Derivatives and an Introduction to Differential Calculus

Having recently posted a new article introducing the concept of derivatives, I came across this old article I wrote earlier, and thought it was probably a more thorough introduction to the concept.

One of the main concepts studied in the field of differential calculus is based on the notion of change – specifically, how one quantity changes compared to another. Perhaps a more succinct version of this physical definition would be “rate of change.” Alternately, a geometric definition could simply be the slope of a curve at a particular point. The underlying key to this branch of mathematics is the concept of the derivative. In this post, I will introduce various aspects of the derivative!

First, let’s consider the derivative in terms of rate of change. And to do that, let’s talk about velocity. We know that velocity is equal to distance per unit time. However, that is a very general description of it. If you drive from your home to the grocery store on the other side of town, you can do the math by dividing your total distance travelled by the time it took you to get there, but what does this number tell you? It actually tells you the average velocity of your trip. Think about it. You had to stop for red lights, stop signs, pedestrians. Maybe you sped up to pass a slow driver. Don’t forget about the actual acceleration of your car from a standstill, and then the deceleration whenever you needed to stop. All of this factors into the calculation of your average velocity, which is simply how far you go in a measured amount of time.

Now, let’s consider how to calculate the average velocity of your car between your home and the first stop sign, while on your way to the grocery store. You no longer have several stops to deal with. You get in your car, accelerate, then as you approach the stop sign, you decelerate to a stop. Your average velocity is calculated from a much shorter interval, and will have much less variation to it. So, your average velocity will be more representative of your actual velocity at any given time.

To extend this demonstration even further, let’s consider the small portion of your trip that is measured between two street lights 10 meters apart that you pass while you are in full motion. That is, let’s assume that we want to measure the velocity without having to calculate stop signs, etc. Our time interval for the measurement is much smaller, and calculating the velocity by dividing the distance by the time it takes to go from one light to the other is even more representative of your velocity at any point between them.

What I am trying to demonstrate is the concept of instantaneous velocity. If you shrink down your time interval of measurement infinitesimally, the two time points approach each other at a single point, and so the average velocity between the two super close points approaches the instantaneous velocity of the single point.


Graphically, this is essentially the same thing you do when you calculate the slope of a tangent line to a curve. You pick two lines on the curve and calculate the slope of the line between them, and then you use limits to make the points get closer and closer together until they are almost the same, and the slope of the line connecting those two infinitely close points is the tangent. (Check out my previous post about using limits to find tangents if you’d like a refresher of this topic:

Because this type of limit occurs so frequently in maths, science, and engineering, it is given the special name of “derivative,” and you calculate derivatives through the process of “differentiation.” So, one interpretation of the derivative is an expression of the instantaneous rate of change (velocity) at a particular point on the curve – a large derivative corresponds to a high rate of change (a steep curve), and conversely a small derivative corresponds to a low rate of change (a relatively flat curve). As a specific example, if you actually have a graph of position (displacement) of an object vs. time, the derivative of the curve at any time point represents the velocity of that object at that specific time. This may take a little practice to become comfortable with the concept, but suffice it to say at this point that learning how to use derivatives is incredibly important to be able to work out more complex concepts relatively easily.

Let’s look at this now in the more formal terms of mathematical symbols and equations. Consider any curve y = f(x).

Now, let us identify the point P on the curve f(x) for when x = a. That is to say, the point (a, f(a)).

Now, let’s go a step further, and identify a point Q that is h units away from a on the x-axis. If it is h units away from a, we can call it “a + h”. (If this is confusing, think about it with numbers instead. Start at, say, x = 3 (instead of a). Now we want to know what is going on 5 units (instead of h) away from x = 3. In other words, we have 3, and we have 3 + 5.) As such, we can therefore identify a point Q ((a + h), f(a + h)).

