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Algebra

# Algebra Common Misconceptions: Level 3 Challenges

$\large \sqrt{-2}\times\sqrt{-3}=\ ?$

In this problem, the square root is a function from the complex numbers to the complex numbers.

If $$x$$ and $$y$$ are non-zero numbers such that $$x>y$$, which of the following is always true?

(A) $$\dfrac{1}{x}<\dfrac{1}{y}$$

(B) $$\dfrac{x}{y}>1$$

(C) $$|x|>|y|$$

(D) $$\dfrac{1}{xy^2}>\dfrac{1}{x^2y}$$

(E) $$\dfrac{x}{y}>\dfrac{y}{x}$$

Calvin has a collection of special weighted dice that all share special properties:

• They're all 4 sided dice
• Any one die has distinct positive integers on each of its faces
• No pair of dice have all their numbers exactly the same
• The probability of rolling number $$x$$ on any one of the dice is $$\frac {1}{x}$$

Let $$a$$ be the maximum number of dice in the collection and let $$S$$ be the sum of all the faces of all the dice in the maximum collection size.

Find $$a+S$$.

I claim that $$1 < -1$$ using the proof below but I was just told that I might have just committed a small mistake.

A. $$- ( i^4 - i^3 - i - 1)< 1 -i^2$$
B. $$i^3 - i^4 + i + 1 < 1-i^2$$
C. $$(i^3+i)(1-i) < (1-i)(1+i)$$
D. $$i^3 + i < 1+i$$
E. $$i^2(1+i) < 1+i$$
F. $$i^2 < 1$$
G. $$i^2 < i^4$$
H. $$1 < i^2$$
I. $$1 < -1$$

Where did I go wrong? At which step(s) did I commit a fallacy?

Note: $$i = \sqrt {-1}$$.

What is the wrong step in the following proof that $$1 = -1$$?

1. Let $$w$$ be a complex number such that $$(w + 1)^3 = (w - 1)^3$$.

2. Solving this equation gives $$w = \pm \frac{i \sqrt{3}}{3}$$.

3. Since $$(w + 1)^3 = (w - 1)^3$$ for our previously mentioned values of $$w$$, cube rooting both sides gives $$w + 1 = w - 1$$.

4. Subtracting $$w$$ from both sides gives $$1 = -1$$.

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