Imaginary numbers were introduced to represent the square root of negative numbers, since there is no real number solution. René Descartes defined the imaginary unit i as the square root of negative one. Powers of i follow a repeating pattern, where i raised to a power that is a multiple of 4 is 1, and powers increase i by one each time the remainder is 1 higher than the previous power.

1. Since Square Roots of Negative Numbers have no solution , in other words, there´s no real number that works as solution of the square root of a negative number. … Have no solution or representation within the Set of Real Numbers XVII Century René Descartes Imaginary Numbers . He defined the unit of these numbers as , which is represented with the letter i. Since i represents the square root of negative 1, we can conclude that… Imaginary Numbers

2. Powers of i The powers of i , are based in the definition of the imaginary unit and its square. We know that Power and its simplification Result i -1 -i 1 i -1 -i 1 i

3. We could indefinetly keep simpliflying powers. But after simplifying them, what do we get???? 1 raised to any power is always 1. , where n is the original power and r is the remainder of the division Powers of i EXAMPLES Solve with your teacher the following powers for i: i 30 , i 17 , i 34 , i 120

4. Exercises: Simplify the following imaginary numbers. Imaginary Numbers