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Preparing to study electrical engineering on Khan Academy

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A summary of math and science preparation that will help you have the best experience with electrical engineering on Khan Academy. Written by Willy McAllister.

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In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively.

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Solving Quadratic Equations by Factoring - Basic Examples

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solving quadratics by factoring - YouTube

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Chemical reactions and stoichiometry | Chemistry | Science | Khan Academy

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We are now going to delve into the heart of chemistry. We learn ways of representing molecules and how molecules react. To do this, we'll even think about "how many" of a molecule we have using a quantity called a "mole".

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Preparing to study electrical engineering on Khan Academy

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Monterey Institute for Technology and Education

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The Algebra 2 course, often taught in the 11th grade, covers Polynomials; Complex Numbers; Rational Exponents; Exponential and Logarithmic Functions; Trigonometric Functions; Transformations of Functions; Rational Functions; and continuing the work with Equations and Modeling from previous grades. Khan Academy's Algebra 2 course is built to deliver a comprehensive, illuminating, engaging, and Common Core aligned experience!

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One of the questions people ask about traditional logic is why it doesn't teach truth tables. Modern logic, the most common kind of logic encountered in high school and college, uses them, so why does traditional logic ignore them?

Many people encounter a smattering of logic in high school math courses. Here, more than likely, they will encounter simple truth tables. Truth tables were invented by philosopher Ludwig Wittgenstein. He invented them to accompany the calculus into which modern analytic philosophers had transformed logic. They were a way to quickly solve for the truth of complex logical propositions in the modern system.

Let's take the statement:

There are 7 days in the week and 24 hours in a day.

In the modern system of logic we would want to immediately reduce this down to its formal elements. Let's say:

If we did this, then we could represent the statement as follows:

How do we find out whether the statement P and Q is true? In modern logic, this is determined by the statement's elements-in this case, P and Q. P and Q is true only when both the statements represented by P and Q are themselves true-in other words, if it is true to say that there are seven days in the week and that there are twenty-four hours in a day. If either one or both of these statements are false, then the combined statement would be false.

Using truth tables, we would set forth all of the truth possibilities of P and Q so we could see clearly when P and Q is true and when it is false:

We don't really need to go to all this trouble to verify that a simple statement like P and Q is true. But what if you had a statement like P and (Q or (If R, then S))?

When statements become much more complex, truth tables can be an easier way to calculate their truth value.

So if truth tables make the determination of the truth of statements easier, then why doesn't traditional logic teach them?

The first reason is that they are not practically useful in actual argument or discussion, since most statements used in everyday speech and even in academic conversation never get to the level of complexity that would require a truth table to figure them out. They are helpful in certain scientific applications and for computer programming (modern logic's most practical application), but outside those fields, they are seldom used or needed.

I have not only taught logic, but have engaged in private and public debate for over 20 years. While I have repeatedly made use of William of Sherwood's traditional mnemonic verse of the 19 valid syllogism forms and the rules for syllogism reduction (both of these are covered in Traditional Logic, Book II), I have never had to resort to a truth table.

Another reason for the absence of truth tables from traditional logic has to do with the philosophical differences between the two systems of logic. To state it baldly, traditional logic doesn't believe in the truth tables.

The reason they are used in one system and not the other has to do with a concept called "truth functionality." "Truth functionality" simply means that the truth or falsity of a statement's parts will tell us the truth or falsity of the whole statement.

In the statement P and Q, we can tell its truth from its component parts, P and Q. P and Q is called a "conjunctive proposition"-it conjoins antecedent P and consequent Q. Traditional logicians believe that conjunctive statements are the only kind of statements whose truth can be "solved" in a truth table. No other kinds of logical statements ( P or Q, If P, then Q, etc.) are truth functional in this way. You cannot tell their truth value merely from the truth value of their parts.

But this does not stop the proponents of modern logic from trying to quantify such statements, even if such quantification results in some strange anomalies. Take the following example:

If it rains, then my dog will get wet

In modern logic, we would "solve" for the truth of this statement using a truth table. If we assume:

... then the truth table would look like this:

This kind of statement is considered true in every possible case except when P is true and Q is false (line 2). Let's say my dog is an outside dog and has no protection from the rain. In that case, when it rains, my dog gets wet-both P and Q would be true, and therefore, it would be true to say (line 1) that the entire statement If it rains, my dog gets wet (If P, then Q) is true.

Now let's say it was raining, but my dog was in the garage, dry and cozy. In that case, it would be true to say that it was raining, but false to say that if it rains, then my dog gets wet (line 2). It rains, but my dog does not get wet. The statement would therefore be false.

