Introduction to Mathematics

If you have 3 apple pies and 19 people, how should you slice the pies so that each person gets an equal share? Each person should get 3/19 ≈ 0.1579 pies, but if you make each pie into 6 slices, that’s only 18 slices for 19 people. You have to slice each pie into 6 and 1/3 slices, with each slice being equal except the 1/3 being smaller, and then give the three 1/3 slices to the 19th person.

What if you have 1 pie and the Half-Blood Prince from Harry Potter and the Half-Blood Prince is the only person dining? Then you have 2 pies per person, assuming he’s half a person. In the expression 1/(1/2), move the denominator to the numerator and flip the ex-denominator, making the reciprocal and thereby converting division to multiplication. 1/(1/2) becomes 1*(2/1) which is just 2, because any number without a denominator has a denominator of 1.

Your neighbor lends you $8000 at 3.75% interest compounded annually, with no payments being required for 25 years and the full balance and interest being required to be repaid at that time. What is the payment? $8000*1.0375^25 ≈ $8000*2.5102 = $20,081.34.

What if you want to make a graph of the increasing amount owed on the Cartesian coordinate system where y is the number of years and x is the dollar amount in thousands? Use the equation y = 8*(1.0375^x).

PROBLEM: Your truck gets 15 miles per gallon in the city and 19 mpg on the highway. Your destination is 38 miles away as the crow flies, 42 miles away through the cities, and 49 miles away if you take Interstate-95. However, taking the interstate requires 5 miles of city driving. How many gallons of gas will be used on each route, which one uses the least gas, and how much does each trip cost if gas is $4.15 per gallon?

SOLUTION: First, note that there are only two routes: 42 miles through the cities and 5 miles through the cities PLUS 44 miles on I-95. Then, do this:
42/15 = 2.8 gallons –> 2.8*4.15 = $11.62 in gas
5/15 + 44/19 ≈ 2.65 gallons –> 2.65*4.15 = $10.99 in gas

NOTE that with the variables provided, the longer route is in fact the cheaper route. However, in reality there would be an inordinate number of variables, such as your engine’s efficiency, time of day, traffic patterns, traffic lights, and unforeseen events. For example, traveling an extra 7 miles may necessitate an earlier oil change or some other maintenance. If you get into an accident on the interstate at 80 miles per hour, you might be instantly killed rather than being only wounded if crashing at a much lower speed in a city. Highway driving vs. city driving requires different mental concentration and one may appeal over the other depending on your upbringing and psychological makeup. In the city, you are more likely to be pulled over by policemen and ticketed. The road surface may be smoother on the interstate, which will prevent your tires from wearing down as quickly. If you break down on the interstate, you may be stranded if you don’t have a cell phone. All math problems simplify.

PROBLEM: The 2012 Presidential election is coming up, and observing that Ron Paul (R) and Barack Obama (D) have won the primaries, General Electric is deciding how much to donate to each candidate’s campaign. GE’s budget is $4,000,000, and they estimate Paul has an 8% chance of winning and Obama has a 92% chance of winning. GE estimates donations to Paul have a political worth three times greater than donations to Obama to secure support from the Constitutionalist movement. How much will GE donate to each campaign?

3*0.08 = 0.24
1*0.92 = 0.92
0.24+0.92 = 1.16
4,000,000/1.16 = 3,448,275.862
3,448,275.862 * 0.24 = $827,586.21 to Ron Paul
3,448,275.862 * 0.92 = $3,172,413.79 to Barack Obama

PROBLEM: Steve Jobs is developing the iPad 3 for release November 28, 2012 and must choose between Foxconn’s 64GB isolinear-NAND flash chip and Foxtrot’s 59GB neo-EEPROM flash chip. Foxconn’s chip costs $28.78 and has a five-year failure rate of 2.8%. Foxtrot’s chip costs $26.55 and has a five-year failure rate of 2.1%. Both chips are functionally identical in form factor, read/write speed, power consumption, resiliency, and failure potential, both technologies are equally reliable, both companies use slave labor, and both companies are of equal capacity, reputability, and geographic location.

Apple estimates the market value of an extra 5GB (64GB vs. 59GB) of storage capacity to be $14.50 per unit, and estimates that each five-year failure will have an effective cost of $895.88 on Apple’s image, future sales, and support network. Which chip should Steve Jobs choose?

($28.78 – 14.50) = $14.28 effective cost per 64GB Foxconn chip
$26.55 = $26.55 relative cost 59GB per Foxtrot chip
$895.88 * 0.028 = $25.08 failure cost per 64GB Foxconn chip
$895.88 * 0.021 = $18.81 failure cost per 59GB Foxtrot chip
$14.28 + 25.08 = $39.36 total cost per 64GB Foxconn chip
$26.55 + 18.81 = $45.36 total cost per 59GB Foxconn chip

Jobs should choose the 64GB Foxconn chip, even though it is 0.7% more likely to fail in the first five years, because it’s easier to market a 64GB device than a 59GB device so the 64GB Foxconn chip has an effective cost of $6.00 less than the 59GB Foxtrot chip, given the variables.

PROBLEM: General Motors Company is developing a new type of engine that improves fuel economy by 100%, but has discovered that 0.000097% of the engines blow up when reaching a speed of 88 miles per hour, instantly killing everyone in the vehicle and seriously wounding everyone in a 100 foot radius. GMC is considering including this engine in its new SUV, the ThinkNeighbor Plus, which will get 42 highway miles per gallon instead of the standard 21 highway mpg, will be manufactured in a quantity of 5 million, and will sell for $38,000. GMC estimates only 0.5% of ThinkNeighbor Pluses will ever reach a speed of 88 miles per hour, and estimates the Public Relations costs of each explosion will be $8 million. The U.S. State Department has pledged to blame the explosions on domestic terrorist attacks, but only to a limit of three. Should GMC manufacture the ThinkNeighbor Plus?

