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# Oldie but goodie 1=2

## Question

You can multiply by adding, so you can make squares that way, too.

• 12 = 1
• 22 = 2+2
• 32 = 3+3+3
• 42 = 4+4+4+4
• ...
• x2x+x+x+x+ ... +x (x times)

The derivative is

2x = 1+1+1+1+ ... +1 (x times) = x

2 = 1

What's wrong here?

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Spoiler

You cannot use the sum rule of differentiation to determine the derivative of  x + x + x + x + x … (repeated x times) since the x in “x times” is not a number, but a variable. The sum rule in differentiation is

(f1(x) f2(x) + ... + fk(x))'  = f  '1(x) +  f  '2(x) + ... +  f 'k(x), where k is any positive integer.

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Doing the derivative correctly,

Spoiler

Define f(x) = sum (i=1,x) x.

Then f(x+1) = sum (i=1,x+1) (x+1)

And f '(x) =~ { f(x+1) - f(x) } / 1 = { (x+1)2 - x2 } / 1 = ( 2x + 1 ) / 1  ~  2x (when 1 is small compared to 2x)

So the derivative of the last sum is 2x, not x.

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