Common Misconceptions (Algebra)

Algebra Common Misconceptions: Level 3 Challenges


2×3=?\large \sqrt{-2}\times\sqrt{-3}=\, ?

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

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

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

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

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

(D) 1xy2>1x2y\dfrac{1}{xy^2}>\dfrac{1}{x^2y}

(E) xy>yx\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 xx on any one of the dice is 1x\frac {1}{x}

Let aa be the maximum number of dice in the collection and let SS be the sum of all the faces of all the dice in the maximum collection size.

Find a+Sa+S.

I claim that 1<11 < -1 using the proof below but I was just told that I might have just committed a small mistake.

A. (i4i3i1)<1i2- ( i^4 - i^3 - i - 1)< 1 -i^2
B. i3i4+i+1<1i2i^3 - i^4 + i + 1 < 1-i^2
C. (i3+i)(1i)<(1i)(1+i)(i^3+i)(1-i) < (1-i)(1+i)
D. i3+i<1+i i^3 + i < 1+i
E. i2(1+i)<1+ii^2(1+i) < 1+i
F. i2<1 i^2 < 1
G. i2<i4 i^2 < i^4
H. 1<i2 1 < i^2
I. 1<1 1 < -1

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

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

What is the wrong step in the following proof that 1=11 = -1?

  1. Let ww be a complex number such that (w+1)3=(w1)3(w + 1)^3 = (w - 1)^3.

  2. Solving this equation gives w=±i33w = \pm \frac{i \sqrt{3}}{3}.

  3. Since (w+1)3=(w1)3(w + 1)^3 = (w - 1)^3 for our previously mentioned values of ww, cube rooting both sides gives w+1=w1w + 1 = w - 1.

  4. Subtracting ww from both sides gives 1=11 = -1.


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