But what about the other two possibilities? What if it does not rain and my dog gets wet (maybe because I sprayed him with the hose), or if it does not rain and he does not get wet (lines 3 and 4)? In what sense is it meaningful to say that, in these cases, the statement If it rains, my dog gets wet is true?

In fact, it makes no sense at all. How could a statement that is contingent on something happening be true or false if it didn't happen?

In the traditional system, a conditional statement asserts a necessary logical connection between the rain and your dog getting wet. But modern logic can only keep its system clean and efficient if it can solve the truth value of the whole on the basis of the parts. Formal efficiency is a Procrustean bed into which we try to fit language. But things get lost in the process-things like common sense.

But it is not just when the antecedent is false that the truth tables don't make sense. Take this statement:

If ducks can swim, then the moon orbits the earth

In this case, the antecedent is true, as is the consequent. And so the truth table tells us (line 1) that the whole statement is true. But is it? What does the fact that ducks can swim have to do with the fact that the moon orbits the earth? The antecedent and the consequent may individually be true, but there is simply no logical connection between them. In the traditional system, the statement is neither true nor false. It is simply meaningless.

The traditional system captures the actual meaning of conditional statements as we use them in real life: to assert a necessary connection between the antecedent (P) and the consequent (Q). In modern logic the only connection recognized is the happenstance coincidence of the truth or falsity of the individual elements. In the traditional view, we assert that the rain and the dog getting wet have a fundamental metaphysical relation. Modern logic was developed by people who rejected the very idea of metaphysics. Traditional logicians reject the idea that language can be quantified in the way that modern philosophers believe it can. Attempts to quantify language only serve to distort it.

Originally published in The Classical Teacher Late Summer 2014 edition.

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Logic & Reality: Why Traditional Logic Doesn't Use Truth Tables

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One of the questions people ask about traditional logic is why it doesn't teach truth tables. Modern logic, the most common kind of logic encountered in high school and college, uses them, so why does traditional logic ignore them? Many people encounter a smattering of logic in high school math courses.

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A truth table is a handy little logical device that shows up not only in mathematics, but also in Computer Science and Philosophy, making it an awesome interdisciplinary tool. The notation may vary depending on what discipline you're working in, but the basic concepts are the same. This primer will equip you with the knowledge you need to understand symbolic logic. We'll start with defining the common operators and in the next post I'll show you how to dissect a more complicated logic statement.
What is Boolean Algebra?

Boolean Algebra is a branch of algebra that involves bools, or true and false values. They're typically denoted as T or 1 for true and F or 0 for false. Using this simple system we can boil down complex statements into digestible logical formulas.

Unary Operators

Unary operators are the simplest operations because they can be applied to a single True or False value.

Identity

The identity is our trivial case. It states that True is True and False is False.

Negation

The negation operator is commonly represented by a tilde (~) or ¬symbol. It negates, or switches, something's truth value.

We can show this relationship in a truth table. A truth table is a way of organizing information to list out all possible scenarios.

We title the first column p for proposition. In the second column we apply the operator to p, in this case it's ~p (read: not p). So as you can see if our premise begins as True and we negate it, we obtain False, and vice versa.

Logical True and Logical False

These are kinda strange operations. Logical true always results in True and logical false always results in False no matter the premise. These operations are often referred to as "always true" and "always false".

Binary Operators

Binary operators require two propositions. We'll use p and q as our sample propositions.

AND

The AND operator (symbolically: ∧) also known as logical conjunction requires both p and q to be True for the result to be True. All other cases result in False. This is logically the same as the intersection of two sets in a Venn Diagram.

OR

The OR operator (symbolically: ∨) requires only one premise to be True for the result to be True. This is equivalent to the union of two sets in a Venn Diagram.

NOR

Logical NOR (symbolically: ↓) is the exact opposite of OR. It requires both p and q to be False to result in True.

XOR

Exclusive Or, or XOR for short, (symbolically: ⊻)requires exactly one True and one False value in order to result in True.

Conditional Operators

Implication

Logical implication (symbolically: p → q), also known as "if-then", results True in all cases except the case T → F. Since this can be a little tricky to remember, it can be helpful to note that this is logically equivalent to ¬p ∨ q (read: not p or q)*.

Let's create a second truth table to demonstrate they're equivalent. To do this, write the p and q columns as usual. Then add a "¬p" column with the opposite truth values of p. Lastly, compute ¬p ∨ q by OR-ing the second and third columns. Remember to result in True for the OR operator, all you need is one True value.
*It's important to note that ¬p ∨ q ≠ ¬(p ∨ q). In the first case p is being negated, whereas in the second the resulting truth value of (p ∨ q) is negated.
Logical Equality

Also known as the biconditional or if and only if (symbolically: ←→), logical equality is the conjunction (p → q) ∧ (q → p). In other words, it's an if-then statement where the converse is also true.

