(2-7i).jpeg "(2+7i)(2-7i)")
Exploring Complex Number Multiplication: (2 + 7i)(2 - 7i)
This article will delve into the multiplication of complex numbers, specifically focusing on the expression (2 + 7i)(2 - 7i). We'll explore the process and the significance of the result.
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1 (i² = -1).
Multiplication of Complex Numbers
To multiply complex numbers, we use the distributive property, similar to how we multiply binomials.
Let's multiply (2 + 7i)(2 - 7i):
(2 + 7i)(2 - 7i) = 2(2 - 7i) + 7i(2 - 7i)
Expanding this, we get:
= 4 - 14i + 14i - 49i²
Since i² = -1, we can substitute:
= 4 - 49(-1)
= 4 + 49
= 53
The Significance of the Result
The result of multiplying (2 + 7i) by its complex conjugate (2 - 7i) is a real number, 53. This is a general principle: The product of a complex number and its conjugate is always a real number.
Complex Conjugate: The complex conjugate of a complex number a + bi is a - bi.
This property is used in various applications in mathematics, particularly in:
- Simplifying complex fractions: Multiplying both numerator and denominator by the conjugate of the denominator often eliminates complex numbers from the denominator.
- Solving equations: The conjugate can be used to manipulate equations and isolate variables involving complex numbers.
Conclusion
The multiplication of complex numbers, as exemplified by (2 + 7i)(2 - 7i), reveals an important relationship between a complex number and its conjugate. The product always results in a real number, demonstrating a fundamental principle in complex number theory with applications in various mathematical areas.