Complex Numbers Quiz

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Questions and Answers

What is the value of the real part of the complex number $z = 5 - 4i$?

  • 0
  • -4
  • -5
  • 5 (correct)

Given the complex number $z = 7 + 0i$, what is its imaginary part?

  • 0 (correct)
  • 7
  • -7
  • 1

Which of the following complex numbers has an imaginary part of -5?

  • $-7 - 5i$ (correct)
  • $4 + 5i$
  • $-5 + 0i$
  • $3 - 5i$

What can be concluded about the complex number $z = -3i$?

<p>It has a real part of 0. (C)</p> Signup and view all the answers

Which of the following is true for the complex number $z = 0$?

<p>It has a real part of 0 and an imaginary part of 0. (A)</p> Signup and view all the answers

What is the result of the expression $(1+i²)+(1+i²)²$?

<p>3 (B)</p> Signup and view all the answers

In the equation $(x-2) +i(y-1)= (1-3i)(1+i)$, what are the values of x and y respectively?

<p>3, 2 (A)</p> Signup and view all the answers

What does the equation $ar{z}i = x-iy$ represent in terms of complex numbers?

<p>The imaginary unit multiplied by a complex conjugate (A)</p> Signup and view all the answers

What is the first step in solving the problem $(x-2) +i(y-1)= (1-3i)(1+i)$?

<p>Multiply out the right side to simplify (D)</p> Signup and view all the answers

Which operation is crucial when manipulating complex numbers involving variables?

<p>Separating real and imaginary parts (D)</p> Signup and view all the answers

What is the result of simplifying $( rac{1}{2} + i )^2 + ( rac{1}{2} - i )^2$?

<p>$- rac{3}{2}$ (C)</p> Signup and view all the answers

What is the multiplicative inverse of $z = 3 + 4i$?

<p>$ rac{3}{25} - rac{4}{25}i$ (D)</p> Signup and view all the answers

What is the conjugate of the complex number $2 - i^2$?

<p>$1$ (C)</p> Signup and view all the answers

What is the final result of $(2 - i)^2$?

<p>$3 - 4i$ (B)</p> Signup and view all the answers

Which equation is used to equate the real and imaginary parts when finding the multiplicative inverse?

<p>$z^2 = a^2 - b^2 + 2abi$ (C)</p> Signup and view all the answers

What is the denominator when finding the multiplicative inverse of $z = 3 + 4i$?

<p>$9 + 16$ (C)</p> Signup and view all the answers

From the derived equations, what are the values of $a^2$ and $b^2$ when $a^2 - b^2 = 5$ and $2ab = 12$?

<p>$a^2 = 9, b^2 = 4$ (A)</p> Signup and view all the answers

What is the result when applying the equation $(a^2)^2 + (b^2)^2 = (a^2 - b^2)^2 + (2ab)^2$?

<p>$a^4 + b^4 = 169$ (C)</p> Signup and view all the answers

What can be deduced from $2ab = 12$ if $a = 3$?

<p>$b = 2$ (C)</p> Signup and view all the answers

What is the process to determine the square root of a number?

<p>Approximate using a calculator or estimation techniques. (C)</p> Signup and view all the answers

What is the definition of a natural number?

<p>A positive whole number excluding zero. (D)</p> Signup and view all the answers

Which method might be used to find a natural number satisfying an inequality?

<p>Trial and error with various integers. (B)</p> Signup and view all the answers

Which of the following is a step in calculating a square root manually?

<p>Estimate nearby perfect squares. (A)</p> Signup and view all the answers

If you were to find the smallest natural number satisfying the inequality x > 0, what would it be?

<p>1 (B)</p> Signup and view all the answers

Flashcards

Complex Number

A number that can be expressed in the form x + yi, where x and y are real numbers and i is the imaginary unit (i² = -1).

Real Part of a Complex Number

The real part of a complex number z = x + yi is the coefficient of the term x. It is denoted as Re(z).

Imaginary Part of a Complex Number

The imaginary part of a complex number z = x + yi is the coefficient of the term y. It is denoted as Im(z).

Imaginary Unit (i)

The imaginary unit i is a number that, when squared, equals -1. It is represented as i² = -1.

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Real Number (as a complex number)

A complex number where the imaginary part is zero. It can be written as z = x + 0i or simply z = x.

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Modulus of a Complex Number

The result of multiplying a complex number by its conjugate (obtained by changing the sign of the imaginary part).

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Conjugate of a Complex Number

The complex number obtained by changing the sign of the imaginary part of a given complex number.

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Purely Imaginary Number

A complex number with a real part equal to 0. It can be written as bi.

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Multiplication of Complex Numbers

The process of multiplying two complex numbers together, using the distributive property and the fact that i² = -1.

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Multiplicative Inverse of a Complex Number

The multiplicative inverse of a complex number z is another complex number z⁻¹ such that their product is 1. This is similar to finding the reciprocal of a real number.

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Formula for Multiplicative Inverse

To find the multiplicative inverse of a complex number z = a + bi, you can use the formula: z⁻¹ = (a - bi) / (a² + b²). This effectively involves conjugating the complex number in the numerator and squaring the magnitude in the denominator.

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Magnitude of a Complex Number

The magnitude of a complex number z = a + bi is the distance from the origin to the point representing z in the complex plane. It's calculated as |z| = √(a² + b²).

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Squaring a Complex Number

Squaring a complex number z = a + bi involves applying the distributive property: z² = (a + bi)² = a² + 2abi + b²i² = (a² - b²) + (2ab)i. This simplifies to a new complex number with real and imaginary components.

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Simplifying Complex Number Expressions

To simplify complex number calculations, use the properties of complex numbers and the rules of algebra. Remember that i² = -1. By combining real and imaginary terms, you can simplify the expression.

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Square Root

A number that, when multiplied by itself, equals a given number.

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Natural Number

A number that can be written without a fractional component, including zero and positive whole numbers.

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Square Root Calculation

A method used to calculate the square root of a number. Often involves breaking down the radicand into smaller factors and finding the square root of each factor.

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Equation

A mathematical expression that represents the relationship between two or more values.

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Inequality

An expression that uses one or more variables to represent a relationship between quantities.

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