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Anonymous asked in Science & MathematicsMathematics · 1 month ago

# Is there any number that is neither real nor imaginary?

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• Since real and imaginary covers every number a complex number is just both real and imahinary

• According to most writers, 1+i is complex but neither real nor imaginary.  Some will say it's imaginary, calling i and -3i "pure imaginary" numbers; but nobody says it's real.

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• complex

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• Depends what *you* mean by 'number'.

Fundamentally, a 'number' is simply an 'element of a set'.

For instance, given a set with a group (field, ring or other) structure, the elements may be called 'numbers'.

The elements of the set of the 'reals' are 'real numbers'.

The elements of the complex set, are 'complex numbers'.

And there are many other ones typically used/named.

In everyday language, 'number' just refers to your usual real numbers.

Again, words just mean what they mean in a given context; what you meant by it.

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• i dont think so

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• If you define "imaginary number" as a number whose square is a negative real number, then yes.  Any sum of a nonzero real number and an imaginary number would be neither real (because it's not on the real number line) and not imaginary (because it's square is not a negative real number.)

Take the number 1+i as an example.  Its square is:

(1 + i)² = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i

The square of 1+i is not a real number so 1+i is not imaginary.

The reason for the "if you define..." part is that many high school algebra books refer to any non-real solution to a quadratic equation as "imaginary".  That isn't really mainstream, but it's common enough to be wary of.

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• Mathematicians also use hypercomplex numbers, such as quaternions, octonions, sedenions, tessarines. Quaternions are also used in engineering and computer graphics.

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• The fraction 6/0 is one

• Not a number.

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