(3-7i).jpeg "(3+7i)(3-7i)")
Multiplying Complex Numbers: (3 + 7i)(3 - 7i)
This article will guide you through the process of multiplying the complex numbers (3 + 7i) and (3 - 7i). We will demonstrate the use of the FOIL method, a common technique for multiplying binomials, and show how the result simplifies to a real number.
Understanding Complex Numbers
Complex numbers are 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).
Using the FOIL Method
FOIL stands for First, Outer, Inner, Last, a mnemonic device for remembering the steps involved in multiplying binomials:
- First: Multiply the first terms of each binomial: 3 * 3 = 9
- Outer: Multiply the outer terms: 3 * -7i = -21i
- Inner: Multiply the inner terms: 7i * 3 = 21i
- Last: Multiply the last terms: 7i * -7i = -49i²
Therefore, (3 + 7i)(3 - 7i) = 9 - 21i + 21i - 49i²
Simplifying the Expression
Notice that the middle terms, -21i and +21i, cancel each other out. Since i² = -1, we can substitute:
9 - 49i² = 9 - 49(-1) = 9 + 49 = 58
Conclusion
The product of (3 + 7i) and (3 - 7i) simplifies to the real number 58. This result highlights a key property of complex numbers: multiplying a complex number by its conjugate (formed by changing the sign of the imaginary part) always results in a real number.