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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. When two complex numbers are divided.
four different numbers: i - -i - 1 - and -1.
Complex Division
How to multiply complex nubers(2+i)(2i-3)
Rational Number

2. Any number not rational
Polar Coordinates - Division
irrational
Rational Number
Argand diagram

3. When two complex numbers are multipiled together.
Every complex number has the 'Standard Form': a + bi for some real a and b.
x-axis in the complex plane
transcendental
Complex Multiplication

4. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
How to find any Power
ln z
rational
Complex Subtraction

5. When two complex numbers are subtracted from one another.
Complex Subtraction
multiplying complex numbers
the complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i

6. ? = -tan?
Polar Coordinates - Arg(z*)
Complex Numbers: Multiply
Subfield
Real and Imaginary Parts

7. When two complex numbers are added together.
natural
sin z
Complex Addition
|z| = mod(z)

8. x + iy = r(cos? + isin?) = re^(i?)
sin iy
Polar Coordinates - z
cosh²y - sinh²y
|z-w|

9. I
Imaginary Unit
z1 / z2
v(-1)
0 if and only if a = b = 0

10. 1
Polar Coordinates - r
i^0
Complex Conjugate
cos z

11. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Irrational Number
The Complex Numbers
Complex Numbers: Multiply
multiplying complex numbers

12. y / r
Euler's Formula
complex
Polar Coordinates - sin?
multiplying complex numbers

13. ½(e^(iz) + e^(-iz))
cos z
Any polynomial O(xn) - (n > 0)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

14. 5th. Rule of Complex Arithmetic
has a solution.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
Every complex number has the 'Standard Form': a + bi for some real a and b.

15. E^(ln r) e^(i?) e^(2pin)
De Moivre's Theorem
zz*
Integers
e^(ln z)

16. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

17. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
0 if and only if a = b = 0
Square Root
the complex numbers

18. A plot of complex numbers as points.
Polar Coordinates - Multiplication by i
ln z
x-axis in the complex plane
Argand diagram

19. For real a and b - a + bi =
Complex Conjugate
Imaginary Numbers
0 if and only if a = b = 0
|z-w|

20. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
standard form of complex numbers
The Complex Numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
sin z

21. No i
Complex Addition
real
z1 ^ (z2)
Real Numbers

22. All the powers of i can be written as
How to add and subtract complex numbers (2-3i)-(4+6i)
four different numbers: i - -i - 1 - and -1.
The Complex Numbers
Complex Addition

23. 2nd. Rule of Complex Arithmetic

24. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
conjugate
a real number: (a + bi)(a - bi) = a² + b²
irrational

25. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
How to add and subtract complex numbers (2-3i)-(4+6i)
zz*
-1

26. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
transcendental
i^0
Complex Multiplication

27. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc
Imaginary Unit
integers
i^1
Rules of Complex Arithmetic

28. A+bi
Complex Number Formula
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler's Formula
Affix

29. (a + bi) = (c + bi) =
Complex Subtraction
(a + c) + ( b + d)i
radicals
rational

30. We can also think of the point z= a+ ib as
Complex numbers are points in the plane
|z| = mod(z)
the distance from z to the origin in the complex plane
the vector (a -b)

31. Equivalent to an Imaginary Unit.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
conjugate
Imaginary number
Subfield

32. All numbers
multiplying complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex
Complex numbers are points in the plane

33. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
real
adding complex numbers
Polar Coordinates - Multiplication by i
e^(ln z)

34. Like pi
Complex Addition
Integers
transcendental
Complex Subtraction

35. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

36. 1
has a solution.
i^4
Integers
ln z

37. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
e^(ln z)
Complex Number
has a solution.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

38. A² + b² - real and non negative
zz*
|z| = mod(z)
z1 ^ (z2)
Complex Division

39. A complex number and its conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.
cos z
transcendental
conjugate pairs

40. Imaginary number

41. 1
Real and Imaginary Parts
Imaginary Unit
i^2
How to solve (2i+3)/(9-i)

42. Given (4-2i) the complex conjugate would be (4+2i)
Complex numbers are points in the plane
adding complex numbers
radicals
Complex Conjugate

43. V(zz*) = v(a² + b²)
Absolute Value of a Complex Number
the vector (a -b)
adding complex numbers
|z| = mod(z)

44. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
i^2
Polar Coordinates - z?¹
Liouville's Theorem -
Roots of Unity

45. (e^(iz) - e^(-iz)) / 2i
sin z
Polar Coordinates - Arg(z*)
standard form of complex numbers
Polar Coordinates - r

46. Starts at 1 - does not include 0
Irrational Number
i^2
multiplying complex numbers
natural

47. The field of all rational and irrational numbers.
conjugate
i^3
Euler Formula
Real Numbers

48. I = imaginary unit - i² = -1 or i = v-1
Rational Number
complex numbers
Liouville's Theorem -
Imaginary Numbers

49. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
cos z
irrational
Absolute Value of a Complex Number
Any polynomial O(xn) - (n > 0)

50. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Imaginary Unit
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate pairs
De Moivre's Theorem