Larger? i or 2i
which is larger, i or 2i?
Interesting question! Actually, inequalities aren't applied to complex
numbers (as they are to the real numbers.) Rather, one would "compare"
absolute values or norms. For a given complex number z, the absolute value
of z = the square root(the square of the x-component + the square of
So...abs(i) = 1 and abs(2i) = 2, which shows 2i has the greater magnitude.
It cannot be overemphasized, however, that the complex numbers themselves
do not obey inequality properties.
2i is larger than i, but this statement is only meaningful on the imaginary axis.
We normally draw real numbers on a horizontal axis (positive = right), and imaginary
numbers on a vertical axis (positive = upwards). On that vertical axis, 2i is
"greater" than i. The imaginary numbers are not really translatable to real numbers,
except for the fact that multiplying or dividing one imaginary number by another
produces a real number.
You can't place these numbers on an everyday ruler, but their being "imaginary"
does not diminish their importance. They are used routinely in electronics and
mechanical engineering, where they are useful in perfectly describing vibrating
or oscillating devices. They also show up in the Schrödinger equation, a basis of
quantum physics. This is another domain that appears to "make no sense", and yet the
non-intuitive techniques accurately predict the way the world works to the limits of
our ability to measure things.
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Update: June 2012