Textbook Question
Find each sum or difference. Write answers in standard form. (3+2i) + (9+3i)
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1. Equations & Inequalities
Complex Numbers
1:55 minutesProblem 58a
Textbook Question
Find each product. Write answers in standard form. (1+3i)(2-5i)
Verified step by step guidance1
Recall that to multiply two complex numbers, we use the distributive property (also known as FOIL for binomials): multiply each term in the first complex number by each term in the second complex number.
Write the expression explicitly: \((1 + 3i)(2 - 5i) = 1 \cdot 2 + 1 \cdot (-5i) + 3i \cdot 2 + 3i \cdot (-5i)\).
Calculate each product separately: \(1 \cdot 2 = 2\), \(1 \cdot (-5i) = -5i\), \(3i \cdot 2 = 6i\), and \(3i \cdot (-5i) = -15i^2\).
Combine the like terms: the real parts \$2\( and \)-15i^2\(, and the imaginary parts \)-5i\( and \)6i\(. Remember that \)i^2 = -1\(, so replace \)i^2\( with \)-1$ in the expression.
Simplify the expression by calculating \(-15i^2\) as \(-15(-1)\), then add the real parts and the imaginary parts separately to write the final product in standard form \(a + bi\).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers
Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. They extend the real number system and are used to represent quantities involving the square root of negative numbers.
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Dividing Complex Numbers
Multiplication of Complex Numbers
To multiply complex numbers, use the distributive property (FOIL method) to expand the product, then combine like terms. Remember to apply i² = -1 to simplify terms involving i² into real numbers.
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Standard Form of Complex Numbers
The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. After multiplication, simplify the expression so it clearly separates the real and imaginary components in this form.
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