(7-2i).jpeg "(7+2i)(7-2i)")
Exploring Complex Number Multiplication: (7+2i)(7-2i)
This article delves into the multiplication of complex numbers, specifically the product of (7+2i) and (7-2i).
Understanding Complex Numbers
Complex numbers are numbers that extend the real number system by introducing the imaginary unit, denoted by 'i', where i² = -1. They take the form a + bi, where 'a' and 'b' are real numbers.
Multiplication of Complex Numbers
The multiplication of complex numbers follows the distributive property, similar to multiplying binomials in algebra. We multiply each term in the first complex number by each term in the second complex number:
(a + bi)(c + di) = ac + adi + bci + bdi²
Since i² = -1, the expression simplifies to:
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Applying the Formula
Let's apply this formula to our example:
(7 + 2i)(7 - 2i) = (7 * 7 - 2 * -2) + (7 * -2 + 2 * 7)i
Simplifying the expression:
(7 + 2i)(7 - 2i) = (49 + 4) + ( -14 + 14)i
Therefore:
(7 + 2i)(7 - 2i) = 53
Interesting Observation
Notice that the imaginary terms cancel out in this case, leaving us with a purely real number result. This is because (7+2i) and (7-2i) are complex conjugates of each other. Complex conjugates are formed by changing the sign of the imaginary part. The product of a complex number and its conjugate always results in a real number.
Summary
The multiplication of (7+2i) and (7-2i) showcases the process of multiplying complex numbers. We see that the product of a complex number and its conjugate results in a purely real number. Understanding these concepts is crucial for working with complex numbers in various mathematical and scientific applications.