I know, I know. This sounds utterly ridiculous. Secret math tricks? Multiplying by 1? Adding 0?
Why on Earth would we do that?
But it’s legit. Here we go.
The Reason These Secret Math Tricks Are Useful
We’re not allowed to just change math problems to make them easier for ourselves, because then we would be doing a different math problem and we wouldn’t get the answer to the original problem. But if we technically didn’t change it? Now that’s allowed.
Multiplying by 1 and adding 0 both do exactly nothing.
Which is what makes them so extraordinarily useful.
For example, \(4\cdot1=4\), meaning that multiplying 4 by 1 didn’t change 4.
And \(4+0=4\), meaning that adding 0 didn’t change 4.
We can even generalize these rules to say:
- \(x\cdot1=x\), meaning that multiplying anything by 1 does not change that thing.
- \(x+0=x\), meaning that adding 0 to anything does not change that thing.
Okay, so how is doing exactly nothing a good Secret Math Trick?
It wouldn’t be, except there are a lot of secret ways to do exactly nothing. For instance, doing nothing via multiplying by 1 and adding 0 is the entire reason algebra works.
That’s right: algebra is secretly just a bunch of ways to multiply by 1 and add 0. In other words, algebra is secretly the art of doing exactly nothing.
Before We Get Any Further, Let’s Talk About Secret Ways To Multiply by 1 and Add 0.
I know this sounds silly, but it’s about to start making sense.
Secret Ways to Multiply by 1:
All of the secret ways to multiply by 1 involve fractions, so if you’re iffy on that you may want to go read my fractions post.
If you remember your fractions (or you just went and read the fractions post), then you know that \(\frac{x}{x}=1\), meaning that any fraction with the same thing in the top and in the bottom is secretly just 1.
So if we multiply \(7\cdot\frac{4}{4}\), that’s secretly just \(7\cdot1=7\).
This can become something very complicated and still work just fine. For instance, \(7\cdot\frac{\frac{3\pi}{4}cos\theta – \log_5(\theta)}{\frac{3\pi}{4}cos\theta – \log_5(\theta)}=7\cdot1=7\). (Sidenote: if you have a problem with trig and also base five logs in the same equation, you have my full permission to write your teacher an email explaining that you didn’t do your homework because it was too mean. Unless, like in this problem, it all works out to secretly just be 1.)
Secret Ways to Add 0:
We can all agree that \(4-4=0\). But have you ever considered that if \(4-4=0\), that means that \(5+4-4=5\), which means that adding and also subtracting 4 to any equation is the same as adding 0?
We can generalize that to say \(x-x=0\), meaning that if we add and subtract the same thing, we are secretly just adding 0.
That thing that we add and subtract can be anything, which is what makes adding 0 such a good secret math trick. For instance, we could add and subtract \(\frac{3\pi}{4}cos\theta\) and we would still secretly be adding 0. (Which is to say: \(5+\frac{3\pi}{4}cos\theta-\frac{3\pi}{4}cos\theta=5\) because \(\frac{3\pi}{4}cos\theta-\frac{3\pi}{4}cos\theta=0\).)
Doing Nothing in Creative Ways Lets Us Manipulate Equations
Generally, I don’t advocate for manipulation. But when it comes to math, manipulation is the name of the game — manipulation of equations, that is. Please don’t use math to manipulate people.
But anyways, we can multiply by 1 and add 0 in order to get an equation into a form we can work with. This trick is awesome because it turns seemingly impossible problems into easy problems.
Let’s look at some examples.
Examples of Secret Math Tricks
There are so many good examples of multiplying by 1 and adding 0 in math, but I’m going to focus here on a couple of common ones that come up pretty early in your math career.
1. Completing the Square
Completing the square is a great example of adding 0.
If we have the equation \(x^2+6x=17\) and we need to know the value of \(x\), the easiest way to get there is to complete the square1, which involves adding zero!
I’m not going to go through how to complete the square in detail here, but remember that the goal is to create an equation in the form \(ax^2+bx+c\), where \(a=1\) and we choose \(b\) and \(c\) carefully so that we can easily factor our equation into the form \((x+h)^2\).
In the above example, we need to add 0 by adding and subtracting 9. (We know that it’s 9 from the equation \(c=(\frac{b}{2})^2\), but that’s not important to the point of this example).
Adding 0 (or adding and subtracting 9) is going to look like this:
\(x^2+6x+9-9=17\)
Then we’ll add some helpful parenthesis in order to do what we want here:
\((x^2+6x+9)-9=17\)
And then we factor and move that extra \(-9\) over to the other side:
\((x+3)^2=26\)
Take the square root of both sides:
\(x+3=\pm\sqrt{26}\)
And finally solve for x:
\(x=\pm\sqrt{26}-3\)
If we didn’t know we could add zero creatively, we’d still just be sitting here staring at \(x^2+6x=17\), trying to bend the rules of the square root (which we are not allowed to distribute across addition), and maybe crying about how difficult algebra is when we don’t know the Secret Math Tricks.
And that’s the power of adding 0.
2. Finding the Common Denominator
Picture this: With five minutes left on the test clock you get to the end of a long and miserable set of algebra, only to realize that in order to simplify your results into a final answer you’re going to have to add \(\frac{8}{23}+\frac{19}{11}\). You definitely cannot visualize twenty-thirds or elevenths in your head, and you’re beginning to suspect that your math teacher became a teacher because they enjoy watching people suffer.
Luckily, all you need to do to make this work out nicely is multiply by 1 a couple of times.
If you remember how to find the common denominator, or if you read that fractions post I mentioned earlier, then you know that to find the common denominator, we just use the denominator of each fraction to make a secret form of 1. And then of course multiply each secret 1 by the other fraction.
So in this case, we need to do \(\frac{8}{23}\cdot\frac{11}{11}\) and \(\frac{19}{11}\cdot\frac{23}{23}\). Granted, that is still disgusting, and if your teacher didn’t give you a calculator on this test, you’re probably still feeling a little disgruntled, but multiplying by 1 is going to make this possible!
We end up with \(\frac{88}{253}+\frac{437}{253}=\frac{525}{253}\), which is clearly disgusting, and unfortunately impossible to simplify. But this whole process only took about four minutes.
Thanks to multiplying by 1, you have an extra minute on the clock to start drafting a petition for reasonable computations on future exams.
When to Use the Secret Math Tricks
- Any time you need to get an equation into a specific format.
- Any time you know how to solve the equation if it were just a little bit different. (This is secretly a form of the above statement).
- Any time you’re doing math and you get stuck, look for a way to add zero or multiply by 1. That’s almost always what you need to do!
Happy Mathing!
Footnotes:
- Unless we have the quadratic formula memorized…but let’s imagine we don’t for the sake of this example. There are other cases where we have to complete the square, but they all involve more set up for an example problem and I don’t want to drag you through that. âŠī¸