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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. 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.'
|z| = mod(z)
a + bi for some real a and b.
Complex Number
Complex Addition

2. A subset within a field.
i^4
Polar Coordinates - Division
zz*
Subfield

3. The field of all rational and irrational numbers.
the distance from z to the origin in the complex plane
Real Numbers
a + bi for some real a and b.
can't get out of the complex numbers by adding (or subtracting) or multiplying two

4. 3
i^3
the complex numbers
a real number: (a + bi)(a - bi) = a² + b²
Real Numbers

5. ? = -tan?
0 if and only if a = b = 0
x-axis in the complex plane
Polar Coordinates - Arg(z*)
Polar Coordinates - z?¹

6. ½(e^(-y) +e^(y)) = cosh y
cos iy
cos z
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0

7. (a + bi)(c + bi) =
Subfield
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to find any Power
cos z

8. Where the curvature of the graph changes
Polar Coordinates - r
four different numbers: i - -i - 1 - and -1.
Complex Numbers: Add & subtract
point of inflection

9. Root negative - has letter i
irrational
Absolute Value of a Complex Number
Square Root
imaginary

10. 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.
We say that c+di and c-di are complex conjugates.
non-integers
Any polynomial O(xn) - (n > 0)
How to find any Power

11. A+bi
cos z
Rational Number
Complex Number Formula
Polar Coordinates - Division

12. 1
a + bi for some real a and b.
cosh²y - sinh²y
cos iy
zz*

13. (a + bi) = (c + bi) =
Square Root
Field
(a + c) + ( b + d)i
natural

14. No i
Polar Coordinates - Arg(z*)
real
Polar Coordinates - Multiplication
Imaginary Numbers

15. When two complex numbers are multipiled together.
i^2 = -1
Complex Multiplication
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - z?¹

16. R?¹(cos? - isin?)
Polar Coordinates - z?¹
conjugate
i^3
the vector (a -b)

17. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
sin z
Polar Coordinates - Multiplication
|z| = mod(z)

18. 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
conjugate
Subfield
z1 ^ (z2)

19. A plot of complex numbers as points.
i²
non-integers
Subfield
Argand diagram

20. Any number not rational
imaginary
irrational
Real and Imaginary Parts
|z-w|

21. 1
i^4
Complex Division
Euler Formula
Complex Exponentiation

22. 4th. Rule of Complex Arithmetic
e^(ln z)
a + bi for some real a and b.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)

23. Rotates anticlockwise by p/2
e^(ln z)
the complex numbers
Polar Coordinates - Multiplication by i
Polar Coordinates - r

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

25. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Roots of Unity
Field
transcendental
Imaginary Numbers

26. When two complex numbers are added together.
the complex numbers
Complex Addition
z1 ^ (z2)
How to multiply complex nubers(2+i)(2i-3)

27. To simplify the square root of a negative number
Imaginary Unit
The Complex Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos z

28. Every complex number has the 'Standard Form':
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
a + bi for some real a and b.
multiplying complex numbers
natural

29. z1z2* / |z2|²
z1 / z2
ln z
Subfield
i^3

30. ½(e^(iz) + e^(-iz))
Real Numbers
cos z
Roots of Unity
How to add and subtract complex numbers (2-3i)-(4+6i)

31. A² + b² - real and non negative
'i'
zz*
For real a and b - a + bi = 0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

32. Real and imaginary numbers
complex
complex numbers
Liouville's Theorem -
-1

33. To simplify a complex fraction
complex
a + bi for some real a and b.
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^1

34. (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
cos iy
non-integers
How to multiply complex nubers(2+i)(2i-3)
radicals

35. A complex number and its conjugate
-1
Subfield
conjugate pairs
sin z

36. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z?¹
Polar Coordinates - z
Roots of Unity
Complex Numbers: Multiply

37. Cos n? + i sin n? (for all n integers)
For real a and b - a + bi = 0 if and only if a = b = 0
(cos? +isin?)n
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)

38. A complex number may be taken to the power of another complex number.
i^0
multiplying complex numbers
irrational
Complex Exponentiation

39. I = imaginary unit - i² = -1 or i = v-1
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Numbers
i^2
Complex Multiplication

40. 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
the complex numbers
i^3
x-axis in the complex plane
How to add and subtract complex numbers (2-3i)-(4+6i)

41. We can also think of the point z= a+ ib as
Imaginary Numbers
the vector (a -b)
Every complex number has the 'Standard Form': a + bi for some real a and b.
For real a and b - a + bi = 0 if and only if a = b = 0

42. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - Multiplication by i
Polar Coordinates - Division
Roots of Unity
Euler Formula

43. V(x² + y²) = |z|
z1 / z2
Polar Coordinates - r
v(-1)
Polar Coordinates - Division

44. Numbers on a numberline
adding complex numbers
z - z*
integers
De Moivre's Theorem

45. All numbers
complex
i²
i^1
Rules of Complex Arithmetic

46. x / r
Euler's Formula
Square Root
i^3
Polar Coordinates - cos?

47. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
Polar Coordinates - r
Real Numbers
Subfield
We say that c+di and c-di are complex conjugates.

48. I
z - z*
Imaginary Unit
i^1
Polar Coordinates - Arg(z*)

49. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
How to add and subtract complex numbers (2-3i)-(4+6i)
|z| = mod(z)
Polar Coordinates - cos?

50. 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
four different numbers: i - -i - 1 - and -1.
Rules of Complex Arithmetic
Real and Imaginary Parts
Complex Conjugate