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!

BEAUTY OF MATHEMATICS from PARACHUTES on Vimeo.

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!

The Difference Between Precision and Accuracy

What is the difference between PRECISION and ACCURACY? Upon first glance, many students would say that the two terms mean the same thing. In normal day-to-day usage, you would talk about how precise this is, or how accurate that is, and in both cases you would be comparing “how close” some thing is to the actual thing.

In Math and Science, however, they are slightly different.

ACCURACY has the meaning that I just described: how close a number is to the actual number.

PRECISION, on the other hand, does not refer to how close a number is to an actual number, but rather refers to how many digits and decimal places a number has. (A separate definition for precision also is that it measures how close repeated measurements are to each other, rather than to the actual value.)

As an example, consider a 100 gram block of wood. We are told that this block has a mass of 100 grams. Now, if we put it on balance A, we get a mass of 99.9 grams. When we put it on balance B, we find a mass of 101.350022. We can say that balance A provides us with a more ACCURATE number because it is closer to the known mass of 100 g, but balance B gives us a much more PRECISE number because it displays more decimal places. However, even with the extra decimal places, it isn’t as accurate as balance A. (Maybe there is something wrong with its calibration?)

Do you think that some measurement can be both accurate AND precise? Sure it can.

Let’s suppose we have a bottle of pop. The label says that it is a 1000 mL container. We pour the pop into a graduated cylinder, and determine that there is 998 mL. Since this particular graduated cylinder that we used has no decimal markings, the whole number 998 is as precise as we can get. But that is as precise as the label says. We measure only a 2 mL difference, which isn’t that much (0.2%). So, with this cylinder, we can say that we have a fairly accurate and precise measurement.

If we could determine that the volume in the container was 998.93188 mL, we could say that this number was accurate, and very precise.

Hopefully these examples were able to show you the difference between accuracy and precision. Please comment if you would like some further explanation; this concept always confuses students. I will do my best to clarify. :)

What are Real Numbers? Rational, Irrational, Natural, and Integer Numbers

What are Real Numbers?  When you are first learning to do math with numbers, you never think about what “kinds” of numbers you use.  You just add 2 plus 2, or subtract 10 from 20, and later on begin to multiply (-3) times (-5) (refer to Multiplying Integers here) and other, more complicated functions.  What you don’t even realize is that all the numbers that you are playing with actually belong to a certain set of numbers.  And this set of numbers is called Real Numbers.  Later on, (much later on,) you might begin to study other sets of numbers, such as the Imaginary Numbers… but that is a much different lesson that we’ll save for another day.

So, just what is a Real Number?  Quite simply, a Real Number is any number that you can express in a decimal form.  They are the numbers that you use everyday in your math homework, and which also get used in laboratories, offices, and industries.  Like I said, you don’t even realize that you’ve been using Real Numbers all along.

The set of Real Numbers is often designated with a rather fancy capital R letter, like this:  \mathbb{R} .

If we think of our set of Real Numbers as being numbers that can be expressed as a decimal, we can similarly think of them as being represented on a number line.  As such, it is easy to see that Real Numbers include all the numbers on the number line, whether they are positive, negative, or zero.

Now, if we continue to think of a number line, we can further break down our set of Real Numbers into further categories.

One category is the Rational numbers, which are any numbers that can be expressed as the ratio of two integers where the denominator is not zero (such as 5 (5/1), 2/3 (0.6666…), or 0.87934 which is really 87934/100000…).  The whole numbers, represented by the ticks, are the Integers.  And the Integers themselves are are composed of Natural numbers, which are your regular ‘counting numbers’ (0, 1 apple, 2 apples, 3 apples) and their negatives.

Of course, if there are Rational numbers, we must also have Irrational numbers.  Irrational numbers cannot be expressed as the ratio of two integers, and so their decimal forms extend forever without repeating.  Square roots are sometimes irrational (such as the square root of 2), as well as pi and e (if you haven’t already, you will learn about these special numbers later).

So then, in a nutshell, what is a Real Number?  They are a set of numbers that can be broken down into a few categories.  Rational numbers are all numbers that can be represented as a simple fraction or repeating decimal.  Irrational numbers are all numbers whose decimals continue forever without repeating.  Together, the Rational and Irrational numbers fill in our number line completely, and form the set of Real Numbers.

Hopefully this makes sense and answers the question that so many students arrive here looking for: “What are Real Numbers?”  Also, I hope it gives you a deeper understanding of the numbers that you have always known how to work with anyways!  :)

Working with Polynomials and Monomials, and How to FOIL

Following up on my post explaining Monomials and Polynomials, here I would like to provide a bit more of an explanation about how to actually work with these terms. In particular, first I am going to show you how to multiply a monomial by a polynomial. Then I will similarly show you how to divide a polynomial by a monomial. And then finally, I will explain the very important concept of FOIL for multiplying to binomials. Nothing too difficult, but all key concepts of algebra. Let’s get to it!

