Document Guidelines by Daniel Browning

Transcription

1 Document Guidelines by Daniel Browning Introduction This is a math primer for those who want to better understand the types of fractions and ratios that are used in quantum mathematics. Already I've introduced three words that the reader may be squirming with: fractions, ratios, and quantum. So let's get right into it! Quantum A "quantum" is simply a quantity, or more simply still, a number. Scientists like to assign "quanta" (more than one quantum) to different processes that take place at the atomic level. But really a quantum can be anything: for example, someone's age. Scientists like to build entire theories on quanta, so these numbers don't change very much, because otherwise the theory would fall apart, or if they do change, the changes are just minor adjustments. This brings us to precision. Precision The number is "precise" to seven digits. But someone using a "lower" precision can specify this number as.24 (which is a "rounding up" from.239), and this number is no less valid. Scientists use different precisions depending on the message they want to convey: sometimes precision is important, sometimes it isn't. Getting back to quanta, a "minor adjustment" might change to you see that the number is still basically.24. But sometimes even such minor changes can make a big difference when such a number is part of a long equation -- because it all adds up! Fractions and Ratios Fractions and ratios are really the same thing. A fraction is one number divided by another number: like 2. In decimal notation, is 0.5. One would say "one-half" when talking about 2 this fraction, but one can also say "one over two" or just "point five" if reading the decimal number. The "one" part is called the numerator, and the "two" part is called the denominator. Think of the numerator as being the "nominal" value which is above; then the "de-nominal" is what takes away from the nominal. That's how you can remember that the numerator is the part that's above. A "ratio" is the same thing as a fraction. 2 would be said aloud as "one to two." It's like betting odds. Some horse in a horse race has odds of 7 -- that's a ratio: "seven to one." "Miles per hour" is a ratio: as a fraction it would look like: miles. Sometimes scientists will hour

2 2 try to be cool and write: miles hour -. This means that "hour" is an inverse. What is an inverse? Inverses "Miles per hour" is the same as: miles x. A ratio of "anything over one" or "anything to hour one" is simply that anything. Anything divided by is itself, and anything times is also itself. Basic, right? Now, if you flip the fraction, you get "one over anything" -- that's the inverse. The inverse of.24 is.24. Simple. When someone writes miles hour -, this is said aloud as "miles times hours to the minus one." (Or "miles per hour," but you knew that.) "Minus one" is an exponent or power. You could also say "hours to the minus one power," or "hours to the negative first power." Since "hours to the first power," or hour, is simply one hour, or hour, then "hours to the negative first power" is the inverse, or hour. This may seem confusing, but only because sometimes the simplest examples of certain math operations are sometimes the hardest to "get." Usually negative exponents are used for scientific notation. 2.4 times 0-3 is just 2.4 x 0.00, or A unit to the "minus unit x unit x unit 0 x 0 x 0 three" is:, so 0-3 is. Let's look more closely at exponents. Oops! I hear someone squawking! Let's see what's the matter... Ooh! Ooh-ooh! Bangileto! (This is my friend Bangileto. It's pronounced Bangghee-LETtoe. He's a math monkey.) What's up little friend? Ooh ooh ooh ooh! You want to know, what's a negative number? Wah! The best way to understand negative numbers is with a number line. You see below that anything below zero, or to the left of zero, is a negative number. Actually, negative numbers aren't used all that much in quantum math. Some people like to think of subtraction as the addition of a negative number. For example, 0-6 is really 0 + (-6). The number line below is part of the Cartesian coordinate system, because it was invented by Rene Descartes, a French philosopher. Since zero is at the center, it's called the origin.

