((1 + i)/(1 − i))ⁿ = 1

holds true is ______.

Smallest Positive Integer n for Which ((1 + i)/(1 − i))ⁿ = 1

Understanding the Given Complex Number Problem

This question asks us to determine the smallest positive non-zero integer value of n for which a complex number raised to the power n becomes equal to 1. The given expression contains the imaginary unit i, where i² = −1.

The equation is:

((1 + i)/(1 − i))ⁿ = 1

The most efficient way to solve this problem is to simplify the complex fraction inside the brackets first. Once the fraction is reduced to a simple power of i, the problem becomes a question about the cyclic powers of the imaginary unit.

Step-by-Step Solution

Step 1: Simplify the Complex Fraction

We begin with the complex number:

(1 + i)/(1 − i)

The denominator contains the complex number 1 − i. To simplify the fraction, we multiply both the numerator and denominator by the conjugate of the denominator.

The complex conjugate of 1 − i is 1 + i. Therefore:

(1 + i)/(1 − i) × (1 + i)/(1 + i)

This gives:

(1 + i)² / [(1 − i)(1 + i)]

We now simplify the numerator and denominator separately.

Step 2: Simplify the Numerator

The numerator is:

(1 + i)²

Expanding the square:

(1 + i)² = 1 + 2i + i²

Since i² = −1:

(1 + i)² = 1 + 2i − 1

Therefore:

(1 + i)² = 2i

Step 3: Simplify the Denominator

The denominator is:

(1 − i)(1 + i)

Using the identity (a − b)(a + b) = a² − b², we obtain:

(1 − i)(1 + i) = 1² − i²

Since i² = −1:

1 − (−1) = 2

Therefore:

(1 − i)(1 + i) = 2

Step 4: Reduce the Fraction to Its Simplest Form

Substituting the simplified numerator and denominator, we get:

(1 + i)/(1 − i) = 2i/2

Therefore:

(1 + i)/(1 − i) = i

The original equation now becomes:

iⁿ = 1

Thus, the problem has been reduced to finding the smallest positive non-zero integer n for which i raised to the power n is equal to 1.

Understanding the Cyclic Powers of i

The powers of the imaginary unit i repeat in a cycle of four. Starting with the first positive power:

i¹ = i

i² = −1

i³ = −i

i⁴ = 1

After the fourth power, the same pattern repeats because:

i⁵ = i,   i⁶ = −1,   i⁷ = −i,   i⁸ = 1

Therefore, iⁿ is equal to 1 whenever n is a positive multiple of 4.

Finding the Smallest Positive Value of n

We need the smallest positive non-zero integer n satisfying:

iⁿ = 1

Checking the powers in increasing order, i¹ = i, i² = −1, and i³ = −i. None of these values is equal to 1. The next power is:

i⁴ = 1

Hence, the smallest positive non-zero integer satisfying the equation is:

n = 4

Alternative Solution Using Polar Form

The same result can also be understood using the polar representation of complex numbers. The complex number 1 + i has argument π/4, while 1 − i has argument −π/4. Both complex numbers have the same modulus √2.

Therefore, when we divide them, their moduli cancel and their arguments are subtracted:

(1 + i)/(1 − i) = e^{i[π/4 − (−π/4)]}

Thus:

(1 + i)/(1 − i) = e^iπ/2

Since e^iπ/2 = i, the original equation becomes:

(e^iπ/2)ⁿ = 1

Therefore:

e^inπ/2 = 1

A complex exponential is equal to 1 when its angle is an integral multiple of 2π. Hence:

nπ/2 = 2kπ

where k is an integer. Simplifying:

n = 4k

The smallest positive value is obtained by taking k = 1. Therefore:

n = 4

Why n = 4 Is the Smallest Possible Answer

The word “smallest” is important in this question because infinitely many positive integers satisfy the equation. Every positive multiple of 4, such as 4, 8, 12, 16, and so on, makes iⁿ equal to 1.

However, the question specifically asks for the smallest positive non-zero integer. Among all positive multiples of 4, the smallest value is 4. Therefore, no value of n smaller than 4 can satisfy the given equation.

Final Answer

The smallest positive non-zero integer n for which ((1 + i)/(1 − i))ⁿ = 1 is 4.