Powers of the imaginary unit (article) | Khan Academy (2024)

Learn how to simplify any power of the imaginary unit i. For example, simplify i²⁷ as -i.

Want to join the conversation?

Log in

  • satyanash

    8 years agoPosted 8 years ago. Direct link to satyanash's post “How is `i^3 = -i` ?My un...”

    How is i^3 = -i ?
    My understanding leads me to:

    i^3 = i * i * i
    i^3 = sqrt( -1 ) * sqrt( -1 ) * sqrt( -1 ) ; i = sqrt( -1 )
    i^3 = sqrt( -1 * -1 * -1 )
    i^3 = sqrt( -1 )
    i^3 = i

    (54 votes)

    • andrewp18

      8 years agoPosted 8 years ago. Direct link to andrewp18's post “On step 3 you did:√𝑎 • ...”

      Powers of the imaginary unit (article) | Khan Academy (4)

      Powers of the imaginary unit (article) | Khan Academy (5)

      Powers of the imaginary unit (article) | Khan Academy (6)

      On step 3 you did:
      √𝑎 • √𝑏 = √(𝑎𝑏)
      But that is only true if:
      𝑎, 𝑏 > 0
      Which is not the case here. Comment if you want to see a proof of that.

      (156 votes)

  • dylan.forr99

    a year agoPosted a year ago. Direct link to dylan.forr99's post “On hour 9 of study. send ...”

    On hour 9 of study. send help. xD

    (41 votes)

    • Vestige

      a year agoPosted a year ago. Direct link to Vestige's post “sending help your way _h...”

      Powers of the imaginary unit (article) | Khan Academy (10)

      sending help your way
      help sent

      (21 votes)

  • Jonathan Huang

    7 years agoPosted 7 years ago. Direct link to Jonathan Huang's post “Where do you use this in ...”

    Where do you use this in the real world?

    (15 votes)

    • 🎓 Arnaud Jasperse

      7 years agoPosted 7 years ago. Direct link to 🎓 Arnaud Jasperse's post “Physics uses it. Electric...”

      Powers of the imaginary unit (article) | Khan Academy (14)

      Powers of the imaginary unit (article) | Khan Academy (15)

      Physics uses it. Electrical Engineering often have to use the imaginary unit in their calculations, but it is also used in Robotics. There, to rotate an object through 3 dimensions they even result in using Quaternions (which build on the imaginary unit i).

      (32 votes)

  • dollyrauh

    8 years agoPosted 8 years ago. Direct link to dollyrauh's post “Hi, could someone maybe h...”

    Hi, could someone maybe help me with what I did wrong in the last challenge problem? I followed along with their explanation well enough, but I'm scratching my head as to exactly where I messed up the problem. My answer was i, while the correct was -i. Also, by way of explanation, in steps three, four, and five, I am assuming that 1 can be rewritten as 1=sqrt(1) because 1*1= 1, therefore the square root of 1 equals 1.

    i^-1=

    1/i^1 (because properties of negative exponents)
    1/i (because i^1=i)
    1/sqrt(-1) (because i=sqrt(-1))
    sqrt(1)/sqrt(-1) (because the 1 in the numerator can be rewritten as sqrt(1) without changing its value)
    sqrt(1/-1) (to simplify inside the square root)
    sqrt(-1) (because 1/-1 is -1)
    sqrt(-1)=i (because i=sqrt(-1))
    =i

    It's driving me crazy that I can't figure out what I did wrong!! Please help.

    Thanks so much!!

    (14 votes)

    • Dominik.Ehlert

      8 years agoPosted 8 years ago. Direct link to Dominik.Ehlert's post “It seems to me your probl...”

      Powers of the imaginary unit (article) | Khan Academy (19)

      Powers of the imaginary unit (article) | Khan Academy (20)

      It seems to me your problem is the definition i = sqrt(-1). Better use i^2 = -1. You can see, that your last step would get you sqrt(-1) = +/- i, so the right solution is at least within, but the problem is better solved another way:
      i ^ (-1) = 1/i = 1/i * i/i <-- Expand with i
      = i/(i^2) <-- Multiply
      = i/(-1) <-- Definition
      = -i <-- Simplify
      Therefore i^(-1) = -i

      (33 votes)

  • joey mizrahi

    8 years agoPosted 8 years ago. Direct link to joey mizrahi's post “how is i^-1 = -idoesnt i...”

    how is i^-1 = -i
    doesnt it equal 1/i

    (7 votes)

    • Jacky Xie

      8 years agoPosted 8 years ago. Direct link to Jacky Xie's post “Yes, i^-1 = 1/i.Notice t...”

