How to graph complex numbers - SSAT Upper Level Quantitative

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Question

Define an operation $\wedge$ as follows:

For all complex numbers ,

$a \wedge b = a \div (b+ 2i)$

Evaluate $8 \wedge 4$

Answer

$a \wedge b = a \div (b+ 2i) = \frac{a}{b+2i}$

$8 \wedge 4 = \frac{8}{4+2i}$

Multiply both numerator and denominator by the conjugate of the denominator, , to rationalize the denominator:

$8 \wedge 4 = \frac{8(4-2i)}{(4+2i)(4-2i)}$

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

$= \frac{32-16i}{16+4}$

$= \frac{32-16i}{20}$

$= \frac{32 }{20} - \frac{ 16}{20}i$

$= \frac{8}{5} -\frac{ 4}{5}i$

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Question

Raise to the power of 4.

Answer

$(4i)^{4} = 4^{4} \cdot i ^{4} = 256 \cdot 1 = 256$

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Question

Multiply the following complex numbers:

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

Answer

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.

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

$=25 + 10i + 5 i + 2i^{2}$

Recall that i is an imaginary number and by definition . Substituting this into the function is as follows.

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Question

Multiply:

$\left (7 - 2i \right )\left (7 - 3i \right )$

Answer

FOIL the product out:

$\left (7 - 2i \right )\left (7 - 3i \right )$

$= 7 \cdot 7 - 7 \cdot 3i - 7 \cdot 2i + 2i \cdot 3i$

$= 49 -21i - 14i + 6 i^{2}$

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Question

Simplify:

$\left ( 8 - 3i\right ) ^{2}$

Answer

Use the square of a binomial pattern to multiply this:

$\left ( 8 - 3i\right ) ^{2}$

$= 8 ^{2}- 2 \cdot8\cdot3i + \left ( 3i\right )^{2}$

$= 64- 48i + 3^{2}i^{2}$

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Question

Multiply:

Answer

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

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

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

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Question

Multiply:

$\left (5 - 3i \right )\left (5 +3i \right )$

Answer

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

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

$\left (5 - 3i \right )\left (5 +3i \right ) = 5 ^{2} + 3 ^{2}= 25 + 9 = 34$

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Question

Evaluate $(3i)^{-5}$ .

Answer

$(3i)^{-5} = \frac{1}{(3i)^{5} }= \frac{1}{3^{5}i^{5} } = \frac{1}{243 i^{4} \cdot i } = \frac{1}{243 \cdot 1 \cdot i }$

$= \frac{1}{243 i } \cdot = \frac{1\cdot i}{243 i \cdot i } = \frac{i}{243 (-1)} = - \frac{1}{243} i$

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Question

Define an operation $\wedge$ as follows:

For all complex numbers ,

$a \wedge b = a \div (b+ 3)$

Evaluate $7i \wedge 5i$

Answer

$a \wedge b = a \div (b+ 3) = \frac{a}{b+3}$

$7i \wedge 5i = \frac{7i }{5i+3} = \frac{7i }{3+ 5i}$

Multiply both numerator and denominator by the conjugate of the denominator, , to rationalize the denominator:

$7i \wedge 5i = \frac{7i (3-5i)}{(3+ 5i)(3-5i)}$

$= \frac{7i \cdot 3-7i \cdot 5i}{3^{2}+ 5^{2}}$

$= \frac{21i -35 \cdot i^{2}}{9+25}$

$= \frac{21i-35 (-1)}{34}$

$= \frac{21i+35}{34}$

$= \frac{ 35+21i}{34}$

$= \frac{ 35 }{34} + \frac{ 21}{34} i$

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Question

Define an operation as follows:

For all complex numbers ,

$ab = a ^{5}-b^{3}$

Evaluate $\left (2 i \right ) \left (3i \right )$

Answer

$ab = a ^{5}-b^{3}$

$\left (2 i \right ) \left (3i \right ) = \left (2 i \right ) ^{5}-\left (3i \right ) ^{3}$

$= 2^{5} i^{5} - 3^{3} i ^{3}$

$= 32\cdot i^{4} \cdot i - 27 \cdot (-i )$

$= 32\cdot 1 \cdot i +27 i$

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Question

Define an operation as follows:

For all complex numbers ,

Evaluate

Answer

$= 4+7i + 3i \cdot 3- 3i \cdot 8i$

$= 4+7i + 9i - 24i ^{2}$

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Question

Give the number which, when multiplied by , yields the same result as if it were increased by 6.

Answer

Let be the number in question. The statement "\[a number\] multiplied by yields the same result as if it were increased by 6" can be written as

We can solve this for as follows:

$N(-1+i) \div (-1+i) = 6 \div (-1+i)$

$N = \frac{6}{-1+i}$

Rationalize the denominator by multiplying both numerator and denominator by the conjugate of the denominator, which is :

$N = \frac{6(-1- i)}{(-1+i)(-1- i)}$

$= \frac{6(-1)- 6\cdot i}{(-1)^{2}+1^{2}}$

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

$= \frac{-6-6i}{2}$

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Question

Give the number which, when added to 20, yields the same result as if it were subtracted from .

Answer

Let be the number in question. The statement "\[a number\] added to 20 yields the same result as if it were aubtracted from " can be written as

Solve for :

$\frac{1}{2} \cdot 2N = \frac{1}{2} \cdot (-20 + 14 i)$

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Question

Define an operation $\bigstar$ as follows:

For all complex numbers ,

$\bigstar a = a^{2} + a$

Evaluate $\bigstar (6i)$ .

Answer

$\bigstar a = a^{2} + a$

$\bigstar (6i) = (6i)^{2} + 6i$

$= 6^{2} i^{2} + 6i$

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Question

Define an operation $\bigstar$ as follows:

For all complex numbers ,

$\bigstar a = a^{2}i + a$

Evaluate $\bigstar (7i)$ .

Answer

$\bigstar a = a^{2}i + a$

$\bigstar (7i) = (7i)^{2}i + 7i$

$= 7^{2}\cdot i^{2} \cdot i + 7i$

$= 49\cdot (-1) \cdot i + 7i$

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Question

Define an operation $\wedge$ as follows:

For all complex numbers ,

$a \wedge b = a \div (b+ 4)$ .

If $(8i )\wedge N = 16$ , evaluate .

Answer

$(8i )\wedge N = 16$ ,

by our definition, can be rewritten as

$8i \div (N+ 4) = 16$

or

$\frac{8i }{N+ 4}= 16$

Taking the reciprocal of both sides, then multiplying:

$\frac{N+ 4}{8i }= \frac{1}{16}$

$\frac{N+ 4}{8i } \cdot 8i = \frac{1}{16}\cdot 8i$

$N+ 4 = \frac{1}{2}i$

$N+ 4 - 4 = \frac{1}{2}i- 4$

$N =- 4 + \frac{1}{2}i$

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Question

Define an operation $\bigstar$ as follows:

For all complex numbers ,

$\bigstar a = a^{2}i - 4 a$

If $\bigstar N = 0$ and , evaluate .

Answer

If $\bigstar N = 0$ , then

$N^{2}i - 4 N= 0$ .

Distribute out to yield

Either or . However, we are given that , so

$\frac{N i }{i}= \frac{4}{i}$

$N= \frac{4}{i} = \frac{4 \cdot i}{i\cdot i}= \frac{4i}{-1} = -4i$

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Question

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

Answer

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

Multiply this by :

Recall that by definition .

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

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Question

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

Answer

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

$8 \cdot 5i = 40 i$

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Question

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

Answer

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

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

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

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

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