Find value of 1+i raise to 2 +i raise to 4+ i raise to6+i raise to 8...+i raise to 20
Answers
In the last article,we learnt about iota which is a complex number equal to −1−−−√. Now, can we find power of iota (in) when n is any whole number. Lets simply calculate some of them and then I will define some general rule.
i=−1−−−√
i2=(−1−−−√)2=−1
i3=i×i2=i×−1=−i
i4=i2×i2=−1×−1=1
i5=i×i4=i×1=i
i6=i×i5=i×i=i2=−1
i7=i×i6=i×−1=−i
i8=(i2)4=(−1)4=1
i9=i×i8=i×1=i
i10=i×i9=i×i=i2=−1
Now, we can make a general rule which will help us to find power of i for any value of integer.
When we carefully look at the above calculations, we can see that
i4n=1, where n is any whole number. (1)
i4n+1=i, where n is any whole number. (2)
i4n+2=−1, where n is any whole number. (3)
i4n+3=−i, where n is any whole number. (4)
We can explain (1), (2), (3) and (4) in more simpler words with the help of example.
Suppose, we want to evaluate i2001. Just divide 2001 with 4 and note down the remainder. When we divide 2001 by 4 then we get remainder equal to 1. Therefore, it can be placed in category (2). Hence, the answer would be i
Lets take another example, we want to evaluate i1146, we just divide 1146 by 4 and we note that the remainder comes out to be equal to 2. It can be placed in (3) category. Hence, the answer is equal to -1.
What you learnt in this article?
How to find in when n is any whole number.
Filed Under: Complex Numbers
Tagged With: complex numbers
Comments
mirsohail says
October 13, 2013 at 01:27
I find it very useful. It will be great if you also explain general cases of iota power (-n).
Reply
Minhaj Afridi says
May 10, 2017 at 01:27
-n convert denomenator and use the same rules
Reply
Harsh says
May 12, 2018 at 01:27
¡ ki power 2018
Reply
saleem says
April 14, 2020 at 01:27
its answer is -1 .if you want trick then reply me.
Reply
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