Math Lessons

While you might not think this really has anything to do with forensics or other security-related issues, the reality is that math is your friend. And my friend. And when you have to calculate the byte offset on a hard drive to locate the cluster where a particular file is located, you will really want to know a little about the basics of math. 

You may have guessed that the origination of this topic is all of the nonsense spreading around on social networking sites like Facebook. Based on the number of times variations on these math problems show up and the number of times I see wrong answers, it seems as though a large number of folks really could stand a brief math lesson and while I am neither a math instructor in real life, nor do I play one on TV, I am going to take this one on because it will make me feel better. 

The acronym to remember here, and it’s really quite simple, is PEMDAS. Make up whatever mnemonic you want to remember, what it really means is parentheses, exponents, multiplication, division, addition and subtraction. This is the officially approved order of operations. When you see a very long chain of mathematical operations, you might think that you should just work left to right and as a general rule, that’s not a bad instinct. However, in order to come up with a consistent and mathematically accurate answer, you should apply the order of operations first. Then you can move on to left to right. You will also find that it’s generally easier to do a simple replacement. Let’s illustrate with an equation I’ve been seeing recently on Facebook. 

7 + 7 / 7 + 7 * 7 - 7

For those of you unfamiliar with two of those symbols, the / is a division sign for cases where we don’t have the horizontal line with a dot above and below, as in a computer keyboard. The * is a multiplication symbol, which is commonly used in place of a X or an x because those might be confusing in algebraic equations. So, let’s apply the order of operations and then re-write the equation after substituting. 

7 + 1 + 49 - 7

7 divided by 7 is 1, so I swapped in a 1 for the division operation I did. 7 multiplied by 7 is 49 so I swapped that value in. That leaves us with the equation above. There are a couple of ways to do this at this point. I could certainly go left to right and add the first three numbers then subtract the last but you may have noticed that two of them cancel each other out. If I were to re-write the equation above as follows, it quickly becomes a lot easier. 

7 - 7 + 1 + 49

This leaves me with adding 1 to 49 resulting in 50. See how easy that was? Keep in mind that the order of operations is really important. I suppose I could get into the history of why someone determined that multiplication and division were more important than addition and subtraction but it would likely bore you to tears. It would take far more of my time to come up with something coherent than I feel like putting in at the moment, so let’s skip it and move on to series. Let’s say you see the following:

10  =  50

9  =  38

8  =  27

7  =  17

5  =  ?

There are two things you should notice right away. The difference between 50 and 38 is 12. 38 to 27 is 11. 27 to 17 is 10. So, the next in the series should be 8 because we were decreasing the right hand side by one less each time. Since the last difference was 10, the next difference will be 8. 17 - 8 is 9. This leads us to the next thing you really should notice. The value on the left skipped one. The value of 6 should be 8. I’m asking for the value of 5. Keep the series going. I decrease the difference on the right hand side by 1, meaning that as I decrease by one on the left, I will be decreasing by 8 on the right hand side. This means that 5 = 0. When you see a series like this, there is generally a trick. They have skipped a value out of the series. This doesn’t mean that you just assign the correct right hand value (the next one in the series) to the wrong left hand value. It means you apply the right hand series twice and assign that value to the left hand side. 

A little bit of math, folks, will take you a very long way. I hope this has been a little bit of help. I know it’s made me feel better to share it with you.