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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
|z| = mod(z)
Polar Coordinates - cos?
Field
the vector (a -b)

2. Like pi
transcendental
Subfield
Complex Division
radicals

3. 5th. Rule of Complex Arithmetic
four different numbers: i - -i - 1 - and -1.
point of inflection
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin iy

4. Have radical
multiplying complex numbers
radicals
Liouville's Theorem -
Absolute Value of a Complex Number

5. 1
Polar Coordinates - r
Polar Coordinates - Multiplication
Imaginary number
i^4

6. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Affix
Roots of Unity
e^(ln z)
-1

7. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
zz*
transcendental
Absolute Value of a Complex Number
Polar Coordinates - Multiplication by i

8. A + bi
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number
standard form of complex numbers
interchangeable

9. Cos n? + i sin n? (for all n integers)
i^2
cos iy
(cos? +isin?)n
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

10. I = imaginary unit - i² = -1 or i = v-1
complex numbers
Imaginary Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^2 = -1

11. (e^(iz) - e^(-iz)) / 2i
sin z
Polar Coordinates - Multiplication
Imaginary Numbers
cos z

12. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Complex Addition
cos z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
ln z

13. 3
i^3
transcendental
Complex Addition
e^(ln z)

14. A complex number may be taken to the power of another complex number.
Field
|z| = mod(z)
Complex Exponentiation
Polar Coordinates - z?¹

15. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Polar Coordinates - Arg(z*)
Every complex number has the 'Standard Form': a + bi for some real a and b.
Square Root

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

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17. A² + b² - real and non negative
the distance from z to the origin in the complex plane
zz*
conjugate
adding complex numbers

18. When two complex numbers are multipiled together.
Complex Multiplication
De Moivre's Theorem
a real number: (a + bi)(a - bi) = a² + b²
subtracting complex numbers

19. (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
integers
Irrational Number
Complex Multiplication
How to multiply complex nubers(2+i)(2i-3)

20. (a + bi)(c + bi) =
the vector (a -b)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^0
(a + bi) = (c + bi) = (a + c) + ( b + d)i

21. A+bi
Complex Number Formula
a real number: (a + bi)(a - bi) = a² + b²
adding complex numbers
Polar Coordinates - r

22. A plot of complex numbers as points.
Argand diagram
v(-1)
cosh²y - sinh²y
(a + c) + ( b + d)i

23. The modulus of the complex number z= a + ib now can be interpreted as
i²
imaginary
Every complex number has the 'Standard Form': a + bi for some real a and b.
the distance from z to the origin in the complex plane

24. (a + bi) = (c + bi) =
multiplying complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
Absolute Value of a Complex Number
(a + c) + ( b + d)i

25. To simplify a complex fraction
Euler Formula
cos z
multiply the numerator and the denominator by the complex conjugate of the denominator.
e^(ln z)

26. 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 - Arg(z*)
The Complex Numbers
We say that c+di and c-di are complex conjugates.
radicals

27. 1
i^1
Complex Number Formula
Euler's Formula
i²

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
Polar Coordinates - sin?
Argand diagram
subtracting complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two

29. 3rd. Rule of Complex Arithmetic
Complex Numbers: Add & subtract
Real and Imaginary Parts
For real a and b - a + bi = 0 if and only if a = b = 0
ln z

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

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31. I
v(-1)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
|z| = mod(z)
z - z*

32. xpressions such as ``the complex number z'' - and ``the point z'' are now
|z| = mod(z)
interchangeable
Polar Coordinates - Multiplication
real

33. When two complex numbers are divided.
Complex Division
the vector (a -b)
Complex Multiplication
Square Root

34. 2nd. Rule of Complex Arithmetic

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35. 1
0 if and only if a = b = 0
i^0
Complex Numbers: Add & subtract
Irrational Number

36. 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.
Real and Imaginary Parts
i^2 = -1
adding complex numbers
Complex Numbers: Multiply

37. I^2 =
-1
Roots of Unity
Imaginary Unit
For real a and b - a + bi = 0 if and only if a = b = 0

38. To simplify the square root of a negative number
Polar Coordinates - Multiplication by i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
v(-1)
ln z

39. A complex number and its conjugate
Roots of Unity
conjugate pairs
sin z
Rational Number

40. 2ib
z - z*
Imaginary number
the vector (a -b)
e^(ln z)

41. Not on the numberline
Polar Coordinates - z?¹
Euler Formula
'i'
non-integers

42. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Numbers: Multiply
point of inflection
Rational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

43. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Imaginary Numbers
v(-1)
conjugate
complex

44. Multiply moduli and add arguments
i²
How to multiply complex nubers(2+i)(2i-3)
Absolute Value of a Complex Number
Polar Coordinates - Multiplication

45. Rotates anticlockwise by p/2
Complex Subtraction
|z| = mod(z)
Polar Coordinates - Multiplication by i
z + z*

46. 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.
cos iy
multiplying complex numbers
Complex Subtraction
How to find any Power

47. All the powers of i can be written as
multiply the numerator and the denominator by the complex conjugate of the denominator.
imaginary
four different numbers: i - -i - 1 - and -1.
Real Numbers

48. V(zz*) = v(a² + b²)
Complex Subtraction
|z| = mod(z)
Rules of Complex Arithmetic
Real and Imaginary Parts

49. 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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50. Real and imaginary numbers
rational
z + z*
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers