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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. 1
i^4
Rational Number
rational
zz*

2. Starts at 1 - does not include 0
Affix
i^2
natural
Polar Coordinates - z

3. We can also think of the point z= a+ ib as
Complex Multiplication
Square Root
the vector (a -b)
Complex Division

4. Have radical
radicals
integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
imaginary

5. To simplify a complex fraction
i^2 = -1
Complex Numbers: Add & subtract
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - sin?

6. 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.

7. 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
adding complex numbers
(cos? +isin?)n
Real and Imaginary Parts
transcendental

8. 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.
Subfield
How to find any Power
Polar Coordinates - Division
has a solution.

9. (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
How to solve (2i+3)/(9-i)
Complex Addition
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
x-axis in the complex plane

10. E^(ln r) e^(i?) e^(2pin)
point of inflection
natural
e^(ln z)
Polar Coordinates - Arg(z*)

11. (e^(iz) - e^(-iz)) / 2i
Rules of Complex Arithmetic
Complex Conjugate
sin z
i^2

12. Where the curvature of the graph changes
point of inflection
(cos? +isin?)n
complex numbers
Imaginary Numbers

13. When two complex numbers are divided.
i²
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
imaginary

14. Not on the numberline
Polar Coordinates - Multiplication by i
Argand diagram
Rational Number
non-integers

15. V(x² + y²) = |z|
Square Root
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
ln z
Polar Coordinates - r

16. Imaginary number

17. Rotates anticlockwise by p/2
Complex Number
Polar Coordinates - Multiplication by i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
'i'

18. ½(e^(iz) + e^(-iz))
How to multiply complex nubers(2+i)(2i-3)
Complex Addition
cos z
Complex Number Formula

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
interchangeable
How to multiply complex nubers(2+i)(2i-3)
multiplying complex numbers
Complex Division

20. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
How to multiply complex nubers(2+i)(2i-3)
Field
Euler Formula
Polar Coordinates - Multiplication

21. All numbers
complex
cosh²y - sinh²y
De Moivre's Theorem
Field

22. 3
Polar Coordinates - Arg(z*)
i^3
four different numbers: i - -i - 1 - and -1.
sin z

23. I
Imaginary Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z
v(-1)

24. 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
Polar Coordinates - cos?
Argand diagram
Integers

25. The reals are just the
How to solve (2i+3)/(9-i)
x-axis in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
a + bi for some real a and b.

26. A number that cannot be expressed as a fraction for any integer.
Irrational Number
multiplying complex numbers
Rational Number
has a solution.

27. ? = -tan?
adding complex numbers
complex numbers
Liouville's Theorem -
Polar Coordinates - Arg(z*)

28. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
i²
multiply the numerator and the denominator by the complex conjugate of the denominator.
e^(ln z)

29. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
natural
i²
ln z

30. A subset within a field.
x-axis in the complex plane
Subfield
multiply the numerator and the denominator by the complex conjugate of the denominator.
a + bi for some real a and b.

31. V(zz*) = v(a² + b²)
Polar Coordinates - r
|z| = mod(z)
the distance from z to the origin in the complex plane
sin iy

32. Divide moduli and subtract arguments
x-axis in the complex plane
De Moivre's Theorem
complex numbers
Polar Coordinates - Division

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

34. Has exactly n roots by the fundamental theorem of algebra
i^0
z - z*
Any polynomial O(xn) - (n > 0)
Complex Number

35. When two complex numbers are added together.
the distance from z to the origin in the complex plane
natural
Complex Addition
z - z*

36. The complex number z representing a+bi.
Polar Coordinates - Division
Euler Formula
Affix
Absolute Value of a Complex Number

37. The product of an imaginary number and its conjugate is
the distance from z to the origin in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
a real number: (a + bi)(a - bi) = a² + b²
(cos? +isin?)n

38. A + bi
natural
standard form of complex numbers
the distance from z to the origin in the complex plane
integers

39. 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.
z1 ^ (z2)
Complex numbers are points in the plane
i²
Irrational Number

40. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number Formula
four different numbers: i - -i - 1 - and -1.
The Complex Numbers

41. 1
Rational Number
i²
a real number: (a + bi)(a - bi) = a² + b²
Absolute Value of a Complex Number

42. I
Complex Exponentiation
Complex Addition
Subfield
i^1

43. 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
subtracting complex numbers
i²
i^2 = -1
Polar Coordinates - Arg(z*)

44. 1
Complex Numbers: Add & subtract
multiplying complex numbers
a real number: (a + bi)(a - bi) = a² + b²
i^0

45. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Number Formula
conjugate
Real and Imaginary Parts
natural

46. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
0 if and only if a = b = 0
Imaginary Numbers
Argand diagram

47. E ^ (z2 ln z1)
z1 ^ (z2)
Field
Every complex number has the 'Standard Form': a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.

48. R?¹(cos? - isin?)
Complex numbers are points in the plane
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - z?¹
Complex Numbers: Add & subtract

49. 2nd. Rule of Complex Arithmetic

50. Multiply moduli and add arguments
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication
Complex Multiplication
Polar Coordinates - Division