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The 80/20 Principle: Imaginary Numbers On the Math Section of the SAT

Today, we'll discuss the famed 80/20 principle, also known as the Pareto Principle. The Pareto Principle states that that around 80% of results come from 20% of efforts.


That is, of all your studying, only a small percentage is responsible for the bulk of your results. The key to effective studying is identifying what comprises that percentage, so that your studying will reap the best results possible. The mistake that many students make is assuming that all parts of their study are created equal. They invest time and energy in understanding highly specified elements of geometry or statistics which are unlikely to appear on their actual SAT.


It matters where you put your attention, and where you put your time.


Important precepts regarding the creation of SAT questions

There are a few important ideas to keep in mind when preparing for the SAT. They are as follows.


Principle I

Firstly, SAT writers love questions that look difficult, but are easy to solve. They need questions which could be solved quickly by students who know the material well. A perfect score should be attainable by a student with excellent math skills. However, if questions could be solved quickly because they're easy, there would be no filter of higher-skilled students from lesser-skilled students.


One way of working around this difficulty is to give questions which scare off students who aren't confident with the material. The question itself may be easy, but because it touches upon a topic that the student feels insecure about, he / she will get overwhelmed and either waste time, or not answer at all.


This leads us to our second point.


Principle II

The more esoteric and confusing the math topic, the easier the question will be. This is because the fact of being off a confusing topic is enough to scare off most students. As a result, writers of the SAT will ease up on the difficulty of the question itself.



Principle III

According to the 80/20 principle, not all studying is created equal. When debating what to study in your next sessions, consider that material should be reviewed in the following order:


  1. Easy & frequent material

  2. Easy & somewhat infrequent material

and / or

More difficult & very frequent material

  1. Infrequent & difficult material


Easy and frequent material may sound obvious, but may students tend to neglect these questions in favor of more difficult subjects. One obvious example is linear equations. Their algebra is not difficult, and they comprise a huge chunk of questions on the math section of every SAT. For more examples of easy and frequent material, see our y-intercept article, slope article, and excess text article.


Questions that are simple but not that common - for example, questions that appear once per exam, or perhaps once every other exam - are the next thing you should focus on. That's because a little bit of study could snag you an extra 10 points on the SAT; the effort pays off.


Alternatively, place your attention on material that is common on the SAT, but is complicated and may take time to learn. This, too, is worth the effort, because the time you invest will now pay off in a significant amount of correct answers.


One example is the quadratic equation. This subject is multifaceted and can’t easily be learned within minutes, or even hours. But because so much of SAT math material is centered on it, it’s worth your time nevertheless.


The very last thing you should give attention to is material which is both difficult and infrequent.

When you're done with other forms of material, you can move on to this. Such subject matter includes, for example, trigonometry, or elements of geometry which aren't totally basic.


This doesn't mean that you shouldn't touch them at all. You should be doing practice tests and random questions interspersed with focused study sessions on a specific subject. And if difficult + uncommon questions pop up, of course you should solve them and make sure you understand the answer. However, if you see that you're struggling with - for example - geometry, and you're still having trouble with - for example - ratios, geometry shouldn't be the subject you reserve time for.


Imaginary Numbers

Today we'll demonstrate the 80 / 20 principle through the topic of imaginary numbers. These tend to appear on the non-calculator section of the SAT, and they aren't frequent; your chances of getting such a question on your own SAT is about 50 / 50. (They appear in about half of SATs.)


The topic of imaginary numbers seems complicated, but in keeping with Principle II, most of the questions asked about it are simple. A little effort, relatively ample results: that's the 80/20 principle in action.


Mathematical principles for imaginary numbers

With imaginary numbers on the SAT, there are two main mathematical principles to consider:


1. i, the symbol for imaginary numbers, is a variable like any other variable. It should be treated mathematically in the same way.


2. i² = -1. You don't have to understand what that means, conceptually. When you see i², just plug in the value of -1. That's all.


These two principles are enough to answer the vast majority of questions on imaginary numbers on the SAT.


Let's examine what this means for a moment by looking at the following expression.


