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I'm a bit confused about the true meaning of imaginary number. It's my understanding that a complex number is a number of the form $a+ib$, where $a$ and $b$ are real numbers. A pure imaginary number is a complex number where $a=0$. However, when they say that a number is an "imaginary number", without the "pure", what do they mean exactly?

Em.
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set5
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    There is no difference between "pure imaginary numbers" and "imaginary numbers". –  Jan 10 '16 at 19:02
  • You should generally avoid using the word "imaginary" since it suggests that complex numbers with $b≠0$ are less "real", in the ontological sense, than numbers which can be represented on a number line(elements of $\mathbb{R}$). – MathematicsStudent1122 Jan 10 '16 at 19:13

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When people say "imaginary number" they should mean what you call a "pure imaginary number", i.e. one of the form $ib$ where $b$ is real. (However people are occasionally sloppy and say "imaginary" when all they mean is "complex".)

Oscar Cunningham
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  • The usage is nowhere near consistent. One speaks of an “imaginary quadratic field”, even though only some of its elements are purely imaginary. – Lubin Jan 10 '16 at 19:11