Complex Numbers & The Quadratic Formula

Complex Numbers & The Quadratic Formula

You'll probably only use complexes in the context of solving quadratics for their zeroes. (There are many other practical uses for complexes, but you'll have to wait for more interesting classes like "Engineering 201" to get to the "good stuff".)

Remember that the Quadratic Formula solves "ax2 + bx + c = 0" for the values of x. Also remember that this means that you are trying to find the x-intercepts of the graph. When the Formula gives you a negative inside the square root, you can now simplify the zero by using complex numbers. The answer you come up with is a valid "zero" or "root" or "solution" for "ax2 + bx + c = 0", because, if you plug it back into the quadratic, you'll get zero after you simplify. but you cannot graph a complex number on the x,y-plane. So this "solution to the equation" is not an x-intercept. In other words, you can make this connection between the Quadratic Formula, complex numbers, and graphing:





































As an aside, you can graph complexes, but not in the x,y-plane. You need the "complex" plane. For the complex plane, the x-axis is where you plot the real part, and the y-axis is where you graph the imaginary part. For instance, for the complex number 3 – 2i, you would graph it like this:


















This leads to an interesting fact: When you learned about regular ("real") numbers, you also learned about their order (this is what you show on the number line). But x,y-points don't come in any particular order. You can't say that one point "comes after" another point in the same way that you can say that one number comes after another number. For instance, you can't say that (4, 5) "comes after" (4, 3)" in the way that you can say that 5 comes after 3. Pretty much all you can do is compare "size", and, for complex numbers, "size" means "how far from the origin". To do this, you use the Distance Formula, and compare which complexes are closer to or further from the origin. This "size" concept is called "the modulus". For instance, looking at our complex number plotted above, its modulus is computed by using the Distance Formula:








Note that all points at this distance from the origin have the same modulus. All the points on the circle with radius sqrt(13) are viewed as being complex numbers having the same "size" as 3 – 2i.

Operations on Complex Numbers

Operations on Complex Numbers

Complex numbers are "binomials" of a sort, and are added, subtracted, and multiplied in a similar way. (Division, which is further down the page, is a bit different.)


























For this last example, if you learned "FOIL", FOILing works for this kind of multiplication. But whatever method you use, remember that multiplying and adding with complexes works just like multiplying and adding polynomials, except that, while x2 is just x2, i2 is –1. That is, you can use the exact same techniques for simplifying complex-number expressions, but you can simplify even further with complexes than with polynomials, because i2 reduces to the number –1.

Adding and multiplying complexes isn't too bad. It's when you start on fractions (that is, division) that things turn ugly. Most of the reason for this ugliness is actually arbitrary. Remember back in elementary school, when you first learned fractions? Your teacher would get her panties in a wad if you used "improper" fractions. For instance, you couldn't say " 3/2 "; you had to convert it to "1 1/2". But now that you're in algebra, nobody cares, and you've probably noticed that "improper" fractions are often more useful than "mixed" fractions. The problem in the case of complexes is that your professor will get his boxers in a bunch if you leave imaginaries in the denominator. So how do you handle this?

Suppose you have the following problem:







The point here is that they want you to get rid of the i underneath. The 2 is fine, but the i has got to go. To do this, you use the fact that i2 = –1. If you multiply top and bottom by i, then the i underneath will vanish in a puff of negativity:











This was simple enough, but what if you have something more complicated?
















Since you still have an i underneath, this didn't help much. So how do you handle this simplification? You use something called "conjugates". The conjugate of a complex number a + bi is the same number, but with the opposite sign in the middle: a – bi. When you multiply conjugates, you are, in effect, multiplying to a difference of squares:













Note that the i's disappeared. This is what the conjugate, difference-of-squares thing is for. Here's how it is used:
















In the last step, note how the fraction was split into two pieces. This is because, technically speaking, a complex number is in two parts, the real part and the i part. They aren't supposed to "share" the denominator.

Complex Numbers: Introduction

Complex Numbers: Introduction

Up until now, you've been told that you can't take the square root of a negative number. That's because you had no numbers that, when squared, were negative. Every number was positive after you squared it. So you couldn't very well square-root a negative and expect to come up with anything sensible.
Now, however, you can take the square root of a negative number, but it involves using a new number to do it. This new number was invented (discovered?) around the time of the Reformation. At this time, nobody believed that any "real world" use would be found for this new number, other than easing the computations involved in solving certain equations, so the new number was viewed as being a pretend number invented for convenience sake.
(But then, when you think about it, aren't all numbers inventions? It's not like numbers grow on trees! They live in our heads. We made them all up! Why not invent a new one, as long as it works okay with what we already have?)
Anyway, this new number is called "i", standing for "imaginary", because "everybody knew" that i wasn't "real". (That's why you couldn't take the square root of a negative number before: you only had "real" numbers; that is, numbers without the "i" in them.) The imaginary is defined to be:














But this doesn't make any sense! You already have two numbers that square to 1; namely –1 and +1. And i already squares to –1. So it's not reasonable that i would also square to 1. This points out an important detail: When dealing with imaginaries, you gain something (the ability to deal with negatives inside square roots), but you also lose something (some of the flexibility and convenient rules you used to have when dealing with square roots). In particular, YOU MUST ALWAYS DO THE i-PART FIRST!

























