Powers of 2

From my early days as an undergrad math student, I knew that powers of 2, the numbers of the form 2ⁿ, have many properties, and are very important in many areas of mathematics.
However, it was only after I became a math tutor that I realized how common questions involving powers of 2 are in standardized tests.

Before becoming a math tutor I had held positions at colleges and high schools as an associate professor, adjunct professor, math teacher, and teaching assistant. Up to that point I had memorized only the first six natural powers of 2, from 2¹ to 2⁶, these are:
2, 4, 8, 16, 32, and 64, the powers of 2 that are two-digit numbers.

Later, when I became a math tutor, and I realized how often standardized tests present students with questions involving powers of 2, then I memorized larger powers of 2, up to 2¹², or:
128, 256, 512, 1024, 2048, and 4096.

It is really a big advantage at test time, for a student to be able to recognize a power of 2 when they see such a number in a problem, because then they can write that number in its exponential format, and use the algebraic rules of exponents to work out the problem’s math easier and faster than dealing with the number in its decimal form, using only arithmetic operations, because the longer route is more time-consuming, and more attention-intensive, therefore riskier and less efficient.

So, if you are preparing for a standardized test, from the CBEST or the ASVAB to the GMAT or the CSET, know your powers of 2. Memorize them well from 2¹=2 to at least 2¹⁰=1024, and you’ll increase your chances of scoring a few extra points in the exam.

Why is substitution so confusing for so many people?

The short answer is, because it is not a mathematical operation. Not in the sense addition, subtraction, multiplication and division are.

For most people, mathematics is about crunching numbers. When they are doing math homework, or preparing for an exam, they expect to be adding, subtracting, multiplying, dividing,  deriving, or integrating, or calculating square roots, or standard deviations, averages, percentages, or otherwise combining two or more numbers into a final numerical answer.

However, when we are following the step-by-step solution to an algebraic equation, oftentimes substitution is one key step in the sequence, and substitution is an editing operation performed on a line of text. It is a special kind of cut-and-paste, find-and-replace action that has found a proper place as a drop-down menu item inside word processing computer software.

If you use a computer-like device to type letters, notes, or messages, you are probably familiar with that menu item, following the sequence of buttons 

Edit > Find > Find and Replace > Replace All

that you can use to change one word or phrase into another all across the selected part of your document. The computer automatically does that for you, saving you the time, effort, and risk of errors you would take if you did it yourself visually and manually, searching line by line, phrase by phrase, and word by word to make your desired replacement in all places.

That is what substitution is about, replacing one expression by another expression, under the assumption that they represent the same numerical value.

What stumps many people when they stumble upon a substitution step while following the solution to an algebraic equation, is their own expectation that every step had to be “mathematical,” meaning: “number-crunching-y” in one way or another. But substitution is editing, it is cutting and pasting, it is not multiplying, it is finding and replacing, it is not adding.

So, in many people’s minds, substitution is not math. They just don’t see the substitution step because they are not expecting to see it. I mean, they see it but they don’t recognize it as math because it is not math in the same sense addition and multiplication are.

They get so confused by the fact they are seeing something they were not expecting to see, that the surprise does not allow them to see whatever else is going on at that step.

It is as if, in some part of their mind they are going like:

“Math, math, math, not-math, .. wait, what?”

It is O.K. There are some seemingly non-mathematical procedures that are part of math, too, especially logic. You just have to learn to expect them to show up every once in a while, so they won’t take you by surprise.

Linear Algebra - matrix dimensions

A nxm matrix is a rectangular array of numbers in n rows and m columns.
For example:
A 2x3 matrix has two rows and three columns.
A 1x3 matrix is a 3-coordinate vector in horizontal format, just one row with three numbers in it.
A 3x1 matrix is a 3-coordinate vector in vertical format, commonly known as a “column vector.” It is a matrix with a single column and three numbers in it.

For us to be able to add or subtract two matrices, they have to have the exact same dimensions. For example, you can add or subtract a 4x3 matrix with another 4x3 matrix but not with a 4x4, 3x3, 3x4, 4x2, 5x3 or any other type of matrix.

Matrix multiplication has its own, particular requirements on the dimensions of the matrices being multiplied. You can multiply any nxm matrix with any mxk matrix, and the result will be a nxk matrix.
For example:
A 3x2 matrix times a 2x4 matrix will result in a 3x4 matrix.
A 3x2 matrix cannot be multiplied with another 3x2 matrix.
A 3x2 matrix times a 2x3 matrix will result in a 3x3 matrix.
A 3x3 matrix times a 3x3 matrix is again a 3x3 matrix.
A 3x3 matrix times a 3x1 matrix will result in another 3x1 matrix.
A 2x7 matrix times a 7x4 matrix will result in a 2x4 matrix.

Matrix multiplication involves a lot more arithmetic operations than matrix addition, because just to get the number for the entry in the i-th row and j-th column of the product, we have to calculate the dot-product of the i-th row of the first matrix times the j-th column of the second matrix.

All this can be pretty confusing for students when they first read it in a book but after being shown how it’s done in a couple of examples, it becomes clear and then with more practice it becomes easy but it always is a lot of work, and you have to pay close attention to all the little calculations involved because the large number of them makes it easy to make silly mistakes.