How to multiply a Polynomial and a Monomial. It’s actually very easy, probably way easier than it sounds! Let’s step back a bit first, though, and think about how we multiply regular numbers and expressions. Consider the expression (x + y), and say that we want to multiply that by 3. How do we do that? By now, I think this is very straightforward. We first write it out, like this:

(3)(x + y), or 3(x + y)

Next, we multiply each part of the expression (x + y) by 3, and then we can drop the brackets at the end. This is called the Distributive Property, which refers to how we can remove the brackets if we are multiplying (distributing) the one term with all of the terms within the brackets. I don’t think I need to go into more detail than this for this basic operation, but leave me a comment below if you’d like a bit of help with it. Maybe I’ll do a separate post on the Distributive Property. When we’re done, we have this:

3x + 3y

Now let’s extend this method to some more complicated looking terms. Let’s consider multiplying the monomial 5x with the polynomial (3x + 1). I’ll walk through the steps below, and it hopefully will be easy to follow. All we do is abide by the Distributive Property and simply multiply the single term into the brackets with all the other terms.

5x(3x + 1)
5x(3x) + 5x(1)

Now that we have properly distributed the terms, we just have to simplify it and we’re done. The first term becomes 15x2 while the second term becomes simply 5x:

15x2 + 5x

And that is all there is to multiplying a monomial with a polynomial. The key is to follow the Distributive Property and multiply things through where appropriate, and then combine like terms if necessary and simplify!

OK, so now hopefully you followed all that. But what about if we want to go the other way? What if we want to start with a polynomial and then divide by the monomial instead? Again, it’s very similar to operations you are already familiar with. In this case, you might compare it to cancelling and reducing terms. Let’s look at an example and you’ll see what I mean.

Let’s say that we want to divide (8x + 4) by 2. Probably pretty simple, right? We just divide the 2 into all of the terms to reduce them, leaving us with the answer (4x + 1). We can think of this as a big fraction, with the (8x + 4) on the top as the numerator, and the 2 on the bottom as the denominator. Because we are adding the 8x and 4, by the laws of fractions, we can separate everything into two separate fractions with the same denominator: 8x/2 + 4/2. Now, it might be even easier to see how to reduce these to get our final answer.

The same logic applies when we want to divide a polynomial by a monomial. Consider the example of dividing (25x3 + 15x2 – 5x) by 5x. I’ll write it out as a big fraction first (typing these out is not the most obvious way of demonstrating!).

Now, just like with the example above, we can cancel and reduce things. For cancelling, remember that you can only cancel if you are performing the same operation to each term. Or alternately, if you want to split this into the multiple smaller fractions first before simplifying, if that is clearer to you, then that works as well. Either way, you will find that you arrive at the final answer of:

5x2 + 3x – 1

Note that because there is an x term in the denominator, you cancel this out of the numerator for each term by reducing each by x. That is why the powers all reduce by one. The same theory applies to factoring the 5.

Hopefully that explains the two rudimentary concepts of multiplying and dividing monomials and polynomials. Let me know in the comments below if you need any more help!

The last thing I’d like to talk about in this post is FOIL, or how to multiply two binomials. Recall that a binomial is an expression that contains two terms. This is a specific instance of multiplying two polynomials together, though clearly, since polynomials can have many terms, it is best to start simple – so we start here by multiplying two binomials! Larger expressions are not necessarily any harder, though it does become more of a challenge to not mix things up. The key to multiplying any two polynomials is to add up the products of multiplying each term in the first polynomial by each term in the second polynomial.

You have probably heard the expression FOIL when talking about multiplying polynomials, and you will probably see FOIL many times in your studies, so it is best to understand well to what it is referring! FOIL is an acronym that stands for the order of operations that you apply when multiplying two binomials. It only works for binomials, though it still abides by the concept I just outlined for multiplying larger polynomials together. It means FIRST OUTSIDE INSIDE LAST. That probably doesn’t mean much until you see it. It refers to all the pairs of products that you add together when multiplying two binomials.

Here is an example of this:

(2x + 3)(x + 5) = ?

  • First: 2x * x
  • Outside: 2x * 5
  • Inside: 3 * x
  • Last: 3 * 5

Adding these up gives:
2x2 + 10x + 3x + 15

Furthermore, because you are adding polynomials with like terms, you can simplify this expression further:
2x2 + 13x + 15

And that’s the final, reduced answer. You can apply the FOIL principle to any two binomials to arrive at their product. When you have more complicated polynomials, such as those composed of 3, 4, 5, or more monomials, you do the same type of thing. What I find easiest is to take the first term of the first polynomial, and multiply it with every term of the second polynomial. Then do the same for the second term in the first one, multiplying with every term in the second one, and so on.

Like this:

(x2 + x + 5)(x3 + x2 + 1)

First group (multiply x2 with all in the second polynomial)

  • x2 * x3
  • x2 * x2
  • x2 * 1

Second group (x)

  • x * x3
  • x * x2
  • x * 1

Third group (5)

  • 5 * x3
  • 5 * x2
  • 5 * 1

Now, you just add up all these terms, and simplify where you can:

(x2 * x3) + (x2 * x2) + (x2 * 1) + (x * x3) + (x * x2) + (x * 1) + (5 * x3) + (5 * x2) + (5 * 1)

(x5) + (x4) + (x2) + (x4) + (x3) + (x) + (5x3) + (5x2) + (5)

(x5) + 2(x4) + 6(x3) + 6(x2) + x + 5

And that’s it. A little more complicated, but as long as you keep track of what you’re doing and work your way through it, you will arrive at the answer!