Now that we have two arbitrary points, let’s determine the slope of the straight line that would connect the two. We can use the same slope formula that we always use, slope = rise/run, but substitute in our variables that we identified above. So, we have:

Now, imagine that the distance h between the two points is getting smaller and smaller. Or in other words, consider the case of when h approaches 0. By doing this, we calculate the slope of the line connecting two infinitesimally close points – which means that we are actually approaching the slope of the tangent line to the curve at point a. In this case, we would express this slope as a limit in the following way, which actually corresponds to the definition of the derivative of a function f at a number a. The derivative is given the special symbol f’(x), and we say “f prime x”, and we express it like this:

Another way of expressing this can be found if we recognize that a + h is really just any x value. So, we can say x = a + h (and by extension, h = x – a), and modify the above derivative definition accordingly:

With this modified equation, it actually becomes a matter of arithmetic to determine the slope at a point. Here is an example of a kind of question that you will see:

“Find the derivative (or, determine the slope of the tangent) of the function f(x) = x2 – 4 at a number a.”

To do this, write the provided equation into the definition, and reduce until you have an answer. Notice below how I combine terms and recognize the identity of a difference of squares.

What this final result tells you is that for our curve, f(x) = x2 – 4, at any number a along it, the slope of the tangent (AKA, the derivative at that point) is equal to the term 2a. Graph it out and try with several values to convince yourself that it’s true! Consider when x = 5. You can determine from the original equation that we have the point (5, 21). At this point on the curve, the slope of the tangent equals 2 x 5 = 10.

Going back to the definition of the derivative that I gave above, you can also apply the concept of point-slope form to it to get a different way of seeing it. Letting y = f(x), you can rearrange the definition as follows, by simple reorganization of the terms:

Here’s a more visual exercise that you may soon encounter.

“If you are provided a graph of a function f(x) – not necessarily the equation – sketch out what the graph of the derivative f’(x) would look like.”

When you actually have the numbers and equation, this becomes much easier… assuming you know how to easily recognize derivatives from the original equations. However, if provided ONLY the picture of the curve, this becomes a bit more abstract, but not really that challenging. It DOES require you to understand the concept of derivatives and rate of change though. Here is why. Take some random curve that you can draw. Any curve will do for this exercise:

The key is rate of change. We have seen that slope is equal to rate of change, so we want to pay particular attention to the slope at several points. And the easiest points to notice are those where the slope is equal to 0. These are all the peaks and valleys of the curve. What I have done next is highlight with red bars all of the zero-slopes:

Now, to proceed with sketching the graph of the derivative f‘(x) vs x, you can start by plotting the points where f’(x) is equal to zero. From there you can then go on to say where the curve of f(x) has a positive, increasing slope, and then sketch that into your f‘(x) graph accordingly. Similarly, decreasing slopes on the f(x) curve will be negative values on the f’(x) curve. For the sake of this exercise, don’t worry so much about how high or low the slopes are. Just focus on whether they are positive or negative at the various parts of the graph. I have gone ahead and plotted out the actual curve of the derivative below in green, alongside the original curve of f(x). You can see that the f‘(x) curve crosses zero wherever the curve for f(x) has peaks or valleys, and the steeper the f(x) curve, the more extreme the f’(x) curve at that same point.

Of course, having a mathematical definition wouldn’t be any fun if there were no conditions or rules associated with it – and the definition of derivatives is no exception. One such rule states that a function is differentiable at a point a if the derivative f’(a) exists. Seems intuitive enough. If a derivative at a point exists, then the base function is differentiable at that point. Probably one of those rules that doesn’t really even need to be said. :)

I’m not going to graph this one out, but it is for you to think on. Consider the case of f(x) = |x|. Where is it differentiable?

If you consider the derivative of the left hand side, it equals –1. The f‘(x) on the right hand side is 1. This function then is obviously differentiable when x < 0, and when x > 0. But what about when x = 0? In this case, since the right hand limit approaches 1 as x approaches 0 from the right, and the left hand limit approaches –1 as x approaches 0 from the left, one must conclude that f’(0) does not exist because both of the one-sided limits approach different numbers.

An extension of this example actually describes a second rule for limits: if f’(a) exists, then the function f(x) is continuous as a. Recall that continuity of a curve is based on the notion that as you approach a point from both the left and the right, the limit of each side approaches the same value. In the example above, approaching 0 from either side resulted in different limits, and hence the graph is not continuous at 0.

