Which expression is equivalent to i
233
?
A. 1
B. –1
C. i
D. –i
The Correct Answer and Explanation is :
To find the expression equivalent to ( i^{233} ), we first need to understand the behavior of the powers of ( i ), the imaginary unit defined as ( i = \sqrt{-1} ). The powers of ( i ) cycle through a specific pattern:
- ( i^1 = i )
- ( i^2 = -1 )
- ( i^3 = -i )
- ( i^4 = 1 )
After ( i^4 ), the powers repeat every four steps:
- ( i^5 = i )
- ( i^6 = -1 )
- ( i^7 = -i )
- ( i^8 = 1 )
This cyclic pattern shows that the value of ( i^n ) depends on the remainder of ( n ) when divided by 4.
To find ( i^{233} ), we need to determine the remainder of ( 233 \div 4 ):
- Calculate ( 233 \div 4 ):
- ( 233 \div 4 = 58.25 ), which means ( 4 \times 58 = 232 ).
- The remainder is ( 233 – 232 = 1 ).
This tells us that ( 233 \equiv 1 \mod 4 ). According to the established cycle, this means:
[
i^{233} = i^{1} = i
]
Thus, the expression ( i^{233} ) simplifies to ( i ).
Conclusion
The correct answer is C. i.
Explanation
Understanding the powers of ( i ) is essential in complex number operations, particularly in fields like engineering and physics where complex numbers are frequently utilized. By recognizing the repetitive nature of ( i )’s powers, we can simplify expressions involving higher powers without resorting to lengthy calculations. The modulo operation provides a quick method to find the equivalent power. In this case, the calculation reveals that any large exponent can be reduced back to one of the four fundamental values (1, ( i ), -1, or -( i )), thereby streamlining complex number manipulations significantly.