Example #1

If i = i = √(-1), what of the following is the equivalent of the sum 4 + 3i -1 + 2i?


A) 3 + 5i

B) 3 + i

C) 7 + i

D) 7 + 5i


Some students get frightened off by this mathematical expression. They know that i represents an imaginary number, but they don't fully understand what an imaginary number is. Perhaps they remember doing some confusing operations with imaginary numbers in class, and now they think something along those lines has to be applied to this question. Alternatively, they don't remember anything about the concept, and they're scared off entirely.


Understand that, so long as you have an algebraic expression like this - no fractions, no i², i³, or other exponents, just a sum of numbers and i variables - this is an algebraic expression like any other algebraic expression. It should be treated in the same way.


If i is too intimidating, replace the i's with x's:


4 + 3x -1 + 2x = 3 + 5x


Replace the x with an i - at this stage it's just a matter of different names - and you'll get:


3 + 5i


The answer is A). You may be surprised, but for half the questions on this topic on the SAT, that's all their asking.


Now let's look at a question where the second principle of imaginary numbers, the concept of i², comes up. Again, you don't have to understand the deep philosophical meaning of i². When you see it, just plug in -1 and move on.


Example #2

Find the product of (3+i)(2+5i).


A) 6 + 12i

B)` 11 + 16i C) 1 + 17i

D) 5 + 13i


We'll FOIL, just like we would if the variable was x instead of i.


(3+i)(2+5i) = 6 +15i + 2i +5 i²


Following Principle 2), replace i² with -1. We get the following expression:


6 + 15i +2i -5


Following Principle 1), add the numbers and i variables separately. If you need to, replace the i's with x's, and then switch back once you have an answer.


Your final answer should be 1 + 17i.



More complicated questions on imaginary numbers

It's true that not every question on imaginary numbers is as simple as those shown above. For example, sometimes questions involve a complex conjugate. This usually occurs when the question centers on a fraction, such that there's an imaginary number in the denominator.

These take more than a few minutes of practice.


Remember the 80/20 principle: focus your attention on where a little effort will make a large difference. Most SAT questions on imaginary numbers are simple. Work with those. When you feel that you've mastered the basics, you can move on to infrequent and also more difficult material.


Let's look at one more example of the simple question type you'll usually see on the SAT on the topic of imaginary numbers.


Example #3

For i = √(-1), which of the following expressions is equivalent to (7i - 8) - (3i - 2i²)?


A) 5 + 10i

B) -8 + 6i

C) -6 + 3i

D) -10 + 4i


Remember that an imaginary algebraic expression should be solved like any other, and the first step in solving an algebraic expression is always to rid ourselves of existing parentheses. Since the first set of parentheses isn't being multiplied or subtracted, removing the parentheses does nothing. The second set of parentheses is used for subtraction, so removing the parentheses changes the sign of everybody within. We'll get the following expression as a result:


7i - 8 - 3i + 2i²


Please note that the correct use of parentheses is perhaps the most important skill in basic algebra. If you don't know how to use them correctly, it's difficult to advance to more sophisticated math. Unfortunately, a large percentage of high school students don't use them adeptly.


If you struggle with parentheses use, please study it. There's a plethora of internet material on on the subject and we too, may soon publish an article on the topic. But keep in mind that if the 80 / 20 principle applies to any area, it's here. Some degree of effort will correct numerous common mistakes that students make on the SAT, spiking your score.


At this point, we'll use Principle 2) to replace with -1:


7i - 8 - 3i - 2


Following Principle 1), we'll combine like terms:


4i - 10


The answer is D). Note that, due to the associative property of addition, the order of the terms makes no difference. That is, 4i - 10 is the same as -10 + 4i.


(If this isn't immediately obvious to you, cease other forms of study to first sharpen your basic algebra skills.)


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For more tips and methods, or to get help personally tailored to your needs, consider working with me. I've helped people from all over the world get into their dream school. In coaching you, I adopt my methods specifically to your personality, schedule, and learning style. For this reason, studying privately with a skilled coach is the best way to increase your test score.


Happy learning,

Tova