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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. A number that can be expressed as a fraction p/q where q is not equal to 0.
z1 / z2
-1
z1 ^ (z2)
Rational Number

2. E^(ln r) e^(i?) e^(2pin)
subtracting complex numbers
e^(ln z)
Complex Division
Euler Formula

3. Written as fractions - terminating + repeating decimals
For real a and b - a + bi = 0 if and only if a = b = 0
rational
'i'
|z| = mod(z)

4. I
Rational Number
Liouville's Theorem -
0 if and only if a = b = 0
v(-1)

5. I = imaginary unit - i² = -1 or i = v-1
(cos? +isin?)n
rational
Imaginary Numbers
Subfield

6. When two complex numbers are divided.
Polar Coordinates - Arg(z*)
standard form of complex numbers
Complex Division
Polar Coordinates - sin?

7. 1st. Rule of Complex Arithmetic
sin iy
conjugate pairs
i^2 = -1
can't get out of the complex numbers by adding (or subtracting) or multiplying two

8. 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
e^(ln z)
Polar Coordinates - r
i²
the complex numbers

9. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
(cos? +isin?)n
zz*
conjugate
i^4

10. I^2 =
i^0
Polar Coordinates - cos?
Imaginary Numbers
-1

11. Imaginary number

12. Not on the numberline
|z-w|
non-integers
Complex Exponentiation
conjugate pairs

13. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
Complex Numbers: Multiply
Complex numbers are points in the plane
Euler Formula
i²

14. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
multiplying complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Numbers: Multiply
Roots of Unity

15. 2a
i^2
Roots of Unity
natural
z + z*

16. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
integers
Absolute Value of a Complex Number
0 if and only if a = b = 0
cos iy

17. A complex number and its conjugate
conjugate pairs
rational
z - z*
natural

18. Real and imaginary numbers
multiplying complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex numbers
a real number: (a + bi)(a - bi) = a² + b²

19. 3rd. Rule of Complex Arithmetic
0 if and only if a = b = 0
cosh²y - sinh²y
Affix
For real a and b - a + bi = 0 if and only if a = b = 0

20. Has exactly n roots by the fundamental theorem of algebra
cos z
cos iy
Any polynomial O(xn) - (n > 0)
Absolute Value of a Complex Number

21. 3
e^(ln z)
complex
Rational Number
i^3

22. Multiply moduli and add arguments
Polar Coordinates - Multiplication by i
Polar Coordinates - Multiplication
Complex numbers are points in the plane
subtracting complex numbers

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

24. For real a and b - a + bi =
v(-1)
four different numbers: i - -i - 1 - and -1.
complex
0 if and only if a = b = 0

25. No i
x-axis in the complex plane
Roots of Unity
real
point of inflection

26. ½(e^(iz) + e^(-iz))
cos z
Polar Coordinates - sin?
|z-w|
Irrational Number

27. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
rational
Complex Division
point of inflection

28. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Complex Number Formula
Imaginary number
interchangeable

29. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
i^3
Polar Coordinates - cos?
can't get out of the complex numbers by adding (or subtracting) or multiplying two

30. 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 numbers are points in the plane
z - z*
How to find any Power
Imaginary Numbers

31. (a + bi) = (c + bi) =
e^(ln z)
v(-1)
(a + c) + ( b + d)i
Complex Subtraction

32. ? = -tan?
z1 / z2
Polar Coordinates - Arg(z*)
How to multiply complex nubers(2+i)(2i-3)
-1

33. y / r
conjugate
Polar Coordinates - sin?
point of inflection
Rules of Complex Arithmetic

34. 2ib
z - z*
cos iy
Polar Coordinates - z
Real and Imaginary Parts

35. 1
Complex Division
0 if and only if a = b = 0
a real number: (a + bi)(a - bi) = a² + b²
cosh²y - sinh²y

36. Root negative - has letter i
Liouville's Theorem -
imaginary
i^1
Field

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

38. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
the distance from z to the origin in the complex plane
zz*
Integers

39. R?¹(cos? - isin?)
Polar Coordinates - Multiplication
i^0
-1
Polar Coordinates - z?¹

40. (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
standard form of complex numbers
How to multiply complex nubers(2+i)(2i-3)
Real and Imaginary Parts

41. A complex number may be taken to the power of another complex number.
i^2 = -1
Complex Exponentiation
i^0
(cos? +isin?)n

42. A² + b² - real and non negative
zz*
Complex Subtraction
the vector (a -b)
multiply the numerator and the denominator by the complex conjugate of the denominator.

43. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Real Numbers
i^2
Real and Imaginary Parts

44. A subset within a field.
(a + c) + ( b + d)i
four different numbers: i - -i - 1 - and -1.
the complex numbers
Subfield

45. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Polar Coordinates - sin?
the complex numbers
0 if and only if a = b = 0

46. E ^ (z2 ln z1)
z1 ^ (z2)
a real number: (a + bi)(a - bi) = a² + b²
the distance from z to the origin in the complex plane
i^1

47. 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
Polar Coordinates - sin?
|z| = mod(z)
'i'
Complex Numbers: Add & subtract

48. Given (4-2i) the complex conjugate would be (4+2i)
Euler Formula
Complex Conjugate
Any polynomial O(xn) - (n > 0)
a + bi for some real a and b.

49. A number that cannot be expressed as a fraction for any integer.
Irrational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)
Affix

50. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
the distance from z to the origin in the complex plane
a real number: (a + bi)(a - bi) = a² + b²
multiplying complex numbers
Rules of Complex Arithmetic