NO. Since the question is “should GMC manufacture the ThinkNeighbor Plus?,” it isn’t even a math question because “should” is completely subjective.

More next time.


A curious property of CPU architecture is the argument that datasets should be big-endian or little-endian, that is, should the most significant items be listed first, saving the least for last, or should the least significant items be listed first, saving the best for last? What if the transmission of data is cut-off mid-stream? The computer program may be tempted to accept the big-endian dataset because it contains the most significant data, but “the devil is in the details” as they say, so this could be a fatal mistake. Similarly, the computer program may be tempted to drop the little-endian dataset entirely because it contains no significant data, but the unrevealed data at the end may have led to a different conclusion.

For the purposes of computer science, it is tempting to say that endianness is a solved problem and Intel won. However, just because Intel is the largest manufacturer of computer processors and Intel’s X86 and X64 processing architectures are little-endian does not mean that is the best way to go. It’s very possible we could have much further advanced computers now if not for Intel’s choice, or much crappier ones. It’s also possible we could have big-endian computers that are just as advanced as the current ones, and it’s possible those computers would be just as advanced despite Intel, or with Intel’s help, i.e. because big-endian is very superior but Intel’s engineers persevered anyway on the hard road, or because big-endian is very inferior and Intel’s engineers took the easy road, never reaching the full potential of human discernment.

It’s also possible that neither path is correct, both are equal, or some are more equal than others, to quote Planet of the Apes. It’s also possible that different endiannesses are appropriate for different situations, i.e. big-endianness for lossy editing and little-endianness for lossless editing, i.e. the difference between editing photos, videos, and music, or text, respectively. Endianness is also known as byte order, and in a multibyte string, the preference of putting the most significant bytes first or the least significant bytes first is a product of your upbringing, whether it be your parents, grandparents, uncles, aunts, cousins, friends, strangers, environment, culture, heritage, private school, the streets, or state-run public schools. Your preference is also a product of your static and unchangeable genetics, and I may have just now subtly or overtly influenced your view of the world by writing this paragraph.

The difference between choosing big-endianness or little-endianness might just as well be the difference between being male or female, black or white, gay or straight, tall or short, right-handed or left-handed, right-brained or left-brained, or speaking English or Spanish, or both, or neither. The question isn’t “does it matter?,” because it obviously has a significant impact upon your life and behavior. The question is “why does it matter?,” because that is the only question that makes sense and has efficacy. Similarly, you can’t understand the motivations behind big-endianness without understanding the motivations behind little-endianness, and you can’t understand the motivations behind little-endianness without understanding the motivations behind big-endianness. I’m not a computer science student, so I don’t know much about it, but I could wager a bet it has something to do with the properties of silicon.

In truth, endianness is just another of life’s little arguments to either make you incredible sane or incredibly insane. No problem can be solved at the same level it was created, so you either have to choose the path of ignorance or choose the path of knowledge. The former divides you into camps of religious zealots, and the latter unifies you into the camp of humanity, not by transcending the need for endianness, but by recognizing that the worst possible action is to stagnate and refuse to make a choice, and that each road is equally valid and you should choose whatever works best for you and your mind. The choice is yours.


Nobody knows the time or the hour that Jesus will return to Earth, or even what form he will choose. It could be 4000 years from now, a million years from now, or it could be that he never even left. It could be that it’s incumbent upon us to change the world and change ourselves to prepare for his arrival. But one thing’s for sure — destroying the world will not get Jesus to return. Only embracing it will, but not in a physical way — in a mental way, for even a man who has looked upon a woman with lust has committed adultery in his heart.

I grew up not believing in any particular religion, but now I see that religion fulfills a critical need in peoples life — the belief in something permanent and unchanging outside themselves in an impermanent and changing physical world. Trying to tear down religion might be the right step at the level of fear, and ignoring religion might be the right step at the level of hope, but only embracing it is the right step at the level of love, tolerating it at the level of perfection, and loving it at the level of imperfection. As in the words of Data from Star Trek: The Next Generation, “to love him was to know him, and to know him was to love him.”

Therefore, it’s very important to first embrace religion, then reject it, and then recapitulate. Most people only go through this process once or twice in their lives, and if they go through it more than that they feel like a failure, when in fact they are much more successful at their human mission than people who never change mental states.

If you go through this process process too frequently, you might have a mental disease such as bipolar “disorder” or schizophrenia. If you don’t go through the process of recapitulation enough, you might suffer a midlife or even a quarter-life crisis. And if you go through it everyday, you might be a Beethoven, Tolstoy, or Einstein just waiting to emerge.

Hope is a level of consciousness lower than knowledge, but faith is a higher level than both combined. It takes courage to stay true when the whole world seems to be against you, because in truth, the whole world loves you, but this doesn’t mean you should embrace strangers, because the world commits itself to actions discordant with its beliefs. There are rights, and then there are privileges, but you can’t earn privileges — they must be given to you, and all privileges are the work of Satan, not Jesus.

Be careful in your worldly affairs, for nothing in this world can truly determine your fate in the afterlife besides your actions. Normally, your thoughts determine your actions, but in some people this isn’t true, so it may be best to just disregard bad thoughts instead of wasting time analyzing where they came from.