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What is the Imaginary Unit i?

What is the imaginary unit i?

Summary

  1. The symbol over the -1 is the square root symbol
  2. 'r' is a variable representing a real number here
  3. The Product Property of Square Roots says we can rewrite the square root of a negative number as the number's square root times the square root of -1
  4. A number that is multiplied by 'i' is called an imaginary number
  5. i√r and 3i are both imaginary numbers
  6. Memorizing the first four powers of 'i' will make simplifying higher powers of 'i' much easier!

Notes

    1. Even though we represent 'i' with a letter, it is NOT a variable
    2. 'i' is always equal to the square root of -1
    1. You might have previously been told that you cannot take the square root of a negative number
    2. You actually CAN take the square root of a negative number, but you need to use this new imaginary unit 'i'
    3. Remember, 'i' is equal to the square root of -1
    4. An imaginary number is a number multiplied by 'i'
    1. You might have previously been told that you cannot take the square root of a negative number
    2. You actually CAN take the square root of a negative number, but you need to use this new imaginary unit 'i'
    3. Remember, 'i' is equal to the square root of -1
    1. The Product Property of Square Roots says we can rewrite the square root of a negative number as the number's square root times the square root of -1
    2. Remember, 'i' is equal to the square root of -1
    1. It's useful to memorize the first four powers of 'i' because it will make simplifying higher powers of 'i' much easier
    1. Since 'i' is equal to the square root of -1, when we square it we just get -1
    2. i3 is the same as multiplying i•i2
    3. Since we know i2 = -1, -1•i just equals -i
    4. i4 is the same as i2•i2, or -1•-1, which is just 1
    5. Note the pattern when raising 'i' to a power!
    6. Once we reach i5 the cycle starts over, which is why we focus on the first four powers of 'i'
    7. As a way to remember the first four powers of 'i', write out the answers as they sound, ignoring the negative signs: 'I Won I Won'
    8. Then just remember that the negatives go with the two middle words!