The only way we can assert a conditional holds in both directions is if both p and q have the same truth value, meaning they're both True or both False. This is why the biconditional is also known as logical equality.

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Intro to Truth Tables & Boolean Algebra - Math Memoirs - Medium

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A truth table is a handy little logical device that shows up not only in mathematics, but also in Computer Science and Philosophy, making it an awesome interdisciplinary tool. The notation may vary depending on what discipline you're working in, but the basic concepts are the same.

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The expression of numerical quantities is something we tend to take for granted. This is both a good and a bad thing in the study of electronics. It is good, in that we're accustomed to the use and manipulation of numbers for the many calculations used in analyzing electronic circuits. On the other hand, the particular system of notation we've been taught from grade school onward is not the system used internally in modern electronic computing devices, and learning any different system of notation requires some re-examination of deeply ingrained assumptions.

First, we have to distinguish the difference between numbers and the symbols we use to represent numbers. A number is a mathematical quantity, usually correlated in electronics to a physical quantity such as voltage, current, or resistance. There are many different types of numbers. Here are just a few types, for example:

WHOLE NUMBERS: 1, 2, 3, 4, 5, 6, 7, 8, 9 . . .

INTEGERS: -4, -3, -2, -1, 0, 1, 2, 3, 4 . . .

IRRATIONAL NUMBERS:
π (approx. 3.1415927), e (approx. 2.718281828),
square root of any prime

REAL NUMBERS:
(All one-dimensional numerical values, negative and positive,
 including zero, whole, integer, and irrational numbers)

COMPLEX NUMBERS: 3 - j4 , 34.5 ∠ 20o

Different types of numbers find different application in the physical world. Whole numbers work well for counting discrete objects, such as the number of resistors in a circuit. Integers are needed when negative equivalents of whole numbers are required. Irrational numbers are numbers that cannot be exactly expressed as the ratio of two integers, and the ratio of a perfect circle's circumference to its diameter (π) is a good physical example of this. The non-integer quantities of voltage, current, and resistance that we're used to dealing with in DC circuits can be expressed as real numbers, in either fractional or decimal form. For AC circuit analysis, however, real numbers fail to capture the dual essence of magnitude and phase angle, and so we turn to the use of complex numbers in either rectangular or polar form.

If we are to use numbers to understand processes in the physical world, make scientific predictions, or balance our checkbooks, we must have a way of symbolically denoting them. In other words, we may know how much money we have in our checking account, but to keep record of it we need to have some system worked out to symbolize that quantity on paper, or in some other kind of form for record-keeping and tracking. There are two basic ways we can do this: analog and digital. With analog representation, the quantity is symbolized in a way that is infinitely divisible. With digital representation, the quantity is symbolized in a way that is discretely packaged.

You're probably already familiar with an analog representation of money, and didn't realize it for what it was. Have you ever seen a fund-raising poster made with a picture of a thermometer on it, where the height of the red column indicated the amount of money collected for the cause? The more money collected, the taller the column of red ink on the poster.

This is an example of an analog representation of a number. There is no real limit to how finely divided the height of that column can be made to symbolize the amount of money in the account. Changing the height of that column is something that can be done without changing the essential nature of what it is. Length is a physical quantity that can be divided as small as you would like, with no practical limit. The slide rule is a mechanical device that uses the very same physical quantity-length-to represent numbers, and to help perform arithmetical operations with two or more numbers at a time. It, too, is an analog device.

On the other hand, a digital representation of that same monetary figure, written with standard symbols (sometimes called ciphers), looks like this:

$35,955.38

Unlike the "thermometer" poster with its red column, those symbolic characters above cannot be finely divided: that particular combination of ciphers stand for one quantity and one quantity only. If more money is added to the account (+ $40.12), different symbols must be used to represent the new balance ($35,995.50), or at least the same symbols arranged in different patterns. This is an example of digital representation. The counterpart to the slide rule (analog) is also a digital device: the abacus, with beads that are moved back and forth on rods to symbolize numerical quantities:

Let's contrast these two methods of numerical representation:

ANALOG                              DIGITAL
------------------------------------------------------------------
Intuitively understood  ----------- Requires training to interpret
Infinitely divisible -------------- Discrete
Prone to errors of precision ------ Absolute precision

Interpretation of numerical symbols is something we tend to take for granted, because it has been taught to us for many years. However, if you were to try to communicate a quantity of something to a person ignorant of decimal numerals, that person could still understand the simple thermometer chart!

