The Complex Number System - Common Core: High School - Number and Quantity

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Question

Find the product of and its complex conjugate.

Answer

This question tests one's understanding on complex numbers and their corresponding conjugate. It is important to remember that when a complex number is multiplied by its conjugate, the imaginary numbers disappear. A complex number and its conjugate takes the form as follows:

$\(a+bi)(a-bi)=a^2+abi-abi-b^2i^2=a^2-b^2(-1)=a^2+b^2$

Note:

For the purpose of Common Core Standards, "find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers", falls within the Cluster A of "perform arithmetic operations with complex numbers" (CCSS.MATH.CONTENT.HSF.CN.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Using the formula above find the conjugate.

The formula states to keep the real portion and the imaginary portion but switch the sign between them.

Complex number:

Real portion:

Imaginary portion:

Conjugate:

Step 2: Set up the multiplication problem between the complex number and its conjugate.

Step 3: Multiply the two binomials together.

Distribute the first term in the first binomial to both terms in the second binomial. Then distribute the second term in the first binomial to each term in the second binomial.

$\ 2(2)+2(i)=4+2i \-i(2)+-i(i)=-2i-i^2$

From here combine the two new expressions together.

Since the product becomes,

$\4+2i-2i-(-1) \4+1 \5$

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Question

Find the conjugate of the complex number.

Answer

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Question

Find the conjugate of the complex number.

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Question

Find the conjugate of the complex number.

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Question

Find the conjugate of the complex number.

Answer

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Question

Find the product of the complex number and its conjugate.

Answer

$\left ( 4+2i \right )\left ( 4-2i \right )$

$16-8i+8i-4i^{2}$

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Question

Find the product of the complex number and its conjugate.

Answer

$\left ( 12-5i \right )\left ( 12+5i \right )$

$144+60i-60i-25i^{2}$

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Question

Find the product of the complex number and its conjugate.

Answer

$\left ( 1-i \right )\left ( 1+i \right )$

$1+i-i-i^{2}$

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Question

Find the product of the complex number and its conjugate.

Answer

$20i\cdot -20i$

$-400i^{2}$

$-400\cdot -1$

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Question

Simplify.

$\frac{2+i}{4-i}$

Answer

$\frac{2+i}{4-i}\cdot \frac{4+i}{4+i}$

$=\frac{\left ( 2+i \right )\left ( 4+i \right )}{\left ( 4-i \right )\left ( 4+i \right )}$

$=\frac{8+2i+4i+i^{2}}{16+4i-4i-i^{2}}$

$=\frac{8+6i-1}{16-1}$

$=\frac{7+6i}{15}$

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Question

Simplify.

$\frac{3+2i}{9-6i}$

Answer

$\frac{3+2i}{9-6i}\cdot \frac{9+6i}{9+6i}$

$=\frac{\left ( 3+2i \right )\left ( 9+6i \right )}{\left ( 9-6i \right )\left ( 9+6i \right )}$

$=\frac{27+18i+18i+12i^{2}}{81+54i-54i-36i^{2}}$

$=\frac{27+36i-12}{81+36}$

$=\frac{15+36i}{117}$

$=\frac{3\left ( 5+12i \right )}{3\left ( 39 \right )}$

$=\frac{5+12i}{39}$

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Question

Simplify.

$\frac{6-4i}{-3i}$

Answer

$\frac{6-4i}{-3i}\cdot \frac{3i}{3i}$

$\frac{18i-12i^{2}}{-9i^{2}}$

$\frac{18i+12}{9}$

$\frac{12+18i}{9}$

$\frac{3\left ( 4+6i \right )}{3\left ( 3 \right )}$

$\frac{4+6i}{3}$

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Question

What does the following equal?

Answer

This question tests ones ability to recognize that when adding and subtracting complex numbers, the imaginary portions act like a variable during algebraic operations. When adding or subtracting complex numbers recall that it is the combination of like terms. The liked terms of complex numbers include the real portion of the complex number (the 's), and the imaginary portion (the 's) given the expression,

$(a+bi)\pm(a_1+b_1i)$ .

For the purpose of Common Core Standards, "use the relation , and the commutative, associate, and distributive properties to add, subtract, and multiply complex numbers", falls within the Cluster A of "perform arithmetic operations with complex numbers" (CCSS.MATH.CONTENT.HSF.CN.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Distribute the negative to each term within the second set of parentheses.

Recall that subtracting a negative number means adding a positive.

Step 2: Identify like terms.

Real portion:

Imaginary portion:

Note: treat as a variable.

Step 3: Combine like terms into one expression.

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Question

Simplify.

$2i\cdot 3i$

Answer

$2i\cdot 3i=\left ( 2\cdot 3 \right )\left ( i\cdot i \right )$

$=6i^{2}$

$=6\cdot -1$

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Question

Simplify.

$5i\cdot 17i$

Answer

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Question

Simplify.

$-4i\cdot 3i$

Answer

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Question

Simplify.

$2i\cdot 10i\cdot -3i$

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Question

Simplify.

$\sqrt{-2}\cdot \sqrt{2}$

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Question

Simplify.

$\sqrt{-3}\cdot \sqrt{-8}$

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Question

Simplify.

$\sqrt{8}\cdot \sqrt{-2}$

Answer

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