Comparing rectangles by their area and by their diagonal

The other day I was in someone’s office waiting for some paperwork to be completed.
It was a small room, about 12 feet by 12 feet.
The back wall was orange, and its lower part was horizontally covered by a desk.
In the middle of the orange wall there was a picture of a green car. I thought that was an interesting color combination.
Looking at the picture on the wall, I wondered how bigger the visible part of the wall (including the picture) was, in area, in relation to the area of the picture.
The horizontal length of the wall was about two-and-a-half times the length of the picture, while the hight of the visible part of the wall was about three times the height of the picture.
Putting

b = length of the picture
h = height of the picture
a = area of the picture
B = length of the wall
H = height of the visible part of the wall
A = area of the visible part of the wall (including the picture)

we have, approximately

B = 5b/2, and H = 3h

therefore

A = BH = (5b/2)(3h) = 15bh/2 = 15a/2

So the proportion between those two areas is 15/2, 0r 7.5

Then I wondered what the proportion would be between the diagonals of those same two rectangles. As it turns out, when the two rectangles (the wall and the picture) do not have the same shape, the proportion between their diagonals depends not only on how many times the sides of the wall are on relation to the sides of the picture, but on the precise shape of the reference rectangle, as well.
If we use

d = measure of the picture’s diagonal
D = measure of the diagonal of the visible part of the wall

Then we have

D = sqrt(B² + H²)
 = sqrt((5b/2)² + (3h)²)
 = sqrt(25b²/4 + 9h²)

while

d = sqrt(b² + h²)

The coefficients of b² and h² inside the square root in the expression for D are not the same, so we cannot simplify that expression completely just in terms of d.

If we introduce a new constant, k, to represent how many times the height of the picture equals its length, then we can write

b = kh

In that case, the diagonals of the two rectangles can be expressed as

d = sqrt((kh)² + h²)  =
 = sqrt(k²•h² + h²) = 
 = h•sqrt(k² + 1)

D = sqrt(25(kh)²/4+ 9h²) = 
 = sqrt(25k²h²/4 + 9h²) = 
 = h•sqrt(25k²/4 + 9)

This shows the proportion between D and d being dependent on k, the proportion between h and b, because

D/d = sqrt(25k²/4 + 9) / sqrt(k² + 1) 

and this is a function of k that is not constant.

A very cool property of binomial coefficients

Last night I was reading a post in this other blog, by Ben Vitale. His post is about some special types of binomial coefficients.

In case you don’t remember, given two non-negative integers m and r, their binomial coefficient  C(m,r) (also known as “m chose r”) counts the number of subsets of size r out of a set of size m, and it is calculated by the formula:

C(m,r) = m!/(r!•(m-r)!)

In particular, when r=2, we get

C(m,2) = m!/(2!•(m-2)!) = m(m-1)/2

This is an expression for counting the number of all possible unordered pairs formed by two elements of a set of size m.
Somewhere near the end of his post, Ben presents the following formula:

C(kn,2) = k•C(n,2) + (1/2)•(k-1)k•n²

On seeing this formula, I noticed the second term on the right-hand side,

(1/2)•(k-1)k•n²

can be rewritten using the binomial coefficient C(k,2), producing the expression

C(kn,2) = k•C(n,2) + C(k,2)•n²

This formula holds true for any two positive integers n and k, each greater than 1.
In particular, by using the trick of swapping n with k in that expression, we get the following one:

C(nk,2) = n•C(k,2) + C(n,2)•k²

Now, multiplication being commutative, we know kn = nk.

Therefore, C(kn,2) = C(nk,2)

And we can connect their two equivalent expressions, making this formula:

 k•C(n,2) + (n²)•C(k,2) = (k²)•C(n,2) + n•C(k,2)

I don’t know about you but I believe this is just amazing!
Let’s illustrate this property with a couple examples:

a) First, we choose two numbers, let’s say n=5 and k=7

b) Then we calculate their squares: 5² = 25, and 7² = 49

c) Now we calculate their binomial coefficients for 2:

C(5,2) = 5•4/2 = 20/2 = 10
C(7,2) = 7•6/2 = 42/2 = 21

d) Now we plug the numbers into the formula, and verify it:

7•C(5,2) + 25•C(7,2) = 49•C(5,2) + 5•C(7,2)

(7)(10) + (25)(21) = (49)(10) + (5)(21)

70 + 525 = 490 + 105

595 = 595

Let’s check some other numbers, for example n=8 and k=14

Their squares are 8² = 64, and 14² = 196

Their binomial coefficients with 2 are:

C(8,2) = 8•7/2 = 28
C(14,2) = 14•13/2 = 91

and we verify:

14•C(8,2) + 64•C(14,2) = 196•C(8,2) + 8•C(14,2)

(14)(28) + (64)(91) = (196)(28) + (8)(91)

392 + 5824 = 5488 + 728

6216 = 6216

The reason I like this property so much is because both expressions

k•C(n,2) + (n²)•C(k,2)

and

(k²)•C(n,2) + n•C(k,2)

are linear combinations of the same two binomial coefficients. These linear combinations have different coefficients:

k and n² in one case, and k² and n, in the other.

In principle, right off the bat, I would not expect them to turn out equal but they are, every single time! What is more, the only change in the coefficients is that you move the square (the exponent 2) from k to n, and viceversa.
I find all this to be pretty cool. I wanted to share it with you, and many thanks to Ben Vitale for his blog post.

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.