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 e^x, we can remove the logarithm from our equation entirely by raising the number e to the power of our equation:

Now, since ln(e^x) 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!

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!

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)

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

Believe it or not, equations have been devised whose curves on a graph actually show you portraits of famous people! It’s true! Check out the graphs of Steve Jobs and Abraham Lincoln, shown below!

If you want to explore these amazing math masterpieces, Wolfram Alpha has a collection of line art portraits that are created from very complicated parametric functions. I’m sure these are devised by a computer. I’d have a very hard time believing that someone came up with any of these formulas! I’m not even going to pretend to understand some of the functions that are used in the equations. Nevertheless, this is probably one of the coolest graphing examples of complex math expressions that you will ever see.

Mathematica Stack Exchange has a more in depth analysis of how these curves are made, if you want to know a bit more about them. Otherwise, check them out in the gallery at Wolfram Alpha! Which one is your favorite?

Probably one of the most common variables you will ever run across in mathematics is “x.” You have undoubtedly seen it countless times, in math problems asking you to simply “solve for x,” or as one of several different letters in a multivariable equation. Using the letter x to denote unknowns has even gone beyond mathematics – just look at The X Files on TV, where x indicates mysterious and unknown FBI cases, or even to the historical example of naming the x-ray, where x indicated a radiation coming from an unknown source. By now, you probably take for granted that x is the go-to letter when you want to represent some unknown quantity. While any letter can be used, I’m sure that you will agree that x is by far the most common one used. Have you ever stopped to wonder why? Why is x the unknown?

Where did this convention come from to let some seemingly random letter become the single one letter most often used to represent the unknown. Interestingly, it has its roots in language. Specifically, it is related to phonetics used in translating from Arabic, and the lack of corresponding phonetic sounds in Spanish.

It is very interesting, and I will let Terry Moore describe in far more detail in the TED video clip. It’s only a few minutes long, and it builds up to a nice tongue-in-cheek joke at the very end – especially if you have any Spanish friends!

Give this video a watch, and find out why x is the unknown!

I really appreciate your comments and feedback, so feel free to drop me a message in those places or in the comments below. If there’s anything I missed or you want explained better, I can help with that too. Thanks again, and cheers!

On that note, I just read a fascinating perspective by Erlina on the use of equal signs and letter symbols in math, and what these things actually mean. I had never stopped to actually give any thought to this topic, but this may be worth considering before you jump into your math lessons and teach the same way as always. Here’s an excerpt from her post:

“Teachers would oftentimes introduce algebra by telling their learners that x stands for an unknown number. It is not incorrect but that’s not all. Some teachers also introduce the word variable by saying that x can take any value that’s why x is called a variable. Again, it is not incorrect but that’s not all. I have heard teachers that say that in an equation, the x is an unknown, but in an algebraic expression, the x is a variable because it can take any value. Is it this simple?”

If nothing else, this post may give you something to think about. It’s a quick read and I recommend it, actually for anyone with an interest in mathematics and letter symbols.

Mobius strip template

There are many more observable examples of math in our world. You can likely find something inspired by math nearby to you. My point is that math is not just a hard subject in school that your teachers are forcing you to study. It has purpose and a place in our world, and some of our greatest achievements are solely based on a solid understanding of the field of mathematics.