The powers of i ( in ) are proven within the desk under and these could be computed fairly simply when n > 0 and n < 0.

When you didn’t fairly perceive the data within the desk, hold studying to see the logic behind it!

## Powers of i when n > 0

i1 = i

i2  = – 1

i3 = i2 × i = -i

i4 = i2 × i2 = 1

i5 = i4 × i = i

i6 = i4 × i2 = -1

i7 = i4 × i3 = -i

i8 = i4 × i4 = 1

Discover the sample i, -1, -i, 1, … repeats after the primary 4 advanced numbers. Generally, if n is an integer greater than zero, the worth of in could be discovered by dividing n by 4 and analyzing the rest.

Did you make the next observations in regards to the powers of i?

• For i4 and i8, the rest is 0 after we divide 4 and eight by 4.
• For i3 and i7, the rest is 3 after we divide 3 and seven by 4.
• For i2 and i6, the rest is 2 after we divide 2 and 6 by 4.
• For i1 and i5, the rest is 1 after we divide 1 and 5 by 4.

Conclusion

Let n > 0 and R is the rest when n is split by 4

If R = 1, in = i

If R = 2, in = -1

If R = 3, in = -i

If R = 0, in = 1

## Powers of i when n < 0

i-1  =  1 / i  = (1 × i) / (i × i) = i / i2 = i / -1 = -i

i-2 = 1 / i2 = 1 / -1 = -1

i-3 = 1 / i3 = 1 / -i = (1 × i) / (-i × i) = i / 1 = i

i-4 = 1 / i4 = 1 / 1 = 1

i-5  =  1 / i5  = 1 / i = -i

i-6  =  1 / i6  = 1 / -1 = -1

i-7  =  1 / i7  = 1 / -i = i

i-8  =  1 / i8  = 1 / 1 = 1

i-1 =-i

i-2 = -1

i-3 = i

i-4 = 1

i-5 =-i

i-6 = -1

i-7 = i

i-8 = 1

Discover the sample -i, -1, i, 1, … repeats after the primary 4 advanced numbers. Generally, if n is an integer smaller than zero, the worth of in could be discovered by dividing n by 4 and analyzing the rest.

Did you make the next observations in regards to the powers of i?

• For i-4 and i-8, the rest is 0 after we divide 4 and eight by 4.
• For i-3 and i-7, the rest is -3 after we divide -3 and -7 by 4.
• For i-2 and i-6, the rest is -2 after we divide -2 and -6 by 4.
• For i-1 and i-5, the rest is -1 after we divide 1 and 5 by 4.

Conclusion

Let n < 0 and R is the rest when n is split by 4

If R  = -1, in = -i

If R = -2, in = -1

If R = -3, in = i

If R = 0, in = 1

## Just a few examples displaying the right way to discover the powers of i.

Instance #1:

i67

67 divided 4 provides a the rest of three. Since n is optimistic, i67 = -i

Instance #2:

i-67

-67 divided 4 provides a the rest of -3. Since n is unfavourable, i-67 = i

Instance #3:

i36

36 divided 4 provides a the rest of 0. Since n is optimistic, i36 = 1

Instance #4:

i-36

-36 divided 4 provides a the rest of 0. Since n is unfavourable, i-36 = 1

Take pleasure in this web page? Please pay it ahead. This is how…

Would you like to share this web page with others by linking to it?

1. Click on on the HTML hyperlink code under.
2. Copy and paste it, including a notice of your personal, into your weblog, a Internet web page, boards, a weblog remark, your Fb account, or wherever that somebody would discover this web page beneficial.