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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. x + iy = r(cos? + isin?) = re^(i?)
Rules of Complex Arithmetic
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - z
multiplying complex numbers

2. Where the curvature of the graph changes
Complex Exponentiation
0 if and only if a = b = 0
point of inflection
imaginary

3. In this amazing number field every algebraic equation in z with complex coefficients
conjugate
Complex numbers are points in the plane
Polar Coordinates - sin?
has a solution.

4. Real and imaginary numbers
Field
complex numbers
z - z*
Complex Addition

5. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
Real and Imaginary Parts
0 if and only if a = b = 0
irrational
natural

6. 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 Numbers: Add & subtract
sin iy
Euler Formula
conjugate

7. A plot of complex numbers as points.
Argand diagram
Rational Number
We say that c+di and c-di are complex conjugates.
i^2

8. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z + z*
Square Root
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

9. Starts at 1 - does not include 0
i²
natural
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Field

10. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
rational
real

11. A complex number and its conjugate
(cos? +isin?)n
0 if and only if a = b = 0
interchangeable
conjugate pairs

12. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Argand diagram
Integers
The Complex Numbers
Rules of Complex Arithmetic

13. 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 Addition
i^4
cos iy
How to find any Power

14. 2a
Roots of Unity
point of inflection
z + z*
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

15. All numbers
four different numbers: i - -i - 1 - and -1.
Argand diagram
Polar Coordinates - Division
complex

16. When two complex numbers are multipiled together.
Complex Multiplication
e^(ln z)
v(-1)
Affix

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

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18. Numbers on a numberline
How to add and subtract complex numbers (2-3i)-(4+6i)
i²
integers
'i'

19. 2ib
z - z*
radicals
For real a and b - a + bi = 0 if and only if a = b = 0
the vector (a -b)

20. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
Polar Coordinates - Division
Complex Number Formula
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - z

21. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate
radicals
e^(ln z)

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

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23. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Real and Imaginary Parts
z + z*
z1 / z2

24. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Integers
Field
point of inflection
imaginary

25. V(x² + y²) = |z|
Polar Coordinates - r
a real number: (a + bi)(a - bi) = a² + b²
Liouville's Theorem -
Polar Coordinates - Division

26. 1st. Rule of Complex Arithmetic
i^2 = -1
0 if and only if a = b = 0
i^3
zz*

27. I^2 =
How to find any Power
-1
Field
e^(ln z)

28. 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
i^4
subtracting complex numbers
Rules of Complex Arithmetic
Roots of Unity

29. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the complex numbers
a + bi for some real a and b.
Complex Exponentiation

30. A+bi
cos z
Complex Number Formula
rational
|z-w|

31. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Complex Multiplication
integers
natural
The Complex Numbers

32. 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.'
Polar Coordinates - cos?
Complex Addition
e^(ln z)
Complex Number

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

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34. Divide moduli and subtract arguments
Polar Coordinates - Division
Complex Number Formula
Polar Coordinates - Multiplication
Square Root

35. Written as fractions - terminating + repeating decimals
standard form of complex numbers
i^0
Affix
rational

36. I
How to add and subtract complex numbers (2-3i)-(4+6i)
Argand diagram
v(-1)
'i'

37. E ^ (z2 ln z1)
Subfield
z1 ^ (z2)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z1 / z2

38. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Subtraction
0 if and only if a = b = 0
Absolute Value of a Complex Number
z1 / z2

39. ? = -tan?
rational
Polar Coordinates - Arg(z*)
-1
0 if and only if a = b = 0

40. Cos n? + i sin n? (for all n integers)
We say that c+di and c-di are complex conjugates.
(cos? +isin?)n
Polar Coordinates - sin?
Complex Number

41. No i
the vector (a -b)
z1 ^ (z2)
real
-1

42. Any number not rational
'i'
multiplying complex numbers
Euler's Formula
irrational

43. 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
Rules of Complex Arithmetic
|z-w|
Complex Numbers: Multiply
Rational Number

44. (e^(-y) - e^(y)) / 2i = i sinh y
Imaginary Numbers
v(-1)
sin iy
Complex Subtraction

45. Rotates anticlockwise by p/2
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Multiplication by i
Real Numbers
Complex Conjugate

46. Imaginary number

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47. The reals are just the
x-axis in the complex plane
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable
The Complex Numbers

48. Derives z = a+bi
the complex numbers
cos z
Euler Formula
Affix

49. 5th. Rule of Complex Arithmetic
Complex Exponentiation
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(cos? +isin?)n
Rules of Complex Arithmetic

50. Have radical
a + bi for some real a and b.
Complex Conjugate
Polar Coordinates - z
radicals