Keep this in mind as you see various graphs of functions. Curves that have a sharp point will not be differentiable at the point, for the reason given above. Similarly, discontinuous curves (i.e. curves with gaps in them) will not have a derivative at the break point either because the one-sided limits do not agree. If f(x) is not continuous as point a, then f’(a) does not exist. A third condition to watch out for is where a graph has a vertical tangent line, in which case the slope is infinite.

Now, I’m going to wrap up this mammoth of a math post with something a bit easier to talk about: notation of derivatives. I have already described a few ways to express these values. I talked about expressing them as limits, and using infinitesimally smaller intervals, and that is a good way to work through them. Symbolically, I said that you can write f’(x) to denote the derivative of the functions f(x). This will likely be the easiest way for you to use it and to recognize it, though here are a few others that mean the same thing:

Each of these terms means the exact same thing. In particular, the D and d/dx are specifically called the differentiation operators, and you can see they have a few variations. Similarly, dy/dx is symbolic of derivatives for historical reasons. Read up on Gottfried Wilhelm Leibniz to learn more about the origins of calculus, where you will see that he introduced this way of representing it. Sometimes, you may see dy/dx referred to as “Leibniz notation.”

And with that final tidbit of mathematical goodness, I am going to end this post. I intend to follow this with another post in the near future that introduces differentiation methods and strategies. Much like the exponent rules, there are also several differentiation rules, and I hope to be able to explain them for you as well. If you have made it to this point of my post, thanks for reading, and please be sure to click the Facebook Like button below or at the top and share with your friends!

What is a Derivative in Calculus

When you are introduced to the study of calculus in your math class, one of the first concepts you will deal with is derivatives. Derivatives do not sound like your typical math functions that you would be familiar with up this point, but they are a core component of calculus and you will work with them in many ways. In this post, I am going to give you a brief introduction to the concept of the derivative, hopefully answering your question “what is a derivative in calculus,” and then I will follow this up in future posts with more detailed examples of using derivatives.

Consider a curve on a graph – any curve will do. Now, pick a point on the graph, and draw a tangent to the graph at that point. Recall that a tangent to a curve is a line which touches the curve at only a single point. This tangent line is a representation of the derivative of that curve at that particular point on the curve. Naturally, every point on your curve will have its own tangent line. Here is a rough example, where I have generated some curve, and the red lines indicate the tangent lines to the curve at the points where the lines touch the curve:

By using the derivative of the equation of your curve, you are able to precisely calculate what the tangent line will look like at any point on your curve. In fact, the derivative of your equation will very likely be another equation, and you can graph this new equation out to see the graph of the derivative, which itself can be interpreted and used in many ways in calculus. You can even find the derivative of the derivative, etc.

Let’s look at a more practical example to hopefully convey the usefulness and significance of calculating derivatives.

If we consider two points A and B, and we know it takes some time to travel from A to B, we can calculate the “average” velocity of this trip by dividing the distance between the two points by the time taken to travel between them. This should be a familiar notion. However, consider this: if we are driving in a car from home to the market, then we do not have the same velocity for the entire trip. We have to start accelerating from being parked, and then decelerate when we come to stop signs, and then accelerate again, and decelerate to a stop when we reach our destination. You can calculate the average velocity of this trip in the way I mentioned above, but the “instantaneous” velocity at any particular point of the trip might be what you are more interested in. How might you go about doing this? Find the derivative! A graph of our trip, where we plot distance travelled against time, might be useful in this case. We can determine the tangents to this curve at any particular points. We know from previous graphing experience that these tangent lines have a slope of rise over run, and distance over time (the y and x axes on this graph) is velocity. So therefore, the tangent line (specifically, its slope) at any particular point on our curve actually represents the instantaneous velocity at that point. So, we can find out exactly how fast we are going at any point of our trip. We just demonstrated this graphically, but by calculating the derivative of our “trip equation” means that we can calculate how fast we were going any any point without actually graphing our equation at all!

This is just an introduction to the concept of derivatives in calculus, but in future posts, I’ll go into more details so that hopefully you can become more familiar with them. The concept might be new and different, but once you begin working with them, you will see that they are not that difficult to work with at all.

