Math, asked by prnagre143, 8 months ago

3) Show that 1 + ¡10 + i20 + i30 is a real number.​

Answers

Answered by UmangChaurasia12
3

Answer:

So, we can write it in the form of i(4×7+2). Hence we get the values of i10=−1,i20=1,i30=−1. We know that 0 is a real number. Therefore, it is proved that 1+i10+i20+i30 is a real number.

Step-by-step explanation:

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Answered by rajaniseelam2006
3

Step-by-step explanation:

Hint: As we know that a real number includes all rational and irrational numbers. It also includes positive numbers, negative numbers and the number zero. For the above we will have to know about the properties of iota (i)(i) given as follows:

i4n=1i4n+2=−1i4n+1=ii4n+3=−ii4n=1i4n+2=−1i4n+1=ii4n+3=−i

Here n can be 0, 1,2, 3…

Complete step-by-step answer:

We have been given to prove 1+i10+i20+i301+i10+i20+i30 is a real number.

Now let us consider the case of i10i10 first. The power is 10, so it can be expressed in terms of multiple of 4 as 8 + 2. So, we can write it in the form of i(4×2+2)i(4×2+2).

We know that,

i4n+2=−1i10=i(4×2+2)=−1i4n+2=−1i10=i(4×2+2)=−1

Again, we have the term i20i20. We know that the power 20 is a multiple of 4. So, we can write it in the form of i4×5i4×5.

We know that,

i4n=1i20=i4×5=1i4n=1i20=i4×5=1

Now, we have the last term as, i30i30. The power is 30, so it can be expressed in terms of multiple of 4 as 28 + 2. So, we can write it in the form of i(4×7+2)i(4×7+2).

We know that,

i4n+2=−1i30=i(4×7+2)=−1i4n+2=−1i30=i(4×7+2)=−1

Hence we get the values of i10=−1,i20=1,i30=−1i10=−1,i20=1,i30=−1.

On substituting the values of i10,i20,i30i10,i20,i30 in the given expression, we get,

1+i10+i20+i30=1+(−1)+(1)+(−1)=01+i10+i20+i30=1+(−1)+(1)+(−1)=0

We know that 0 is a real number.

Therefore, it is proved that 1+i10+i20+i301+i10+i20+i30 is a real number.

Note: Be careful while calculating the value of i10,i20,i30i10,i20,i30. Also remember that ‘i' is known as iota and it is equal to −1−−−√−1. The most common mistake is thinking that the even power will be positive, but that is not the case here. So, we must be very careful in that regard.

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