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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.
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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. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

2. When two complex numbers are divided.
Complex Division
interchangeable
Imaginary Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

3. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

4. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
Complex Number Formula
Complex Numbers: Add & subtract
Complex Division
irrational

5. Has exactly n roots by the fundamental theorem of algebra
interchangeable
Any polynomial O(xn) - (n > 0)
Polar Coordinates - sin?
De Moivre's Theorem

6. V(x² + y²) = |z|
i²
-1
Polar Coordinates - r
0 if and only if a = b = 0

7. Derives z = a+bi
rational
Real and Imaginary Parts
Irrational Number
Euler Formula

8. 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
i^1
complex numbers
Rules of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0

9. Cos n? + i sin n? (for all n integers)
Polar Coordinates - Multiplication by i
Polar Coordinates - Multiplication
(cos? +isin?)n
Complex Numbers: Add & subtract

10. A + bi
standard form of complex numbers
Imaginary Numbers
ln z
Polar Coordinates - z?¹

11. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - Division
i^2
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z

12. Equivalent to an Imaginary Unit.
For real a and b - a + bi = 0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z + z*
Imaginary number

13. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
Imaginary Unit
subtracting complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - z

14. 1
zz*
Integers
standard form of complex numbers
i^2

15. Starts at 1 - does not include 0
natural
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Division
Any polynomial O(xn) - (n > 0)

16. When two complex numbers are multipiled together.
We say that c+di and c-di are complex conjugates.
Complex Multiplication
(cos? +isin?)n
How to multiply complex nubers(2+i)(2i-3)

17. 1
i^0
|z| = mod(z)
the distance from z to the origin in the complex plane
integers

18. Multiply moduli and add arguments
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

19. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Polar Coordinates - sin?
Polar Coordinates - Multiplication by i
subtracting complex numbers
multiplying complex numbers

20. V(zz*) = v(a² + b²)
x-axis in the complex plane
How to find any Power
|z| = mod(z)
i^2

21. A plot of complex numbers as points.
i^2
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Addition
Argand diagram

22. The product of an imaginary number and its conjugate is
integers
Complex Number Formula
standard form of complex numbers
a real number: (a + bi)(a - bi) = a² + b²

23. 2ib
v(-1)
Polar Coordinates - cos?
Field
z - z*

24. When two complex numbers are subtracted from one another.
four different numbers: i - -i - 1 - and -1.
Rational Number
cosh²y - sinh²y
Complex Subtraction

25. All the powers of i can be written as
Complex Multiplication
How to add and subtract complex numbers (2-3i)-(4+6i)
multiply the numerator and the denominator by the complex conjugate of the denominator.
four different numbers: i - -i - 1 - and -1.

26. In this amazing number field every algebraic equation in z with complex coefficients
transcendental
How to solve (2i+3)/(9-i)
Imaginary number
has a solution.

27. I
i^1
i^4
The Complex Numbers
Real and Imaginary Parts

28. 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.
Complex Subtraction
the complex numbers
'i'
How to find any Power

29. ? = -tan?
|z| = mod(z)
Field
Polar Coordinates - Arg(z*)
z1 / z2

30. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Argand diagram
Roots of Unity
(cos? +isin?)n

31. Root negative - has letter i
How to solve (2i+3)/(9-i)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
adding complex numbers
imaginary

32. Have radical
Real Numbers
radicals
the distance from z to the origin in the complex plane
'i'

33. A number that cannot be expressed as a fraction for any integer.
Affix
Absolute Value of a Complex Number
Irrational Number
cosh²y - sinh²y

34. A+bi
Complex Number Formula
real
z + z*
the distance from z to the origin in the complex plane

35. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
Polar Coordinates - z
the vector (a -b)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

36. Divide moduli and subtract arguments
standard form of complex numbers
i^2
Polar Coordinates - Division
real

37. Where the curvature of the graph changes
Polar Coordinates - Division
the vector (a -b)
point of inflection
Polar Coordinates - z

38. 1st. Rule of Complex Arithmetic
e^(ln z)
De Moivre's Theorem
i^2 = -1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

39. 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.'
Complex Number
irrational
non-integers
Subfield

40. Not on the numberline
the distance from z to the origin in the complex plane
Complex Numbers: Multiply
Square Root
non-integers

41. To simplify the square root of a negative number
adding complex numbers
Polar Coordinates - Division
standard form of complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

42. Like pi
transcendental
'i'
Polar Coordinates - r
Imaginary Numbers

43. I = imaginary unit - i² = -1 or i = v-1
e^(ln z)
Euler Formula
Rational Number
Imaginary Numbers

44. A² + b² - real and non negative
z1 / z2
Imaginary Numbers
Subfield
zz*

45. 5th. Rule of Complex Arithmetic
Irrational Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
zz*
Complex Numbers: Multiply

46. Imaginary number

47. 1
i²
Polar Coordinates - Division
i^1
cosh²y - sinh²y

48. 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
a real number: (a + bi)(a - bi) = a² + b²
How to add and subtract complex numbers (2-3i)-(4+6i)
adding complex numbers
z1 / z2

49. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Real and Imaginary Parts
Real Numbers
i^0
The Complex Numbers

50. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Rational Number
rational
a + bi for some real a and b.
Roots of Unity