The infinitely divisible vs. discrete and precision comparisons are really flip-sides of the same coin. The fact that digital representation is composed of individual, discrete symbols (decimal digits and abacus beads) necessarily means that it will be able to symbolize quantities in precise steps. On the other hand, an analog representation (such as a slide rule's length) is not composed of individual steps, but rather a continuous range of motion. The ability for a slide rule to characterize a numerical quantity to infinite resolution is a trade-off for imprecision. If a slide rule is bumped, an error will be introduced into the representation of the number that was "entered" into it. However, an abacus must be bumped much harder before its beads are completely dislodged from their places (sufficient to represent a different number).

Please don't misunderstand this difference in precision by thinking that digital representation is necessarily more accurate than analog. Just because a clock is digital doesn't mean that it will always read time more accurately than an analog clock, it just means that the interpretation of its display is less ambiguous.

Divisibility of analog versus digital representation can be further illuminated by talking about the representation of irrational numbers. Numbers such as π are called irrational, because they cannot be exactly expressed as the fraction of integers, or whole numbers. Although you might have learned in the past that the fraction 22/7 can be used for π in calculations, this is just an approximation. The actual number "pi" cannot be exactly expressed by any finite, or limited, number of decimal places. The digits of π go on forever:

3.1415926535897932384 . . . . .

It is possible, at least theoretically, to set a slide rule (or even a thermometer column) so as to perfectly represent the number π, because analog symbols have no minimum limit to the degree that they can be increased or decreased. If my slide rule shows a figure of 3.141593 instead of 3.141592654, I can bump the slide just a bit more (or less) to get it closer yet. However, with digital representation, such as with an abacus, I would need additional rods (place holders, or digits) to represent π to further degrees of precision. An abacus with 10 rods simply cannot represent any more than 10 digits worth of the number π, no matter how I set the beads. To perfectly represent π, an abacus would have to have an infinite number of beads and rods! The tradeoff, of course, is the practical limitation to adjusting, and reading, analog symbols. Practically speaking, one cannot read a slide rule's scale to the 10th digit of precision, because the marks on the scale are too coarse and human vision is too limited. An abacus, on the other hand, can be set and read with no interpretational errors at all.

Furthermore, analog symbols require some kind of standard by which they can be compared for precise interpretation. Slide rules have markings printed along the length of the slides to translate length into standard quantities. Even the thermometer chart has numerals written along its height to show how much money (in dollars) the red column represents for any given amount of height. Imagine if we all tried to communicate simple numbers to each other by spacing our hands apart varying distances. The number 1 might be signified by holding our hands 1 inch apart, the number 2 with 2 inches, and so on. If someone held their hands 17 inches apart to represent the number 17, would everyone around them be able to immediately and accurately interpret that distance as 17? Probably not. Some would guess short (15 or 16) and some would guess long (18 or 19). Of course, fishermen who brag about their catches don't mind overestimations in quantity!

Perhaps this is why people have generally settled upon digital symbols for representing numbers, especially whole numbers and integers, which find the most application in everyday life. Using the fingers on our hands, we have a ready means of symbolizing integers from 0 to 10. We can make hash marks on paper, wood, or stone to represent the same quantities quite easily:

For large numbers, though, the "hash mark" numeration system is too inefficient.

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The expression of numerical quantities is something we tend to take for granted. This is both a good and a bad thing in the study of electronics. It is good, in that we're accustomed to the use and manipulation of numbers for the many calculations used in analyzing electronic circuits.

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Up until now, you've probably considered mathematics as a set of rules for using numbers. The study of logic as a branch of mathematics will require you to think more abstractly then you are perhaps used to doing.

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This story is a supplement to the feature "Quantum Computing with Ions" which was printed in the August 2008 issue of Scientific American.

Entanglement: Spooky Action At a Distance
The "ambiguous cube" ( a) is like an ion in a superposition state-a measurement of the ion will lock it into one of two definite states (0 or 1). When two ions are in an entangled superposition ( b), a measurement will force both ions into the same state (either 0 or 1) even though there is no physical connection between them.

Truth Table
A trapped-ion computer would rely on logic gates such as the controlled not (CNOT) gate, which consists of two ions, A and B. This truth table shows that if A (the control bit) has a value of 0, the gate leaves B unchanged. But if A is 1, the gate flips B, changing its value from 0 to 1, and vice versa. And if A is in a superposition state (0 and 1 at the same time), the gate puts the two ions in an entangled superposition. ( Their state is now identical to the one shown in the box above.)
Illustrations: Jen Christiansen

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This story is a supplement to the feature "Quantum Computing with Ions" which was printed in the August 2008 issue of Scientific American. Entanglement: Spooky Action At a Distance The "ambiguous cube" ( a) is like an ion in a superposition state-a measurement of the ion will lock it into one of two definite states (0 or 1).

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It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not, A and B".

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intel 8255a programmable peripheral interface

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