\\cos(t)$ and $y(t)=\\sin(t)$.

Given dimensions $a$,$b$, how would we find minimum area of rectangle that can be inscribed in the outer rectangle I realise that the answer is zero if the outer rectangle is a square, and i wonde. The square root of i is (1 + i)/sqrt (2) [try it out my multiplying it by itself.] it has no special notation beyond other complex numbers