(2-7i)-in-a%2Bbi-form.jpeg "(2+7i)(2-7i) In A+bi Form")
Expanding Complex Numbers: (2 + 7i)(2 - 7i)
This article explores the expansion of the complex number product (2 + 7i)(2 - 7i) into the standard a + bi form.
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as √-1. The key property of i is that i² = -1.
Expanding the Product
To expand the product (2 + 7i)(2 - 7i), we can use the distributive property:
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Multiply each term in the first set of parentheses by each term in the second set: (2 + 7i)(2 - 7i) = 2(2 - 7i) + 7i(2 - 7i)
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Distribute: = 4 - 14i + 14i - 49i²
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Simplify using i² = -1: = 4 - 49(-1)
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Combine real and imaginary terms: = 4 + 49 = 53
Therefore, (2 + 7i)(2 - 7i) = 53.
The Result in a + bi Form
The result 53 can be expressed in the standard a + bi form as 53 + 0i. This shows that the product of a complex number and its conjugate is always a real number.
The Conjugate of a Complex Number
The conjugate of a complex number a + bi is a - bi. Notice that in our example, the two factors (2 + 7i) and (2 - 7i) are conjugates of each other.
Key Takeaways
- The product of a complex number and its conjugate results in a real number.
- Expanding complex products involves using the distributive property and simplifying using the definition of i².
- Any real number can be expressed in the complex form a + 0i.