School math, multimedia, and technology tutorials.
Uhm, maybe they give their girlfriends some valentine ‘stuff’ with some hidden math on it. For example, a greeting card with a Sierpinski Triangle decorated with heart fractals such as the one shown below.
Or maybe some GeoGebra flower applet.
***
Photo Credit: XKCD
If you liked my post on Wolfram Alpha, then you will probably like Google Squared. Google Squared is a semantic search engine developed by Google which displays search results in tabular format. The results in tables can be customized, saved, and shared. The table shown below was the result of the search phrase “20th century mathematicians” entered in Google Squared.
Really? Lewis Carroll was a mathematician? I didn’t know that.
Note: You can enlarge the figure above by clicking it.
One of the elementary observations about real numbers is that if we add the two numbers, regarding of the order, their sum is unique. We know that 4 + 3 = 3 + 4. This property is also the same when we multiply. If we have two numbers, regarding of the order, their product is also unique. For example, 6 x 3 = 3 x 6.
In general if we have real numbers a and b, we have a + b is always equal to b + a. Also, if we have real numbers c and d, c x d is always equal to d x c. These are called the commutative property of addition and multiplication, two of the axioms of algebra.
Although axioms, as we have discussed, are statements that we accept without proof, some axioms can be explained intuitively. In this post, we are going to discuss the two axioms mentioned above intuitively using geometric representations.
The geometry of a + b
One way of understanding why a + b is always equal to b + a is by representing the two numbers as lengths. For instance, we can represent the addition of integers above (4+3 = 3 + 4) as the sum of the lengths of two segments. In geometric representation, reversing the order of the numbers in the operation is just like rotating the segment 180 degrees about the center as shown below. Of course, rotating the segment way will never change its length.
Now, since that we can substitute any positive numbers to a and b we are sure that the commutative property is true for all positive numbers. » Read more