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The square of any real number is non-negative. So there is no number whose square is negative. But there are times when it is mathematically helpful to be able to take the square root of a negative number. This is where i comes to the rescue.
i is a number such that i2 = -1.
Then, if you want to find the square root of -4, for example,
-4 = 4*(-1)
So that sqrt(-4) = sqrt[4*(-1)] = sqrt(4)*sqrt(-1) = +/- 2*i
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Can a number be imaginary but not pure imaginary?
Yes. The number 1 + i is imaginary but not pure imaginary, while 5i is pure imaginary.
Is the difference of a complex number and its conjugate a real imaginary or pure imaginary number?
It is a pure imaginary number.Since (a+bi)-(a-bi) = 2bi, it is a pure imaginary number (it has no real component).
Is -23 an imaginary number?
No, but √-23 is. An imaginary number is the square root of a negative number.
Why is the square root of a negative number imaginary?
Any real number, when squared will give a positive real number. This is true weather you have a negative real number and square it, or positive. The concept of imaginary numbers was invented to handle equations which needed to use the square root of a negative number, in order to solve them.At first, they were not considered useful beyond this, so they were named imaginary numbers. Through the work of Euler and others, it became evident that imaginary and complex numbers could be useful, especially when analyzing waves, such as electrical waveforms.
Are imaginary and complex numbers the same?
No. A complex number is a number that has both a real part and an imaginary part. Technically, a pure imaginary number ... which has no real part ... is not a complex number.