      Powers of the imaginary unit (article) | Khan Academy (24)

      Yes, i^-1 = 1/i.
      Notice that you can multiply 1/i by i/i, which gives you i/-1, or -i, which is much easier to comprehend than 1/i.

      (18 votes)

  • Rob

    8 years agoPosted 8 years ago. Direct link to Rob's post “will i^0=1?”

    will i^0=1?

    (9 votes)

    • Abhishek Kumar

      5 years agoPosted 5 years ago. Direct link to Abhishek Kumar's post “Hey, the explanation to y...”

      Hey, the explanation to your query is "any number other than 0 taken to the power 0 is defined to be 1".
      So i=√-1, "i" here is a real number. So i^0=1.
      It is explained in great detail at this website-->
      (though it requires some math, though eventually, you get the idea)
      http://mathworld.wolfram.com/Power.html
      However, "zero to the power of zero" (or 0^0) is undefined.

      (6 votes)

  • Peu255

    a year agoPosted a year ago. Direct link to Peu255's post “where are the powers of t...”

    where are the powers of the imaginary unit? I've seen no powers. Does it throw fireballs?

    (13 votes)

  • umairansari16

    7 years agoPosted 7 years ago. Direct link to umairansari16's post “what Will be i to the pow...”

    what Will be i to the power i 899999

    (4 votes)

    • kubleeka

      7 years agoPosted 7 years ago. Direct link to kubleeka's post “i^899999(i^900000)/i1/i...”

      Powers of the imaginary unit (article) | Khan Academy (33)

      i^899999
      (i^900000)/i
      1/i
      i/(i^2)
      i/(-1)
      -i

      (12 votes)

  • O Madhu 💞

    a year agoPosted a year ago. Direct link to O Madhu 💞's post “I understand how to solve...”

    I understand how to solve for i using simple negative exponents, but I don't know how you would solve for more larger negative exponents.

    (1 vote)

    • Tanner P

      a year agoPosted a year ago. Direct link to Tanner P's post “I’ll walk you through an ...”

      Powers of the imaginary unit (article) | Khan Academy (37)

      I’ll walk you through an example.

      Say you want to evaluate i^-207

      1. Divide by four and find the remainder
      207/4 = 51 R 3

      Since the remainder is 3, i^-207 = i^-3

      To see why this works: Using exponent properties, you can rewrite the problem as
      (i^4)^-51 * i^-3
      =1^-51 * i^-3
      =1 * i^-3
      =i^-3

      2. Now, this is something you know how to solve
      i^-3
      =1/i^3
      =1/-i
      =1/-i * i/i
      =i/-i^2
      =i/-(-1)
      =i/1
      =i.

      i^-207 = i

      (18 votes)

  • owengeorge

    a year agoPosted a year ago. Direct link to owengeorge's post “Is there any reason to us...”

    Is there any reason to use multiple of 4 over multiple of 1,2 etc or is just arbitrary?

    (4 votes)

    • Kim Seidel

      a year agoPosted a year ago. Direct link to Kim Seidel's post “It must be 4. It is not a...”

      Powers of the imaginary unit (article) | Khan Academy (41)

      It must be 4. It is not arbitrary. The powers of i rotate thru 4 values, not something else.
      i^1 = i
      i^2 = -1
      i^3 = -i
      i^4 = 1
      Then this pattern repeats
      i^5 = i
      i^6 = -1
      i^7 = -i
      i^8 = 1
      etc.

      Hope this helps.

      (11 votes)

Powers of the imaginary unit (article) | Khan Academy (2024)
Top Articles
Latest Posts
Article information

Author: Rubie Ullrich

Last Updated:

Views: 5449

Rating: 4.1 / 5 (72 voted)

Reviews: 95% of readers found this page helpful

Author information

Name: Rubie Ullrich

Birthday: 1998-02-02

Address: 743 Stoltenberg Center, Genovevaville, NJ 59925-3119

Phone: +2202978377583

Job: Administration Engineer

Hobby: Surfing, Sailing, Listening to music, Web surfing, Kitesurfing, Geocaching, Backpacking

Introduction: My name is Rubie Ullrich, I am a enthusiastic, perfect, tender, vivacious, talented, famous, delightful person who loves writing and wants to share my knowledge and understanding with you.