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An imaginary number is one that includes the square root of -1, written √(-1). While one might think that such a number could not possibly exist, it plays an important role in quantum mechanics.

What number could be the square root of -1? It would have to be a number that when squared, that is, multiplied by itself, equals -1. This is a number that is not equal to 0, nor less than 0, nor greater than 0. No ordinary number fits the bill. This is why it’s called an “imaginary number” or “impossible number” and symbolized with the letter i. 

What if we’re calculating, and√(-1) shows up as the answer to an algebra problem? Could happen. For example, the equation x2 = -4 has the solution 2(√(-1)), that is, 2i. For students in many algebra classes, it would be reason to check for a calculation error. Algebra teachers don’t generally give students problems that involve imaginary numbers. And if the student finds that all her calculations are correct, and she still gets such an answer, she may be told that the answer isn’t allowed. Just as dividing by 0 isn’t allowed.

But in more advanced math, mathematicians and physicists work with i without blinking an eye (so to speak). Over the last couple of centuries, both mathematicians and physicists have found i to be useful when solving equations.

An entire set of imaginary numbers can be created by multiplying i times an ordinary number, for example:

3i          4½ i        -7.35 i

Difference between imaginary numbers and real numbers

Ordinary numbers are called “real numbers.” These are all the numbers that could be used to express distances measured by a ruler, for example:

0    1     4½     7.9543  or  -1    -2¾      -5.6947   

A real number can be combined with an imaginary number, for example:

3 + i     -14 + 4i     5 – 2.7i

Imaginary Numbers and Complex Numbers

A combination of both real and imaginary numbers is called a “complex number.” Thus, -14 + 4i is a complex number. It means -14 plus the quantity 4 times the square root of -1. Many calculations of quantum mechanics involve complex numbers.


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