How to graph complex numbers

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SSAT Upper Level Quantitative › How to graph complex numbers

Questions 1 - 10

Multiply the complex conjugate of 8 by . What is the result?

None of the other responses gives the correct product.

Explanation

The complex conjugate of a complex number is . Since , its complex conjugate is itself. Multiply this by :

$8 \cdot 5i = 40 i$

Give the product of and its complex conjugate.

The correct answer is not given among the other responses.

Explanation

The product of a complex number and its conjugate is

$\left (a+bi \right )\left ( a-bi\right ) = a^{2}+b^{2}$

which will always be a real number. Therefore, all four given choices, all of which are imaginary, can be immediately eliminated. The correct response is that the correct answer is not given among the other responses.

Multiply the complex conjugate of by . What is the result?

None of the other responses gives the correct product.

Explanation

The complex conjugate of a complex number is . Since , its complex conjugate is .

Multiply this by :

Recall that by definition .

$-7i \cdot 3i = -7 \cdot 3 \cdot i \cdot i = -21 (-1) = 21$

Subtract from its complex conjugate. What is the result?

Explanation

The complex conjugate of a complex number is , so the complex conjugate of is . Subtract the former from the latter:

$\left ( 7+9i\right ) - \left ( 7-9i\right ) = 7 + 9i - 7 + 9i = 7 - 7 +9i + 9i = 18i$

Multiply the complex conjugate of by . What is the result?

Explanation

The complex conjugate of a complex number is , so the complex conjugate of is . Multiply this by :

$(3-7i) \cdot 4i = 3 \cdot 4i - 7 i \cdot 4i$

$= 12i - 7 \cdot 4 \cdot i \cdot i$

$= 12i -28 \cdot (-1)$

Multiply:

Explanation

This is a product of an imaginary number and its complex conjugate, so it can be evaluated using this formula:

$\left (a + bi \right )\left (a -bi \right ) = a^{2} + b ^{2}$

$\left (1.1 + 0.6i \right )\left (1.1 -0.6 i \right ) = 1.1^{2} + 0.6 ^{2} = 1.21 + 0.36 = 1.57$

Multiply the following complex numbers:

$\left (5 + i \right )\left (5 + 2i \right )$

Explanation

FOIL the product out:

$\left (5 + i \right )\left (5 + 2i \right )$

To FOIL multiply the first terms from each binomial together, multiply the outer terms of both terms together, multiply the inner terms from both binomials together, and finally multiply the last terms from each binomial together.

$=5 \cdot 5 + 5 \cdot 2i + 5 \cdot i + i\cdot 2i$