Is 1+i>-1+i?

This is part of a series on common misconceptions.

True or False?

\[1+i>-1+i\]

Why some people say it's true: Subtract \(i\) from both sides, and we will have \(1>-1\), which is obviously true.

Why some people say it's false: (Explanation)

The statement is \( \color{red}{\textbf{false}}\).

Proof:
Let's assume it's true, i.e. \(1+i>-1+i.\)
Squaring both sides gives \(1+i^2+2i>-1+i^2-2i,\) which on further simplification gives \(2i>-2-2i.\)
Rearranging, \(4i>-2.\)
Dividing by 2 throughout gives \(2i>-1.\)
Squaring LHS and RHS again, we have \(4i^2>1,\) which gives \(-4>1.\)
This is clearly false, hence a contradiction to our original statement.
Therefore, the given statement is false. \(_\square\)

Rebuttal: This doesn’t make sense.
Reply: Imaginary numbers don’t operate in the same way as real numbers

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See Also

• List of Common Misconceptions