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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. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
i^2 = -1
Argand diagram
v(-1)

2. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
interchangeable
i^1
rational

3. V(x² + y²) = |z|
non-integers
sin iy
Rules of Complex Arithmetic
Polar Coordinates - r

4. 1
i^4
Polar Coordinates - Multiplication by i
z + z*
How to multiply complex nubers(2+i)(2i-3)

5. 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
imaginary
subtracting complex numbers
Real and Imaginary Parts
z - z*

6. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Complex Conjugate
Polar Coordinates - Multiplication by i
z - z*

7. The product of an imaginary number and its conjugate is
(a + bi) = (c + bi) = (a + c) + ( b + d)i
|z| = mod(z)
the vector (a -b)
a real number: (a + bi)(a - bi) = a² + b²

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
Complex Numbers: Add & subtract
the complex numbers
i^2 = -1
Complex Conjugate

9. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Euler's Formula
Liouville's Theorem -
De Moivre's Theorem

10. 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)
Polar Coordinates - cos?
-1
(a + bi) = (c + bi) = (a + c) + ( b + d)i

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

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12. A complex number and its conjugate
i^2 = -1
Argand diagram
conjugate pairs
Complex Division

13. Divide moduli and subtract arguments
Polar Coordinates - Division
How to add and subtract complex numbers (2-3i)-(4+6i)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Every complex number has the 'Standard Form': a + bi for some real a and b.

14. Numbers on a numberline
has a solution.
|z| = mod(z)
Polar Coordinates - sin?
integers

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

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16. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - Multiplication by i
standard form of complex numbers
Euler Formula
interchangeable

17. When two complex numbers are multipiled together.
Complex Multiplication
cos z
a + bi for some real a and b.
-1

18. We can also think of the point z= a+ ib as
point of inflection
rational
the vector (a -b)
Complex numbers are points in the plane

19. 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.
the distance from z to the origin in the complex plane
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex numbers are points in the plane

20. 1st. Rule of Complex Arithmetic
real
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^2 = -1
Polar Coordinates - z?¹

21. A+bi
rational
Complex Multiplication
Complex Number Formula
radicals

22. 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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23. 3
sin iy
Square Root
i^3
the distance from z to the origin in the complex plane

24. (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
How to multiply complex nubers(2+i)(2i-3)
For real a and b - a + bi = 0 if and only if a = b = 0
z - z*
Imaginary number

25. R^2 = x
Rational Number
'i'
Square Root
Real and Imaginary Parts

26. ½(e^(-y) +e^(y)) = cosh y
cos iy
zz*
Complex numbers are points in the plane
Imaginary number

27. Given (4-2i) the complex conjugate would be (4+2i)
Imaginary number
The Complex Numbers
Complex Conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.

28. The field of all rational and irrational numbers.
Polar Coordinates - Arg(z*)
Polar Coordinates - Multiplication
Complex Exponentiation
Real Numbers

29. (e^(-y) - e^(y)) / 2i = i sinh y
multiply the numerator and the denominator by the complex conjugate of the denominator.
four different numbers: i - -i - 1 - and -1.
adding complex numbers
sin iy

30. Have radical
non-integers
radicals
Imaginary Numbers
Complex Exponentiation

31. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
How to multiply complex nubers(2+i)(2i-3)
Complex Number Formula
conjugate
Euler's Formula

32. Written as fractions - terminating + repeating decimals
transcendental
|z-w|
How to add and subtract complex numbers (2-3i)-(4+6i)
rational

33. (e^(iz) - e^(-iz)) / 2i
Complex Conjugate
Polar Coordinates - z
sin z
z + z*

34. y / r
'i'
irrational
Polar Coordinates - sin?
cos z

35. V(zz*) = v(a² + b²)
integers
|z| = mod(z)
Argand diagram
(a + bi) = (c + bi) = (a + c) + ( b + d)i

36. A + bi
-1
i²
standard form of complex numbers
Subfield

37. A² + b² - real and non negative
zz*
multiplying complex numbers
i²
-1

38. Root negative - has letter i
has a solution.
Polar Coordinates - Multiplication by i
Complex Numbers: Multiply
imaginary

39. 1
i^0
i^2
Integers
Polar Coordinates - sin?

40. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
(a + bi) = (c + bi) = (a + c) + ( b + d)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
ln z
complex numbers

41. 2nd. Rule of Complex Arithmetic

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42. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos z
Real Numbers
Square Root

43. Where the curvature of the graph changes
i^0
point of inflection
rational
Rational Number

44. z1z2* / |z2|²
z1 / z2
Any polynomial O(xn) - (n > 0)
Real and Imaginary Parts
Irrational Number

45. The modulus of the complex number z= a + ib now can be interpreted as
natural
How to add and subtract complex numbers (2-3i)-(4+6i)
multiplying complex numbers
the distance from z to the origin in the complex plane

46. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers
|z-w|
integers

47. 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.
How to find any Power
(cos? +isin?)n
Polar Coordinates - r
De Moivre's Theorem

48. Rotates anticlockwise by p/2
|z| = mod(z)
Polar Coordinates - Multiplication by i
non-integers
Complex Division

49. R?¹(cos? - isin?)
Polar Coordinates - z?¹
transcendental
Complex Conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.

50. Like pi
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z + z*
transcendental
e^(ln z)