(2-2i).jpeg "(2+2i)(2-2i)")
Exploring the Product of Complex Conjugates: (2+2i)(2-2i)
In the realm of complex numbers, we often encounter scenarios where we need to multiply complex numbers together. One such scenario involves multiplying a complex number by its conjugate. This article delves into the product of (2+2i) and its conjugate (2-2i), exploring its implications and significance.
Understanding Complex Conjugates
A complex number is of the form a + bi, where a and b are real numbers, and i is the imaginary unit (i² = -1). The conjugate of a complex number is formed by simply changing the sign of its imaginary part. Therefore, the conjugate of a + bi is a - bi.
Multiplying (2+2i) and (2-2i)
Let's proceed with the multiplication:
(2+2i)(2-2i) = 2(2-2i) + 2i(2-2i) = 4 - 4i + 4i - 4i²
Since i² = -1, we can substitute:
= 4 - 4(-1) = 4 + 4 = 8
Significance of the Result
The result of multiplying a complex number by its conjugate is always a real number. This is a key property of complex conjugates and has significant applications:
- Finding the Modulus: The modulus of a complex number (a + bi) is its distance from the origin in the complex plane and is calculated as √(a² + b²). Interestingly, the product of a complex number and its conjugate equals the square of its modulus: (a + bi)(a - bi) = a² + b². In our example, the modulus of (2+2i) is √(2² + 2²) = √8, and indeed, (2+2i)(2-2i) = 8.
- Simplifying Complex Fractions: When we have a complex number in the denominator of a fraction, we can multiply both the numerator and denominator by the conjugate of the denominator to simplify the expression and eliminate the imaginary unit from the denominator.
Conclusion
The product of (2+2i) and its conjugate (2-2i) results in the real number 8. This demonstrates a crucial property of complex conjugates where their multiplication always yields a real number. This property is invaluable in various applications, including determining the modulus of complex numbers and simplifying complex fractions.