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Fractional Exponents & Imaginary Numbers











































(-9)^1/2 has no solution within the set is a real numbers

How are imaginary numbers used for this type of mathematical problem?

Explain why imaginary numbers are not imaginary in the sense that you can only imagine them.




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Feb 24, 2011
Fractional Exponents & Imaginary Numbers
by: Staff


The question:

(-9)^1/2 has no solution... please explain.


The answer:

(-9)^1/2 has a solution, but you need to employ a new type of number.


(-9)^1/2

= sqrt(-9)

= sqrt(-1*9)

= sqrt(9)*sqrt(-1)

= 3*sqrt(-1)

i stands for the square root of -1. i is called an imaginary number.

The square root of (-1) has two solutions: +i and -i

= 3*(-i)


= 3i


“i” is used to represent the square root of -1 because the Real Number System cannot represent the square root of -1.

However, imaginary numbers are not imaginary in the sense that you can only imagine them. (Naming them “imaginary” was probably not the best choice of terminology.)

Imaginary numbers are simply a special type of number. (Other special number types would include fractions or negative numbers.)

Imaginary numbers were invented so equations such as x^2 + 4 = 0 can be solved.

Imaginary numbers allow roots to be calculated for every polynomial.

Solving many polynomials used in electrical engineering, mechanical engineering, and physics would be impossible without using imaginary numbers.

In conclusion, let me illustrate how the invention of imaginary numbers compares with the invention of integers.

Natural numbers (counting numbers) cannot be used solve the following type of problem:

You have $100, but spend $150. What is your balance?

Your balance is -$50, but Natural numbers only include the positive numbers {1, 2, 3, 4, …}.

You cannot use natural numbers to calculate a -$50 value. The -$50 is a value of $50 less than nothing.

How can you calculate less than nothing? To calculate a -$50 value, you need a new type of number. That is why integers {…-4, -3, -2, -1, 0, -1, 2, 3, 4, …} were invented.

Similarly, imaginary numbers were also invented to deal with a certain type of problem.

Imaginary numbers were invented so equations such as x^2 + 4 = 0 can be solved.





Thanks for writing.


Staff
www.solving-math-problems.com


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