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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 / r
Complex Exponentiation
Polar Coordinates - cos?
We say that c+di and c-di are complex conjugates.
Complex Addition

2. 1st. Rule of Complex Arithmetic
multiplying complex numbers
subtracting complex numbers
natural
i^2 = -1

3. Any number not rational
irrational
Euler's Formula
Polar Coordinates - r
z1 / z2

4. 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
i²
subtracting complex numbers
the complex numbers
Absolute Value of a Complex Number

5. 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
a real number: (a + bi)(a - bi) = a² + b²
cosh²y - sinh²y
Complex Numbers: Add & subtract
ln z

6. R^2 = x
a + bi for some real a and b.
Square Root
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?

7. Multiply moduli and add arguments
For real a and b - a + bi = 0 if and only if a = b = 0
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication
cosh²y - sinh²y

8. 3
i^3
transcendental
subtracting complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

9. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
a + bi for some real a and b.
(cos? +isin?)n
How to solve (2i+3)/(9-i)
The Complex Numbers

10. All numbers
the distance from z to the origin in the complex plane
conjugate
Complex Addition
complex

11. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to add and subtract complex numbers (2-3i)-(4+6i)
a real number: (a + bi)(a - bi) = a² + b²
Absolute Value of a Complex Number

12. 4th. Rule of Complex Arithmetic
0 if and only if a = b = 0
Euler Formula
(a + bi) = (c + bi) = (a + c) + ( b + d)i
sin iy

13. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
complex numbers
conjugate
Rational Number
How to solve (2i+3)/(9-i)

14. (a + bi)(c + bi) =
i^0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate pairs
We say that c+di and c-di are complex conjugates.

15. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Subfield
v(-1)
z + z*

16. Have radical
radicals
cos z
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication

17. 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
adding complex numbers
Square Root
Polar Coordinates - sin?
point of inflection

18. 2nd. Rule of Complex Arithmetic

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19. 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
How to multiply complex nubers(2+i)(2i-3)
Real and Imaginary Parts
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Multiplication by i

20. A complex number and its conjugate
ln z
rational
conjugate pairs
z1 / z2

21. 1
x-axis in the complex plane
i^4
a + bi for some real a and b.
i^2

22. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos iy
Absolute Value of a Complex Number
Polar Coordinates - z

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

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24. Equivalent to an Imaginary Unit.
interchangeable
-1
Argand diagram
Imaginary number

25. A + bi
Affix
standard form of complex numbers
real
For real a and b - a + bi = 0 if and only if a = b = 0

26. When two complex numbers are added together.
ln z
Complex Number
Complex Addition
e^(ln z)

27. A+bi
'i'
Complex Number Formula
Any polynomial O(xn) - (n > 0)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

28. Numbers on a numberline
integers
Euler's Formula
Euler Formula
Real Numbers

29. 2ib
z - z*
Complex Numbers: Add & subtract
Complex numbers are points in the plane
Complex Conjugate

30. (a + bi) = (c + bi) =
Imaginary Unit
(a + c) + ( b + d)i
Absolute Value of a Complex Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

31. E ^ (z2 ln z1)
Roots of Unity
standard form of complex numbers
z1 ^ (z2)
subtracting complex numbers

32. When two complex numbers are subtracted from one another.
(cos? +isin?)n
the vector (a -b)
Affix
Complex Subtraction

33. 5th. Rule of Complex Arithmetic
integers
-1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z-w|

34. 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
cosh²y - sinh²y
natural
i^4

35. I^2 =
-1
z + z*
x-axis in the complex plane
the complex numbers

36. A complex number may be taken to the power of another complex number.
i^3
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Exponentiation
Rational Number

37. Rotates anticlockwise by p/2
0 if and only if a = b = 0
Polar Coordinates - r
z - z*
Polar Coordinates - Multiplication by i

38. No i
standard form of complex numbers
Polar Coordinates - z
i^0
real

39. V(zz*) = v(a² + b²)
natural
Square Root
|z| = mod(z)
i^2 = -1

40. Imaginary number

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41. The product of an imaginary number and its conjugate is
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos iy
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a real number: (a + bi)(a - bi) = a² + b²

42. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
Polar Coordinates - Division
Complex Addition
Rules of Complex Arithmetic
Complex numbers are points in the plane

43. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
real
non-integers
Complex Addition

44. A number that can be expressed as a fraction p/q where q is not equal to 0.
has a solution.
i^2 = -1
Rational Number
Complex Exponentiation

45. Not on the numberline
non-integers
Liouville's Theorem -
Absolute Value of a Complex Number
Every complex number has the 'Standard Form': a + bi for some real a and b.

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

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47. xpressions such as ``the complex number z'' - and ``the point z'' are now
'i'
interchangeable
Absolute Value of a Complex Number
Complex Multiplication

48. Like pi
transcendental
How to multiply complex nubers(2+i)(2i-3)
the distance from z to the origin in the complex plane
integers

49. 1
cosh²y - sinh²y
How to add and subtract complex numbers (2-3i)-(4+6i)
Any polynomial O(xn) - (n > 0)
Imaginary Numbers

50. For real a and b - a + bi =
0 if and only if a = b = 0
How to find any Power
v(-1)
(a + c) + ( b + d)i