Logarithm Rules

In my previous post, I introduced the concept of logarithms to you. I explained how logarithms and exponents are connected, and then showed you a quick trick to help you remember how to convert between the two. Now in this post, I’m going to go a little bit deeper and explain a few rules of logarithms to help you actually do math with them. Considering how you now already know that logs and exponents are related, it should come as no surprise that, just like there is a set of exponent rules, there is also a set of logarithmic rules.

As you look at these logarithm rules, keep in mind that by convention, if you write logs without the subscript number to indicate their base, it is assumed that you are dealing in base–10. For simplicity, this is the convention that I am going to use in this post, though these rules certainly apply when dealing with logs of other bases.

With that intro out of the way, let’s get to it.

The first law of logarithms is the product rule. If you are familiar with the product rule of exponents, then this logarithm law should be a piece of cake for you. Where the exponent rule says that when multiplying exponential expressions with the same base, you simply add the exponents, this same thing applies when multiplying logarithms of the same base. Therefore, the rule states that the logarithm of a product is equal to the sum of the logarithms.

This rule is very commonly used, and it is important to recognize that you can use it in either direction. That is, the logarithm of a product converts to a sum of logarithms, and vice versa.

The next logarithmic law is the quotient rule. Again, this law can easily be derived by applying your knowledge of the exponent quotient rule (though I will leave that for you as an exercise). However, it does appear to look different. This rule states that the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. Sounds like a mouthful, but the expression is probably much simpler to understand.

Again, watch for opportunities to use this relationship in either direction!

The third law of logs is the power rule. This one is surprisingly simple to remember, and again can be found by manipulating exponent and logarithm laws. Quite simply, this law says that when you have a logarithm of an exponential expression, the exponent can be “brought out” of the log and used as a coefficient for the log.

The last of the rules of logarithms that I’m going to discuss here today is called the base change rule. Recall that I stated above that all of my examples that I’ve used so far in this post use the convention of an assumed base–10. If I wanted to change my expression to utilize a different base, this rule helps us to do that. So then, if I have my log in some base of a number, and I want to express this in terms of a different base, I simply take the log in my new base of the original number and divide that by the log in my new base of the original base. Sounds wordy, but again, a picture is worth a thousand words:

Here, my original base is B, and my new base that I want to express things in is X.

That is all I have to say about the rules of logarithms in this short introduction to them. They are fairly straight forward themselves, though can be used in very complex equations. I will try to do a separate post soon outlining some examples of all of these rules, though I do think that the general forms that I’ve noted above are pretty self-explanatory.

If you are interested in learning more about logarithms, there is a much more thorough summary of logarithms at the Learning and Teaching Math blog, which I highly recommend (for this and other math topics!)

Decimal, Binary, Octal, and Hexadecimal Number Systems Explained

When we talk about numbers, in most cases, we are referring to numbers composed of the digits 0–9. We don’t really put any thought into it, and just accept that this is how numbers are represented, because this is probably what we were taught when we were really young. However, this is just one possible numeral system that could be used to represent quantities. There are other systems that you could use, and in this post, I’m going to give brief introductions to the decimal, binary, octal, and hexadecimal number systems.

The decimal numeral system is the counting system most widely used around the world. You may also hear of it referred to as a base ten system. In fact, this is where the word “decimal” comes from; Latin “decem” means ten. When we say it is a base ten system, that means that each digit position has ten possibilities, or more specifically, can be any of the digits 0 – 9. So, if we count out 15 apples, we know that this breaks down to a 5 in the ones column and a 1 in the tens column, and so we automatically equate this as 1 times ten and 5 times one to make 15. This also shows the importance of each position in the number. In a base ten system, there is a column for ones, tens, hundreds, thousands, etc. All of this is probably very familiar already, so I’m not going to go into any further details of the decimal system.

