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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. The square root of -1.
Imaginary Unit
i^1
z - z*
a real number: (a + bi)(a - bi) = a² + b²

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

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3. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Polar Coordinates - z?¹
Absolute Value of a Complex Number
Polar Coordinates - Division
Complex Conjugate

4. The field of all rational and irrational numbers.
How to solve (2i+3)/(9-i)
Real Numbers
Complex Numbers: Add & subtract
imaginary

5. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Numbers: Add & subtract
How to add and subtract complex numbers (2-3i)-(4+6i)
transcendental
|z| = mod(z)

6. 3
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - r
z1 / z2
i^3

7. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate pairs
multiplying complex numbers
i^0

8. y / r
|z-w|
Polar Coordinates - sin?
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^2

9. z1z2* / |z2|²
Rational Number
De Moivre's Theorem
has a solution.
z1 / z2

10. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
zz*
i²
i^1

11. (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
Field
How to multiply complex nubers(2+i)(2i-3)
i^2 = -1

12. When two complex numbers are added together.
Complex Multiplication
How to add and subtract complex numbers (2-3i)-(4+6i)
(cos? +isin?)n
Complex Addition

13. All numbers
complex
i^2 = -1
the complex numbers
Complex Addition

14. Any number not rational
irrational
cos z
Absolute Value of a Complex Number
standard form of complex numbers

15. 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
i^1
Complex Numbers: Add & subtract
z1 ^ (z2)
x-axis in the complex plane

16. (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
Every complex number has the 'Standard Form': a + bi for some real a and b.
How to solve (2i+3)/(9-i)
subtracting complex numbers
v(-1)

17. 2nd. Rule of Complex Arithmetic

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18. In this amazing number field every algebraic equation in z with complex coefficients
subtracting complex numbers
The Complex Numbers
has a solution.
How to solve (2i+3)/(9-i)

19. All the powers of i can be written as
a real number: (a + bi)(a - bi) = a² + b²
Roots of Unity
four different numbers: i - -i - 1 - and -1.
Irrational Number

20. Real and imaginary numbers
complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
four different numbers: i - -i - 1 - and -1.
How to multiply complex nubers(2+i)(2i-3)

21. 1
Square Root
'i'
i^0
Complex Multiplication

22. 1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.
multiply the numerator and the denominator by the complex conjugate of the denominator.
i²

23. A² + b² - real and non negative
De Moivre's Theorem
adding complex numbers
zz*
Argand diagram

24. Numbers on a numberline
i^0
has a solution.
integers
Imaginary Numbers

25. 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
subtracting complex numbers
rational
Complex Subtraction

26. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Euler's Formula
x-axis in the complex plane
Real Numbers

27. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
-1
Square Root
(a + c) + ( b + d)i

28. A number that cannot be expressed as a fraction for any integer.
Imaginary Unit
Every complex number has the 'Standard Form': a + bi for some real a and b.
Irrational Number
Liouville's Theorem -

29. ½(e^(iz) + e^(-iz))
Euler Formula
cos z
Complex Number Formula
multiplying complex numbers

30. Imaginary number

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31. When two complex numbers are multipiled together.
Complex numbers are points in the plane
interchangeable
cosh²y - sinh²y
Complex Multiplication

32. 4th. Rule of Complex Arithmetic
i^2 = -1
Integers
i^3
(a + bi) = (c + bi) = (a + c) + ( b + d)i

33. Every complex number has the 'Standard Form':
z1 / z2
-1
Real Numbers
a + bi for some real a and b.

34. I^2 =
Complex Exponentiation
z1 ^ (z2)
e^(ln z)
-1

35. 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.

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36. A+bi
i^1
the complex numbers
Complex Number Formula
|z| = mod(z)

37. (a + bi)(c + bi) =
Any polynomial O(xn) - (n > 0)
real
Polar Coordinates - z?¹
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

38. A plot of complex numbers as points.
Complex Exponentiation
Polar Coordinates - Division
Rational Number
Argand diagram

39. I
cos z
four different numbers: i - -i - 1 - and -1.
i^1
z1 ^ (z2)

40. 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
e^(ln z)
Complex Multiplication
Liouville's Theorem -

41. Root negative - has letter i
the complex numbers
Polar Coordinates - Multiplication by i
imaginary
Imaginary number

42. To simplify the square root of a negative number
sin z
interchangeable
Complex Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

43. Divide moduli and subtract arguments
Absolute Value of a Complex Number
Complex numbers are points in the plane
Rational Number
Polar Coordinates - Division

44. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
the distance from z to the origin in the complex plane
Any polynomial O(xn) - (n > 0)
subtracting complex numbers

45. A + bi
standard form of complex numbers
sin iy
the distance from z to the origin in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.

46. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
We say that c+di and c-di are complex conjugates.
Field
Complex Multiplication

47. V(zz*) = v(a² + b²)
z + z*
How to add and subtract complex numbers (2-3i)-(4+6i)
|z| = mod(z)
Polar Coordinates - Division

48. To simplify a complex fraction
Liouville's Theorem -
Polar Coordinates - Multiplication
multiply the numerator and the denominator by the complex conjugate of the denominator.
a + bi for some real a and b.

49. E^(ln r) e^(i?) e^(2pin)
Complex Exponentiation
complex numbers
0 if and only if a = b = 0
e^(ln z)

50. 1
Polar Coordinates - Multiplication by i
(a + c) + ( b + d)i
Absolute Value of a Complex Number
i^2