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Michael A.
Recent Applied Math Graduate From University of California, Davis
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Pre-Calculus
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Question:

Why do we sometimes state that the solution to a problem is "all real numbers"? What does that mean? Aren't all numbers real?

Michael A.

This is actually a very good question with somewhat of an abstract answer, but it is important to know the distinction between the different sets, or collections, of numbers. In pre-calculus and calculus, we primarily deal with real numbers - the set of numbers that we've been accustomed to dealing with thus far in mathematics like $$4$$, $$3/2$$, $$-2.5$$, or even $$\pi$$. The set of real numbers (often denoted by $$\mathbb{R}$$), is composed of a few different sets of numbers which we generally take for granted. Let's take a look at some of these sets. We'll start with the integers. The integers (denoted by $$\mathbb{Z}$$) refers to the set of all positive and negative whole numbers, as well as zero. In set notation, we use the curly brackets '{' and '}' to refer to items that belong to a set. Using this notation, the set of integers is as follows: $$\mathbb{Z} = \{... -2, -1, 0, 1, 2, ...\}$$, where the dots imply that the list continues infinitely beyond those numbers. The integers are the numbers we used when we first started to learn basic math and should be very familiar. Note that all integers are whole numbers, meaning that fractions are NOT integers. Another very closely related set to the integers is the set of natural numbers (denoted by $$\mathbb{N}$$), which is almost the same as the integers, except it only includes the non-negative integers. In other words, $$\mathbb{N} = \{0, 1, 2, 3, 4, ...\}$$. Building upon our previous definition, the rational numbers (denoted by $$\mathbb{Q}$$) refers to the set of numbers that can be expressed as a ratio of integers. In other words, for any two integers (recall that this means any two whole numbers numbers belonging to $$\mathbb{Z}$$) $$n$$ and $$m$$, a rational number $$q$$ is one that can be written as $$q = n/m$$. An easy example of this is any fraction you can think of. The numerator is $$n$$, and the denominator is $$m$$. However, the rational numbers span more than just fractions - they actually also include the integers and rational numbers. If you let $$m = 1$$, then it is clear that $$q = n$$, meaning that any positive or negative number, as well as zero, is by definition also a rational number. So the rational numbers make up a lot of the numbers we use in arithmetic without really thinking about it. Now, you might think that this compromises of all the numbers we use, but in fact this is not true! Not all numbers we use can be expressed as a rational number using a ratio of integers. For a very classical example, think of $$\sqrt2$$. Can you think of 2 integers $$n$$ and $$m$$ such that $$\sqrt2 = n/m$$? The answer is no because no such integers exist! Thus, $$\sqrt2$$ is not a rational number but instead belongs to a set of numbers we call irrational numbers. Sometimes, this set is denoted by $$\mathbb{I}$$ and refers to these numbers that cannot be expressed as a ratio of integers. These numbers have decimal expansions that never repeat or end, and can be commonly generated by taking the square root of numbers. Some very common irrational numbers that we take for granted include $$\sqrt2$$, $$\pi$$, and $$e$$. Now, in terms of these sets, what is the set of all real numbers, $$\mathbb{R}$$? Well, it is the combination of the set of rational numbers $$\mathbb{Q}$$ and the set of irrational numbers $$\mathbb{I}$$. That just about accounts for all the conventional numbers we use on a daily basis. But now you're probably asking, what is NOT a real number, then? Well, that is something you probably dealt with in algebra II. One such set of non-real numbers is the set of imaginary numbers - the numbers whose squares are negative. You probably recall seeing the identity $$i = \sqrt{-1}$$. This is one such imaginary number. Ultimately, if you multiply any real number by $$i$$, you will obtain an imaginary number, and these numbers have very interesting properties that are typically discussed in a university setting. Although you have dealt with algebra involving these imaginary numbers and have been able to treat them as though they were real, they behave quite differently than the real numbers we are used to dealing with, and for that reason the topic branches off into a completely different and exciting region of mathematics. In summary, not all numbers are real, but for our study of pre-calculus and further into calculus, we will mainly utilize the real numbers as defined above. The importance of distinguishing between the different sets of numbers may not be completely clear here, but there are certainly times where you will need to understand the difference between them. One such example comes up in calculus when you deal with sequences, which are functions whose independent variables are the natural numbers $$\mathbb{N}$$, not the real numbers like we're used to seeing with function formulas like $$f(x)$$ where $$x$$ is taken to be a real number. This difference is very subtle and hard to recognize without understanding the above classification of numbers.

Pre-Algebra
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Question:

How do you isolate for a variable? For example, how do you solve $$3x = 9$$?

Michael A.