Now, let’s look at the binary numeral system. While the decimal system is the most commonly used system, probably a very high percentage of people are at least aware of the binary system, though they probably don’t fully understand it. The binary system is a base two system, usually denoted as 0 and 1. In fact, the reason why binary is historically associated with computer programming is because this 0 and 1 can physically be represented in computer circuitry as being “open” and “closed” states – way more than I’m going to explain here, but interesting nonetheless. If we consider a base two system, we quickly see that counting is an entirely different experience. Unlike the base ten system with ones, tens, hundreds, since the base two system only uses two digits, instead it has digit positions for ones, twos, fours, eights, sixteens, etc. Each position is a factor of 2 greater than the position to its right, much like in base ten where each position is a factor of 10 greater than the position on its right.

Let’s count apples again to see binary counting in action:

  • 0 apples = 0 in binary
  • 1 apple = 1 in binary
  • 2 apples = 10 in binary
  • 3 apples = 11 in binary
  • 4 apples = 100 in binary
  • 5 apples = 101 in binary
  • 6 apples = 110 in binary
  • 7 apples = 111 in binary (= 1 one + 1 two + 1 four)
  • 8 apples = 1000 in binary

The important thing to realize is that physically, we have the same quantity of apples, regardless of the numeral system we are using to represent them. So to speak of 8 (base ten) apples or 1000 (base two) apples, we are really talking about the same thing.

We can apply these counting concepts to the other numeral systems as well. Let’s look at the octal numeral system first. As you can probably tell from its name, octal is a base eight system. So, each number position can have any of the digits 0–7, and the labelling of the positions goes up by a factor of 8 for each position. So, we have ones, eights, sixty-fours, five hundred twelves, etc. Let’s look at some apple quantities again, but in octal this time. You can figure out what 0–7 apples looks like in octal notation, but what about 8 or higher:

  • 7 apples = 7 in octal
  • 8 apples = 10 in octal
  • 9 apples = 11 in octal (= 1 one + 1 eight)
  • 10 apples = 12 in octal
  • 11 apples = 13 in octal (= 3 ones + 1 eight)
  • 20 apples = 24 in octal
  • 24 apples = 30 in octal
  • 30 apples = 36 in octal (= 6 ones + 3 eights)
  • 100 apples = 144 in octal (=4 ones + 4 eights + 1 sixty-four)
  • 750 apples = 1356 in octal (= 1 five hundred twelve + 3 sixty-fours + 5 eights + 6 ones)

Finally, let’s look at the hexadecimal number system. This system, also used frequently in computer programming, is a base sixteen system (as can be seen by its name: hexa- mean six and deci- means ten = 6+10 = 16). This counting system may take the most work to understand, but again, it follows the same rules. In this case, however, we run into a wrinkle. So far, we have looked at decimal, binary, and octal systems. All of these utilize at least some of the familiar digits 0–9. But how do we represent a number between 10–15 by only using a single digit? We only know of these quantities as two-digit numbers! Is there anything else we might use that is only a single character, without having to make up a brand new one? This is where hexadecimal is slightly different from the others. It uses 0–9 to represent those digits, though for 10–15 it uses the letters A-F. So, to write 10 in hexadecimal, it is simply A. Similarly, 11 is B, and so on. As a result, the number positions again are larger than the previous systems, so we have ones, sixteens, two hundred fifty-sixes, four thousand ninety-sixes, etc.

So counting apples again!

  • 13 apples = D apples in hex
  • 15 apples = F apples in hex
  • 16 apples = 10 apples in hex
  • 26 apples = 1A apples in hex
  • 100 apples = 64 apples in hex
  • 124 apples = 7C apples in hex
  • 50000 apples = C350 apples in hex

By including letter characters as digits, hexadecimal may be the most challenging of these number systems to wrap your head around. The bigger the numbers get, the more complicated they may seem. However, they all follow the same concepts and aren’t too difficult to figure out, once you get used to thinking in a number system other than base ten!

Logarithms Explained – What is a Logarithm?

Logarithms sound like very advanced math concepts, but they’re not really that much more difficult than many other mathematical concepts. The word “logarithm,” like the word “algorithm,” gives the impression of being a very complex topic. However, much like how an algorithm is nothing more complicated than a set of rules or processes to follow, a logarithm can also be very easy to work out. The first step to clearing up any confusion, though, is to explain what is a logarithm?

