The danger of complex number i

The complex number i = √has a danger that some people may not be aware of. We use H = -1, see here.

For, consider:

-1 = i²  = (H) (H) = (H H) = 1 = 1

Professor of mathematics Edward Frenkel states in his book, intended for the general audience, and thus giving false information to that general audience:

“Note that it is customary to denote √-1 by i (for “imaginary”), but I chose not to do this to emphasize the algebraic meaning of this number: it really is just a square root of -1, nothing more and nothing less. It is just as concrete as the square root of 2. There is nothing mysterious about it.” (E. Frenkel, “Love & Math”, p101-102)

Observe the factual error and the error in didactics:

  1. The factual error is to say that the symbol √ has the same meaning in √-1 as in √2.
  2. Didactically, it is writing that conveys the algebraic meaning better, not writing √-1.

It took William Rowan Hamilton (1805-1865), the hero of Irish mathematics, a major part of his time to discover that = {0, 1}, i.e. the point in the two-dimensional plane where x = 0 and y = 1. Stepping into another dimension is not the same as staying in the same dimension. If you treat those at the same then you get above deduction that -1 = 1. The conclusion is that i is an operator and not a common number. The step (√H) (√H) = (H H) is forbidden since it concerns an operator, with a different rule for √. We can only call i a “(complex) number” if we adapt the notion of “number” to include it.

Let us look a bit more at the reason why i was mysterious and imaginary. Consider the quadratic equation, and let us “complete the square” on the left hand side

a x²  + b x + c = 0                                                  (formula for a vertical parabola)

x²  + b aH x       = – c aH                 (bring c to the right and multiply by aH= 1 / a)

b aH 2H) ²  = (b aH 2H) ²  – c aH                                    (using  2H 2H =  1)

+ b aH 2 =  ±  √ ((b aH 2H) ²  – c aH )                 (discriminant = √(b² – 4 a c) )       

(– b  ±  √ (4 a c ))  (2 a)H                            (the quadratic formula)

From wikipedia: this formula covering all cases was found by Simon Stevin in 1594, who also gave us the decimal dot. The present form was given by Descartes in 1637. In the past people were calculating every step. Having the final formula allows you to reduce the actual number of calculations you have to do.

There will be an intersection with the horizontal axis (above equation has a root) only if D ≥ 0. Otherwise there is no intersection.

It is an option to interprete i = √H as a number too. In that case the problem is redefined to have existed in the complex plane all along, and then there is always a solution. This explains where the mystery comes from: you have to grow aware that your original problem was not one-dimensional but two-dimensional.

Frenkel’s approach “there is nothing mysterious about it” kills this last insight. He claims to draw you to the beauty of mathematics, comparable to masterpieces of art, but at the same time he says that you should not be worried since it is as common as bread and butter. There is a difference between admiring a masterpiece and making one yourself. The professor is seriously confused. It is better that students understand the quadratic equation and the complex plane, and then admire their own understanding too.

Parabolic jump (Source: Jarek Tuszynski, wikimedia commons)

Parabolic jump (Source: Jarek Tuszynski, wikimedia commons)


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