Often times you'll find that you'll be given some sort of equation that contains a variable, and the question will ask you to find the value of that variable. In other words, it is asking you to isolate, or solve, for the variable. The equation itself can be as complicated or simple as the situation calls for, and methods will vary, but to begin, we will consider a relatively simple class of equations in Algebra called linear equations. Let's exemplify this with the proposed example: $$3x = 9$$. Now, what is this equation telling us? It says that if we multiply $$3$$ and some particular number $$x$$, we get $$9$$. Now, if we stopped to think about this in our heads for awhile, we most likely would be able to come up with an answer. Since we are well versed in our multiplication tables, after some guessing and checking, we could come up with the answer $$3$$ - we know that $$3 * 3 = 9$$, so based on the original equation, that must mean that $$x = 3$$. While this method is a very logical and there's nothing wrong about it, it doesn't express any specific strategy of reaching the answer. Therefore, if we were given an equation that was more involved, it would not be immediately obvious what $$x$$ would be. This is why we need to develop a process by which we isolate the variable $$x$$. Let's return to $$3x = 9$$. Let's build a strategy to obtain the same answer we found earlier. First, we need to understand that an equation signifies a fact, and whatever we do to this equation must respect that fact. In other words, we need to preserve the equality. We cannot add or multiply something to one side of the equation without performing the same action to the other side, or else we would end up with a statement that isn't true, and therefore the equality would not be held. Thus, one rule of thumb is that whenever you do something to one side of the equation, repeat the action to the other side. Since we want to find out what $$x$$ is, we will want to get it by itself on one side of the equation (hence, isolating for $$x$$). What stands in our way? Well, on the left side we have $$3x$$ when we want $$x$$. In order to remedy this, we can divide $$3x$$ by $$3$$ to get $$x$$, because dividing by $$3$$ undoes the multiplication of $$x$$ and $$3$$. But now we need to make sure we preserve the equality. Since we divided the lefthand side of the equation by 3, we must also make sure to divide the right hand side of the equation by $$3$$. In other words, we must divide $$9$$ by $$3$$, which is $$3$$. So now we have reduced our original equation $$3x = 9$$ to $$x = 3$$, which gives us the same answer we obtained by our guess and check method. To recap, let's run through the steps we followed to isolate for $$x$$. First, we started with the original equation, $$3x = 9$$ And then we decided to divide the lefthand side by 3 to get rid of the multiplication by 3. Since we are dealing with an equation, we had to perform this action to both sides in order to preserve the equality: $$3x/3 = 9/3$$ To finally obtain $$x = 3$$. Now, we've dealt with our first basic linear equation. As we progress in algebra, these equations will become trickier, but as long as we remember to respect the equality and use our pocketed arithmetic skills wisely, we will be able to find the value of $$x$$ no matter the linear equation. This is a stepping stone to solving equations, which will prove to be a skill you will use extensively throughout your education in mathematics.

Calculus
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Question:

If we have a function $$f(x)$$, what is its derivative $$f'(x)$$ and what are the implications of it? Why is it important?

Michael A.

Well, you can think of a function very generally as a map from the input you feed it $$x$$, to the value it spits out at you $$f(x)$$. Graphically, this will typically appear as some sort of curve in the $$xy$$ plane traced out by the coordinates corresponding to $$(x, f(x))$$. When you derive $$f(x)$$ or take the derivative of $$f$$ with respect to $$x$$, the processor mechanism you use to derive will heavily depend on the nature of the actual function $$f$$. As we cover the wide variety of functions we can encounter in math, we will explore the general rules and trends used to find derivatives, so that will come with time. However, graphically, the derivative has a particularly useful interpretation. You can think of the derivative of $$f(x)$$, $$f'(x)$$, as the rate of change of $$f$$ at a specified value of $$x$$. This means that for any particular $$x$$, $$f'(x)$$ measures the slope of the line tangent to $$f$$ at $$x$$. In this sense, the derivative can be defined as an instantaneous rate of change. Now, why should we care about the derivative? Well, in nature we can mathematically model many phenomena by relating a measured value, such as position, to an independent variable that makes sense for the particular study, such as time. So when you study the relationship between position and time, you'll see that the position $$x$$ is actually a function of time, $$t$$, so we can express this relationship by virtue of some function $$f$$, $$x = f(t)$$, where $$f$$ will also depend on the study. When we take the derivative of $$f$$ with respect to time, $$f'(t)$$ will then represent the instantaneous rate of change of position at any time $$t$$. In other words, the derivative of $$f$$, in this case, gives us the velocity of the body we are studying - a very useful quantity in scientific disciplines. Now, while this was a very specific example, the takeaway message here is that the derivative allows us to study how measured quantities change. Many times in life we will want to know how things change and by how much. How can we accomplish this? By using derivatives!

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