So then, just what is a logarithm?

A logarithm is related to exponents. Actually, it’s kind of like working backwards with exponents. A logarithm (or “log” for short) of a number is the exponent that another value (the base) must be raised to equal that number. Still sounds complicated, right? Looking at it another way, a logarithm tells us how many of one number we multiply together to get another number. That sounds a bit more manageable. Now let’s look at an example so that you can really see how easy this is.

Let’s consider the number 81. How many 3’s do we multiply together to equal 81?
3 x 3 = 9
3 x 3 x 3 = 27
3 x 3 x 3 x 3 = 81
As you can see, and easily work out yourself, you multiply 3 four times to produce 81. Another way to write this is with exponents, like this:

Now, what does any of this have to do with logarithms? In this example, we can say that the log with base 3 of 81 is 4. Or, as we found above, we multiply 3 (our base) 4 times to get 81. It is more common to say this as “the base 3 log of 81 is 4.”

What would a new math concept be without introducing a new expression? Of course there is a simple way of writing this down on paper, rather than longhand with words. Here is what our expression looks like:

Let’s compare our two expressions (the exponential notation and the log notation), and then I will show you a trick to help you work with them. This trick got me through years of logs in school! Notice first that all of the numbers are used in both expressions, though they’re in different places. In this case, the 3 is the base value, which is raised by the exponent 4 to produce the result 81. Similarly, with the logarithm notation, the base value is again 3, and we want to know what number this base is raised to give our results of 81. So, as you can see, the solution to the logarithm actually tells us what the exponent is!

Here is the simple arrangement of these components that help when working with logs. It can basically be boiled down to a simple rearrangement. The 3 moves over and bumps the 4 up to the exponent level, you remove the “log” word, and you’re left with your exponential notation of the expression.

Hopefully this post has helped you to understand what a logarithm is, and how logarithms are related to exponents. As I explained at the start, the concept is not that difficult. Now, much like the exponents they are related to, there are a bunch of rules that you need to follow when doing any kind of mathematical operations involving them. In my next post, I’ll begin to explain some of these logarithm rules.

Thanks for reading my post! Please remember to like me on Facebook if this was helpful for you. Of course, as always, feel free to leave me some feedback in the comment below.

The Merry Christmas Equation

This is a fun post for the holiday season that I first read over at my friend Guillermo’s page, Math and Multimedia. You often hear that everything can be described by mathematics. In this case, we’re going to use some math to convey a special message for this time of year!

To begin, let’s start with this equation:
Now let’s rearrange a bit by multiplying both sides by r2:

We currently have in this expression a natural logarithm. Since the natural logarithm ln(x) is defined as the inverse function of the exponential function ex, we can remove the logarithm from our equation entirely by raising the number e to the power of our equation:

Now, since ln(ex) is equal to x, we can apply this to our equation to leave only what was in the parentheses on the right side:

To remove the denominator, we have to multiply both sides by m:

Finally, with a little bit more rearranging, we are left with our final expression:

Merry Christmas and Happy Holidays to all of my readers, and best wishes for the New Year!

The Beauty of Mathematics

For all those people who question why we must study mathematics, here is a video for you. While math is a great way to stretch your mind, to learn important problem solving skills that are transferable to other aspects of your life, math goes so much further than just that. The beauty of mathematics is all around us. From the very small scale of chemical reactions that give rise to biological life, all the way to the grand scale of physics that dictate gravitational theory or star formation, everything has some sort of formula or concept that helps explain what is going on, or predicts what is going to happen. It’s not a long video, but it has some cool imagery that makes you stop and recognize that some things that we take for granted everyday are really just complicated mathematical systems. Give it about watch and let me know what you think in the comments below!


Google’s Free Online Calculator

It’s probably happened to you before: you go to do your math homework and you suddenly realize that you don’t have your calculator handy. What if you have a test tomorrow and you absolutely need your calculator right now to help you study? If you have your smartphone or computer with you, you can easily access Google’s free, built-in calculator, and your problem is solved!

