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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. For real a and b - a + bi =
a + bi for some real a and b.
0 if and only if a = b = 0
complex numbers
Complex Exponentiation

2. 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.'
adding complex numbers
Complex Number
rational
Roots of Unity

3. A complex number and its conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i
'i'
conjugate pairs
sin iy

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.
Euler Formula
Complex Subtraction
How to find any Power
The Complex Numbers

5. All numbers
standard form of complex numbers
irrational
complex
-1

6. 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
sin z
Real Numbers
adding complex numbers
standard form of complex numbers

7. 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
Polar Coordinates - Arg(z*)
Rules of Complex Arithmetic
De Moivre's Theorem
Any polynomial O(xn) - (n > 0)

8. Divide moduli and subtract arguments
Polar Coordinates - Multiplication by i
Polar Coordinates - Division
radicals
Complex Number

9. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
cosh²y - sinh²y
the vector (a -b)
real
Absolute Value of a Complex Number

10. R^2 = x
Square Root
standard form of complex numbers
|z| = mod(z)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

11. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - z
Imaginary Unit
The Complex Numbers
sin iy

12. xpressions such as ``the complex number z'' - and ``the point z'' are now
four different numbers: i - -i - 1 - and -1.
i^3
i²
interchangeable

13. The reals are just the
'i'
For real a and b - a + bi = 0 if and only if a = b = 0
subtracting complex numbers
x-axis in the complex plane

14. A + bi
a + bi for some real a and b.
standard form of complex numbers
complex numbers
integers

15. Every complex number has the 'Standard Form':
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - cos?
cos iy
a + bi for some real a and b.

16. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to multiply complex nubers(2+i)(2i-3)
Affix

17. A subset within a field.
Subfield
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Subtraction
integers

18. V(zz*) = v(a² + b²)
sin iy
|z| = mod(z)
Polar Coordinates - cos?
zz*

19. When two complex numbers are multipiled together.
natural
Complex Multiplication
zz*
i^1

20. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
conjugate pairs
Integers
Subfield
Affix

21. A² + b² - real and non negative
v(-1)
irrational
zz*
integers

22. V(x² + y²) = |z|
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - r
integers
Any polynomial O(xn) - (n > 0)

23. Real and imaginary numbers
Euler Formula
Roots of Unity
x-axis in the complex plane
complex numbers

24. To simplify the square root of a negative number
The Complex Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to find any Power
We say that c+di and c-di are complex conjugates.

25. 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
How to find any Power
Complex Numbers: Add & subtract
Rational Number
Complex numbers are points in the plane

26. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
imaginary
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Multiplication by i
Complex Conjugate

27. A plot of complex numbers as points.
i^2 = -1
z1 / z2
imaginary
Argand diagram

28. Have radical
Subfield
point of inflection
z1 / z2
radicals

29. 2a
cos z
i^1
z + z*
real

30. 1
i^4
Polar Coordinates - z
The Complex Numbers
cos iy

31. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
Complex Multiplication
ln z
Polar Coordinates - r

32. x / r
Polar Coordinates - cos?
a + bi for some real a and b.
e^(ln z)
z1 ^ (z2)

33. All the powers of i can be written as
point of inflection
the distance from z to the origin in the complex plane
four different numbers: i - -i - 1 - and -1.
v(-1)

34. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
How to solve (2i+3)/(9-i)
a + bi for some real a and b.
ln z
-1

35. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
v(-1)
i^2 = -1
z1 / z2

36. A complex number may be taken to the power of another complex number.
cosh²y - sinh²y
Complex Exponentiation
real
z1 ^ (z2)

37. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Numbers
adding complex numbers
Polar Coordinates - cos?

38. 1st. Rule of Complex Arithmetic
Square Root
i^2 = -1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^4

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

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40. Derives z = a+bi
Argand diagram
zz*
Euler Formula
radicals

41. Equivalent to an Imaginary Unit.
integers
Polar Coordinates - Multiplication
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary number

42. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
cosh²y - sinh²y
How to find any Power
(cos? +isin?)n

43. The complex number z representing a+bi.
Imaginary number
Affix
i²
Complex numbers are points in the plane

44. When two complex numbers are added together.
the complex numbers
Complex Addition
|z-w|
radicals

45. I
i^1
i^3
cos iy
Every complex number has the 'Standard Form': a + bi for some real a and b.

46. 1
Every complex number has the 'Standard Form': a + bi for some real a and b.
Rules of Complex Arithmetic
i²
cos z

47. 5th. Rule of Complex Arithmetic
conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex numbers are points in the plane
Imaginary Numbers

48. 4th. Rule of Complex Arithmetic
|z-w|
real
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i

49. We can also think of the point z= a+ ib as
Every complex number has the 'Standard Form': a + bi for some real a and b.
cos iy
subtracting complex numbers
the vector (a -b)

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

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