Absolute Value

Absolute value is a tricky thing. Students love it because it is so easy. They probably come away thinking, "Could I have done this right? That just felt way to easy. And even if I did do it right, it seams pretty pointless." There actually are mathematics applications to absolute value. Here are a few.

Average Deviation – There are a number of formulas that measure the variability of data. A common one is the standard deviation. However, average deviation is similar and easier to compute. The average deviation simply finds the average distance each number is from the mean. To find the average deviation, the distance from the mean is found for each piece of data in the set. Those distances are added and then divided by the number of pieces of data. If the mean is 32, we would want 28 and 36 to both be considered positive 4 units away from the mean. Absolute value is used so there are no negative values for those distances.

Example: 

- A set of data is {21, 28, 31, 34, 46}. The mean average is 32.The average deviation is 6.4.

Statistical Margin of Error – As mandated by the U.S. Constitution, every ten years the government is required to take a census counting every person in the United States. It is a huge undertaking and involves months of work. So how are national television ratings, movie box office results, and unemployment rates figured so quickly – often weekly or even daily? Most national statistics are based on collecting data from a sample. Many statistics that are said to be national in scope are actually data taken from a sample of a few thousand. Any statistic that is part of a sample is subject to a margin of error. (In 1998, President Clinton attempted to incorporate sampling in conducting the 2000 census, but this was ruled as unconstitutional.)

Example:

- On October 3, 2014 the government released its unemployment numbers for the month. Overall unemployment was listed at 5.9%. The report also stated that the margin of error was 0.2%. Government typically uses a level of confidence of 90%. Thus there is a 90% chance that the actual unemployment rate for the month was x, where |x-5.9| ≤ 0.2.

Richter Scale Error – The Richter scale is used to measure the intensity of an earthquake. However, like many measurements, there is a margin of error that needs to be considered. Scientists figure that the actual magnitude of an earthquake is likely 0.3 units above or below the reported value. If an earthquake is reported to have a magnitude of x, the difference between that and its actual magnitude, y, can be expressed using absolute value:  |x-y| ≤ 0.3.

Body Temperature – “Normal” body temperature is assumed to be 98.6° F. For any student that has made the case that anything other than 98.6° prevents their attendance at school, there is good news. There is a range surrounding that 98.6 value that is still considered in the normal range and will allow your attendance at school. Your 99.1° temperature is probably just fine. Supposing plus or minus one degree is safe, an expression could be written |x-98.6| ≤ 1.0, which would represent the safe range. Why is the absolute value a necessary part of this inequality? Without it, a temperature of 50 degrees would be considered within the normal range, since 50-98.6 = -48.6, which is, in fact, well less than 1.0.

Ten Cool Things about Laplace

I thought it was time to look at another interesting mathematician. This week it is Pierre-Simon LaPlace.

1. Lived from 1749 to 1827, all in France.
2. Married at age 39 to an 18 year old.
3. Made important contributions to the method of least squares - used to find a best fitting line.
4. Wrote the five volume Celestial Mechanics contributing greatly to a theory of the origins of the universe.
5. Appointed by Napoleon Bonaparte to be Minister of the Interior of France.
6. Later regretting this, Napoleon later stated, "Laplace was not long in showing himself a worse than average administrator."
7. Very possible apocryphal, but Napoleon was speaking to Laplace on the influence of God on a a particular situation to which he replied, "I had no need of that hypothesis."
8. Commenting on this story, Stephen Hawking said, "I don't think that Laplace was claiming that God does not exist. It's just that he doesn't intervene, to break the laws of science."
9. When he died, his brain was removed and displayed.
10. He is buried in Paris in the Pere Lachaise Cemetery along side other star-studded famous residents Balzac, Sarah Bernhardt, Bizet, Maria Callas, Chopin, Joseph Fourier, Yves Montand, Jim Morrison, Marcel Proust, Rossini, and Oscar Wilde.

Sermon Stats

In sermon notes in a church bulletin it stated, "The probability Jesus could have fulfilled even eight of these prophesies is 1 in 10 to the 17th power (1 in 100,000,000,000,000,000)". This was a statistic taken from a book, although I don't know the title. I thought there is a math application in there somewhere.

I thought that small a number might be almost incomprehensible to most. Maybe to everyone. It

reminds my of something David Letterman said once regarding buying a lottery ticket. A particular lottery was at a near record amount and lots of people were buying them. He wanted people to consider that if you buy a ticket, your chance of winning is only slightly more than if you don't buy one. Incidentally, I was in the audience for one of his shows during his final month. Hilarious. I am including a picture for no other reason than I love Dave. Back to math.

I considered a couple of ways to tie this probability to other situations. How does this probability compare with chances in rolling a die? In flipping a coin?

Well, the chances of rolling a "6" are one in six. How many consecutive rolls would correspond to the above probability?

1 / 10¹⁷ = 1 / 6^x
10¹⁷ = 6^x
Taking the common log of each side, we get:
17 = x(log6)
x = 21.85

So, at least 21 consecutive rolls coming of 6.

Similarly with flipping the coin. The coin has only two outcomes, so:

1 / 10¹⁷ = 1/2^x
After a few steps we get x = 56.47

56 heads in a row. Unlikely.

If worried about church vs state issues, a teacher could come up with other kinds of problems. The actually probablility of winning a certain lottery, winning the grand prize in the McDonald's Monopoly Game. For example, I just looked up on-line that the probability of getting the Boardwalk piece - 1 in 602,000,000.

Good Luck.

Evaporation

When I was a lad, I remember looking a drops of rain that had plopped on the sidewalk. It was a light rain so I could make out the individual drops. They gradually evaporated. I noticed that if I used my finger and spread the raindrops, out they evaporated faster. At my tender age I had no idea why. Still not 100% certain, but my guess now is that if you have a drop of water it is losing molecules, i.e., evaporating, from its surface. If you take a drop of water, it is evaporating at a certain rate. If you separate that drop into two drops, it will evaporate faster because there is much more surface area for which it can use to evaporate.

So, let's examine this as a math application. Suppose a drop of water is spherical and has a volume of 10 whatevers. Then suppose we separate that drop into two drops of volume 5 whatevers each. Then we look at their total surface areas. Will they come out the same?

We start with the fact that it has a volume of 10:  (4/3)πr³ = 10
If we solve for r we get a value of r = 1.3365
We can then find its surface area: 4π(1.3365)² = 22.446

Now what if we now have two spheres of 5 each.
Their radii would be:  (4/3)πr³ = 5, so r = 1.0608
The surface area of one drop is 14.140. There are two drops, though, so the total surface area of them would be 28.28

Comparing the two situations, we can in fact state that, since 28.28/22.466 = 1.26, there is 26% more surface area, and so I will postulate a 26% faster drying time for the two drops over the one drop.

That was to be the end of the story, but I thought what if you separate one spherical drop into two? Is it always going to be a 26% greater surface area.

Unfortunately this is beyond my skill level to show all this. You might recall that I only recently found how to write integer exponents. This process involves having the cube root of a fraction all taken to a power of two and other assorted difficulties. So let me map this out leaving a few gaps for you or students to work through. It really is a great problem, though, with the opportunity to review simplifying - some major simplifying.

• Let us say that  (4/3)πr³ = V
• Solve for r
• Substitute this expression into 4πr²
• Now, find the radius for a sphere that his half the original volume:    (4/3)πr³ = (v/2)
• Substitute this r into 4πr²
• Make a ratio of the two radii and simplify 
• You end up with cube root of 16 divided by 2, which is 1.26
• Ta-da. An increase of 26%

Satisfying