Google has included many special features into their search page that aren’t immediately obvious to the users. For example, you can type in New York weather to get a weather report for New York (instead of getting search results for weather websites). Or, if you search for your favourite sports team, you can get the latest score from their game. Of particular interest to this post, however, is the built-in calculator function.

To access Google’s calculator, all you need to do is type “calculator” or enter the mathematical formula into the search box, and you will be presented a result in an online calculator app. From there, you can make changes to your formula, or clear it and start over again. This calculator even goes beyond basic math functions, by providing buttons for trigonometry, logarithms, exponents, etc. You can even change between Radians and Degrees modes, just like on a real scientific calculator.

This calculator isn’t just limited to being available on desktop browsers. You can access it just the same from mobile browsers on your iPhone or other smartphone. The mobile version has a slight variation to it, in that you are shown a basic calculator when you hold your phone in portrait mode, but flip it horizontally into landscape mode, and it turns into the full-size scientific calculator.

Also, if you have Desktop Voice Search enabled on your Google Chrome browser, all you need to do is click the mic icon, and speak your math equation. Google will interpret your words, and return the calculator to you with the result. Related to this Google calculator is the unit conversion trick you can do in the search box. Simply type in something like “3.25 miles in km” and it will do the conversion for you in the result.

These tricks are great time savers if you find yourself without your calculator, or you just want to get a quick answer to a math question without having to bother going to look for your calculator. And the way website browsers are designed now, it’s takes almost no time to turn on the device and get to a search box, where you can enter your math questions!

Multiplying Integers – Why Multiplying Two Negatives is a Positive

I have recently been asked to explain why two negative numbers multiply together to product a positive number. This can be a difficult concept to grasp at first, but I would like to try to simplify an explanation. First, I would like to recommend and give credit to David’s website, at Practic-All. In particular, he has posted a document about “Why the Rules for Multiplying Integers Work,” which I highly recommend.

Briefly, to summarize David’s work, you want to think of the two numbers you are multiplying together as being two “groups.” After that, the key to multiplication is to understand that it is a form of repeated addition. That is to say, if you multiply 2 x 3, you are really adding 2 + 2 + 2, or 3 groups of 2.

So, if you have 5 x 2, it is really saying “add 2 groups of 5,” or 5 + 5. (Of course, it could also mean add 5 groups of 2… they both mean the same thing.)
Similary, if you have 10 x 4, it is telling you to “add 4 groups of 10,” or 10 + 10 + 10 + 10. (Or similarly, 10 groups of 4.)

Multiplying two positive numbers is easy. But what about if one of the signs is negative? What does that mean? Well, if you look at it as groups, and remember the key of repeated addition, you will hopefully be able to understand it.

Take 2 x (–5). Using our language, it is telling us to add 2 groups of (–5). The sum is now a larger group equal to (–10).

Another way to look at this is to consider it this way: 2 x (–5) could be the equivalent of saying “take away 5 groups of 2.” Again, this gives us (–10).

Putting all this together, we can hopefully now see why multiplying two negative numbers gives a positive number. (Keep in mind that subtracting a negative number is the same as adding a positive number.) Take the example of (–5) x (–6). We look at it as “take away 5 groups of (–6).” If I write this out, starting with nothing, I could show this as:

0 – (–6) – (–6) – (–6)- (–6)- (–6)

Of course, taking away a negative is like adding a positive, we can rewrite this as:

0 + 6 + 6 + 6 + 6 + 6 = 30

Try to think of multiplication just as a form of adding over and over again, and remember that taking away a negative is the same as adding a positive, and hopefully you will be able to make sense of it. Once again, I highly recommend that you visit David’s site to see multiplying integers visualized in a very helpful way.

What is Scientific Notation?

So, just what is Scientific Notation?

It is probably one of the first topics you will learn in early physics courses. As such, it is crucial that you understand it and are able to use it. Honestly, without scientific notation, physics problems get INCREDIBLY difficult, and if you don’t get it at first, you should really put in extra effort to figure it out. You won’t regret it… and it really isn’t all that hard. It just sounds crazy.

Scientific Notation is merely a short-hand way of expressing really large or really small numbers. It doesn’t sound all that important, but I will show you how convenient it is.