In computations, you deal with i just as you would with x, except for the fact that x2 is just x2, but
i2 is –1:




























Note this last problem. Within it, you can see that , because i2 = –1. Continuing, you get:




























In other words, to calculate any high power of i, you can convert it to a lower power by taking the closest multiple of 4 that's no bigger than the exponent and subtracting this multiple from the exponent. For example, a common trick question on tests is something along the lines of "Simplify i99", the idea being that you'll try to multiply i ninety-nine times and you'll run out of time, and the teachers will get a good giggle at your expense in the faculty lounge. Here's how the shortcut works:

i99 = i96+3 = i(4×24)+3 = i3 = –i

That is, i99 = i3, because you can just lop off the i96. (Ninety-six is a multiple of four, so i96 is just 1, which you can ignore.) In other words, you can divide the exponent by 4 (using long division), discard the answer, and use only the remainder. This will give you the part of the exponent that you care above. Here are a few more examples:




















Now you've seen how imaginaries work; it's time to move on to complex numbers. "Complex" numbers have two parts, a "real" part (being any "real" number that you're used to dealing with) and an "imaginary" part (being any number with an "i" in it). The "standard" format for complex numbers is "a + bi"; that is, real-part first and i-part last.

Capacitors and Inductors

Capacitors and Inductors

Introduction

• Two more linear, ideal basic passive circuit elements.
• Energy storage elements stored in both magnetic and electric fields.
• They found continual applications in more practical circuits such as filters, integrators, differentiators, circuit breakers and automobile ignition circuit.
• Circuit analysis techniques and theorems applied to purely resistive circuits are equally applicable to circuits with inductors and capacitors.

Capacitors

• Electrical component that consists of two conductors separated by an insulator or dielectric material.
• Its behavior based on phenomenon associated with electric fields, which the source is voltage.
• A time-varying electric fields produce a current flow in the space occupied by the fields.
• Capacitance is the circuit parameter which relates the displacement current to the voltage.

Parallel capacitances

• The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitances.

Series capacitances

• The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.

 

Inductors

• Electrical component that opposes any change in electrical current.
• Composed of a coil or wire wound around a non-magnetic core/magnetic core.
• Its behavior based on phenomenon associated with magnetic fields, which the source is current.
• A time-varying magnetic fields induce voltage in any conductor linked by the fields.
• Inductance is the circuit parameter which relates the induced voltage to the current.

Series inductances

• The equivalent inductance of N series-connected inductors is the sum of the individual inductances.

Parallel inductances

• The equivalent inductance of series-connected inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

Voltage Divider Rule (VDR) and Current Divider Rule (CDR)

Voltage Divider Rule (VDR)

• Whenever voltage has to be divided among resistors in series use voltage divider rule principle.

Current Divider Rule (CDR)

• Whenever current has to be divided among resistors in parallel, use current divider rule principle.

 

 

 

 

 

International System of Units (SI)

International System of Units (SI)

Multiplier	Prefix	Symbol
1018	Exa	E
1015	Peta	P
1012	Tera	T
109	Giga	G
106	Mega	M
103	Kilo	k
102	Hector	h
101	Deka	da
10-1	Deci	d
10-2	Centi	d
10-3	Mili	m
10-6	Micro	m
10-9	Nano	n
10-12	Pico	p
10-15	Femto	f
10-18	Atto	a

Series and Parallel Circuits

Series and Parallel Circuits

Series Equivalent Circuit

• The equivalent resistance for any number of resistors in series connection is the sum of each individual resistor.

• Apparently the single equivalent resistor is always larger than the largest resistor in the series connection.
• Series resistors carry the same current thru them.
• Voltage across each of the resistors obtained using voltage divider rule principle or Ohm’s law.

• The equivalent resistance for any number of resistors in parallel connection is obtained by taking the reciprocal of the sum of the reciprocal of each single resistor in the circuit.

• Apparently, the single equivalent resistor is always smaller than the smallest resistor in the parallel connection.
• Voltage across each resistor must be the same.
• Currents thru each of them are divided according to the current divider rule principle.

Kirchhoff’s Laws

Kirchhoff’s Laws

Kirchhoff's Current Law

• Kirchhoff’s Current Law (KCL) states that the algebraic sum of current entering a node must be equal to that of leaving the same node.
• Applying KCL, we obtain;

i2 + i6 = i1 + i3 + i4 + i5













Kirchhoff's Voltage Law

• Kirchhoff’s Voltage Law states that the algebraic sum of voltage drop in a loop must be equal to that of voltage rise in the same loop. Stated it in a different way is that the algebraic sum of all voltages around a loop must be zero.
• Applying KVL, we obtain;

Loop 1: V1 + V2 + V3 = Vs
Loop 2: V4 + VIs = V2