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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. 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)
How to solve (2i+3)/(9-i)
multiply the numerator and the denominator by the complex conjugate of the denominator.
the distance from z to the origin in the complex plane

2. xpressions such as ``the complex number z'' - and ``the point z'' are now
Affix
The Complex Numbers
interchangeable
Complex Addition

3. Imaginary number

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4. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
How to multiply complex nubers(2+i)(2i-3)
adding complex numbers
radicals
Imaginary Unit

5. 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.
Liouville's Theorem -
i^2 = -1
Complex Number
Complex numbers are points in the plane

6. 1
Complex Numbers: Add & subtract
Euler Formula
cosh²y - sinh²y
adding complex numbers

7. V(zz*) = v(a² + b²)
Complex Multiplication
the vector (a -b)
|z| = mod(z)
Roots of Unity

8. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
multiplying complex numbers
Complex Multiplication
Complex Conjugate
Roots of Unity

9. 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
sin z
|z| = mod(z)
Real and Imaginary Parts
Euler Formula

10. I
i^1
The Complex Numbers
How to multiply complex nubers(2+i)(2i-3)
Complex Numbers: Multiply

11. 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
For real a and b - a + bi = 0 if and only if a = b = 0
We say that c+di and c-di are complex conjugates.
i^2
the distance from z to the origin in the complex plane

12. 1
Polar Coordinates - sin?
i^2
z - z*
For real a and b - a + bi = 0 if and only if a = b = 0

13. Divide moduli and subtract arguments
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Division
transcendental
Irrational Number

14. 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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15. 4th. Rule of Complex Arithmetic
Complex numbers are points in the plane
Polar Coordinates - z?¹
How to find any Power
(a + bi) = (c + bi) = (a + c) + ( b + d)i

16. Equivalent to an Imaginary Unit.
Polar Coordinates - Multiplication by i
Imaginary number
subtracting complex numbers
z1 / z2

17. ½(e^(-y) +e^(y)) = cosh y
Complex Numbers: Multiply
cos z
Any polynomial O(xn) - (n > 0)
cos iy

18. 2nd. Rule of Complex Arithmetic

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19. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the vector (a -b)
Polar Coordinates - r
'i'

20. E ^ (z2 ln z1)
Irrational Number
Polar Coordinates - Multiplication
z1 ^ (z2)
Polar Coordinates - cos?

21. A complex number and its conjugate
We say that c+di and c-di are complex conjugates.
conjugate pairs
Complex Conjugate
Every complex number has the 'Standard Form': a + bi for some real a and b.

22. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Roots of Unity
point of inflection
Integers
How to solve (2i+3)/(9-i)

23. When two complex numbers are multipiled together.
Real Numbers
ln z
Complex Multiplication
|z| = mod(z)

24. When two complex numbers are divided.
Complex Division
Rational Number
Polar Coordinates - z
Imaginary Numbers

25. When two complex numbers are added together.
Complex Addition
i^3
non-integers
For real a and b - a + bi = 0 if and only if a = b = 0

26. 3
i^3
Imaginary number
Imaginary Numbers
Liouville's Theorem -

27. (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
can't get out of the complex numbers by adding (or subtracting) or multiplying two
How to solve (2i+3)/(9-i)
z1 / z2
integers

28. Like pi
natural
transcendental
|z| = mod(z)
Every complex number has the 'Standard Form': a + bi for some real a and b.

29. 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.'
Complex Number
Polar Coordinates - Multiplication
multiplying complex numbers
Square Root

30. The field of all rational and irrational numbers.
Absolute Value of a Complex Number
Roots of Unity
Real Numbers
Polar Coordinates - Division

31. 1st. Rule of Complex Arithmetic
i^2 = -1
i^0
Roots of Unity
interchangeable

32. 1
How to multiply complex nubers(2+i)(2i-3)
sin iy
i²
Square Root

33. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
a real number: (a + bi)(a - bi) = a² + b²
Field
Complex Division
z1 / z2

34. 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.
conjugate pairs
can't get out of the complex numbers by adding (or subtracting) or multiplying two
point of inflection
Complex Numbers: Multiply

35. To simplify a complex fraction
(a + c) + ( b + d)i
Imaginary number
multiply the numerator and the denominator by the complex conjugate of the denominator.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

36. z1z2* / |z2|²
Rules of Complex Arithmetic
z1 / z2
z + z*
i^0

37. 5th. Rule of Complex Arithmetic
the vector (a -b)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^4
e^(ln z)

38. No i
real
Imaginary Numbers
cosh²y - sinh²y
zz*

39. All the powers of i can be written as
Complex numbers are points in the plane
The Complex Numbers
Polar Coordinates - Multiplication
four different numbers: i - -i - 1 - and -1.

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

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41. 1
Complex numbers are points in the plane
i^0
Subfield
Polar Coordinates - sin?

42. 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
the vector (a -b)
How to find any Power
integers

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

44. 2ib
non-integers
Real Numbers
z - z*
adding complex numbers

45. Multiply moduli and add arguments
-1
complex
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Multiplication

46. 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
Square Root
Complex Number Formula
How to solve (2i+3)/(9-i)
subtracting complex numbers

47. A plot of complex numbers as points.
interchangeable
Polar Coordinates - Division
Argand diagram
Polar Coordinates - z?¹

48. The modulus of the complex number z= a + ib now can be interpreted as
standard form of complex numbers
the distance from z to the origin in the complex plane
Irrational Number
Euler Formula

49. A complex number may be taken to the power of another complex number.
Complex Exponentiation
x-axis in the complex plane
De Moivre's Theorem
|z-w|

50. All numbers
For real a and b - a + bi = 0 if and only if a = b = 0
complex
Subfield
subtracting complex numbers