Let’s think about the measure of time. The length of time it takes the Earth to revolve once around the Sun is a year. We are all familiar with this. We also know that there are 12 months in a year, and 365 days (usually) in a year. Let’s go further though… What about hours? Minutes? Seconds? In a year with 365 days, there are:

365 days * 24 hours/day = 8760 hours
8760 hours * 60 minutes/hour = 525,600 minutes
525,600 minutes * 60 seconds/minute = 31,536,000 seconds

The numbers get pretty big. Why don’t you think about how many seconds there are in 100 years: 3,153,600,000 seconds!

Obviously, you don’t want to have to write down THAT number over and over again, and any numbers you calculate from it, in your equations to solve problems. This is where Scientific Notation comes in.

Scientific Notation basically takes the first non-zero numbers and multiplies that by some factor of 10. Each position in a multi-digit number is represented by 1 power of 10. You have ones, tens, hundreds, thousands, ten thousands, and so on…. and on the opposite side of the decimal, you have tenths, hundredths, thousandths, etc.

So, you can see that:
10 can be written as 1 x 101 (one times ten to the power of 1)
100 can be written as 1 x 102 (one times ten to the power of 2)
1000 can be written as 1 x 103 (one times ten to the power of 3)
and so on…
(If you also recall you rules for exponents, you can see that this pattern continues both up and down… 1 is 100)

Scientific Notation is just expressing things as powers of 10. All you have to do is move the decimal place x number of places so that you have one digit before it, and then multiply your number by 10 to the power of the x places you moved the decimal.

36 = 3.6 x 101… because the decimal place moves 1 to the left
189 = 1.89 x 102… because the decimal place moves 2 to the left
5389 = 5.389 x 103… because the decimal place moves 3 to the left

What about for bigger numbers?

22000 = 2.2 x 104… move the decimal 4 places to the left. Also note that you don’t need to record zeroes if they are the last digits.

On the other hand:
22001 = 2.2001 x 104… same power, but the zeroes cannot be ignored, because they set the position for the final digit, the 1.

Let’s look back at our example of time:

8760 hours = 8.76 x 103 hours
525,600 minutes = 5.256 x 105 minutes
31,536,000 seconds = 3.1536 x 107 seconds
3,153,600,000 seconds = 3.1536 x 109 seconds

Now you can appreciate that if you have to rewrite 3.1536 x 109 seconds several times in a calculation, it is simpler to keep track of… and more importantly, there is far less of a chance of accidentally leaving out some of the zeroes and completely getting the wrong answer.

I won’t go into too much detail for extremely small numbers, since it is essentially the same concepts as I have described above. The thing to remember here is that you move the decimal place to the RIGHT this time, and give a negative power of 10. I will, however, leave you with a few examples to work through so that you can hopefully understand it more completely.
0.1 is equal to 1 x 10^(–1) (one times ten to the power of negative 1)
0.01 is equal to 1 x 10^(–2) (one times ten to the power of negative 2)
0.001 is equal to 1 x 10^(–3) (one times ten to the power of negative 3)
and so on…
0.53 = 5.3 x 10^(–1)
0.0687 = 6.87 x 10^(–2)
0.0000873 = 8.73 x 10^(–5)
0.0000000070067 = 7.0067 x 10^(–9)

Another helpful tip is that if your starting number is between 1 and 0 (ie. it is SMALL), it gets a negative power. If it is greater than 1 (ie. it is BIG) it gets a positive power. Positives for big, negatives for small.

Scientific Notation for small quantities is equally as handy as for large numbers. Consider that a virus may be as small as 1 x 10^(–7) m, or that a proton has a mass of 1.7 x 10^(–27) kg. Also imagine what a chemist’s life would be like without Scientific Notation. They would have to write out this proton mass every time they need it in a calculation. It would probably get very tiring, very quickly, and probably with several errors, if they had to write out every time 1700000000000000000000000000 kg. (Of course, with that many digits, one would hope this chemist would include a little bit more precision… but that is a topic for another day!)

As always, let me know if you found this helpful or would like some more clarification!