3 3 Squares and Cubes "mile 2 " is said aloud as "miles squared." This is "mile x mile". Or using a number, 3 2 is 3 x 3, or 9. A power can be any number: 4 3 is said aloud as "four cubed." This is 4 x 4 x 4 -- which is 64. This brings us to square roots. Square Roots (and other Roots) If 3 2 is "three squared," then what is the "square root" of 3 x 3? It's 3. Since 3 2 is 9, the square root of 9 is written as: 9. Somtimes people say "radical nine." You can also take the "cube root" of a number. The symbol for this is: The little "3" in the crook of the root symbol stands for "third power" or "cube." As usual, scientists like to be snazzy and sometimes they use fractions as powers -- to indicate roots! For example, a square root can also be written as:.24 /2. In other words, a square root is the inverse of a square! A cube root can be written in the same way:.24 /3. And getting really fancy, the square root of a cube is:.25 3/2. So now you know... Wait a sec... Yeah, I hear big monkey hoots. Let's check in with Bangileto. Woo woo! EEEE! EEEE! Oh, you want to know why we use the words "square" and "cube"? WAH! Okay, monkey man. Let's explore. The explanation is obviously geometric, and any carpenter will at least know the first part. Take a vertical line of length n, with n representing any number. If you push that line to the right by n units, we get a square, of n x n, as shown below. The area of the square is "n squared." Now, take that entire square, and push it back into the page by n units, corresponding to depth. Now we have a cube of n x n x n, also shown below. The volume of the cube is "n cubed." Ooh!

4 4 Solving for "an Unknown" There's something that crops up all the time, which is solving for an "unknown" in a ratio. For example, let's say you've got the ratio 3, which is a quantum that describes something 2 important, but you've got this other value which describes the exact same thing as 3, except that it's 8! You want to know what the ratio with an 8 in it instead of a 3 will be, but you need to solve for an unknown -- which is what's underneath the 8. (What's that underside of the fraction called? This is a quiz.) Or put it this way: we've got 3 2, but we only have 8, where "n" is an unknown. This means n we must solve for the "unknown." The way you do this is to set the two quanta equal to each other: 3 2 = 8. This is called an equality. n The next step is to combine the two fractions by taking diagonal products: 3 x n = 8 x 2. When we solve as much as we can, we first get 3n = 6. Now we need to isolate the "unknown" on the left-hand-side of the equation. (Whatever's to the left of the "equals" symbol is the "left-hand-side," and whatever's to the right is the "right-hand-side.") We do this isolation by dividing both sides by 3. We get: 3n 3 = 6 3. Now we know that 3 3 is "three over three," and anything over itself is! So the "3's" cancel out on the left-handside, and since times anything is anything, this leaves us with: n = 6. That's the 3 solution to our unknown. Now, this leaves us with a peculiar result: 3 2 = We can either replace 6 3 with a numerical value, in which case our original equality is: 3 2 = 8, or if we want to be 5.33 purists, we can use a little trick which says: "anything over a fraction is equal to anything times the reciprocal of that fraction." Uh-oh. What's a reciprocal?? Bangileto! Why are you standing on your head? Ooh ooh ooh EEEE! EEEE! Oh! You're being a reciprocal! Well aren't you the smart one... WEEE! Wah ooh ooh. Bangileto, do you want to tell everyone your story? Wah! Oooh... Ooh! Ooh ooh ooh! EEEE! EEEE! Ooh ooh wah wah! WAWEE! The whole story, little friend! WAH! WAH! Ooh ooh ooh ooh. OOH! Ooh ooh! EEEE! EEEE! Brings a tear to my eye... WAH!

5 5 Reciprocals A "reciprocal" is almost like an inverse, except it's not limited to " over something." It's a fraction that's been swapped. The reciprocal of 6 3 is 3. Just switch the top and the bottom. 6 Back to the Unknown Now we have: 3 2 = 8 x 3 8 x 3. The right-hand-side is the same as, which turns out to be 6 x But now look what's happened! Instead of 8 we have solved for 24! Not quite what we 6 intended when we set out! We could summarize all this by writing: 3 2 = = 24. All of 6 these ratios are identical because they're all equal to.5. Repeating Numbers This leads to repeating numbers. Notice the number Remember precision? Well, I've cheated and only written this number out to three digits of precision. Actually, the "3's" after the decimal place go on forever: they're repeated out to infinite precision. Instead of writing a million "3's" we can simply write: That little doohickey on top is an "overbar," indicating that whatever's underneath gets repeated to infinite precision. You can put anything under an "overbar": is a perfectly nice repeating number, being ad infinitum. (That's Latin for "to infinity.") Continuation Notation Writing down math equations is a sequence of operations in many cases, but instead of writing: 2 x 3 = / 9 = 4... which is two lines, the mathematical thought can placed on a single line by using "continuation notation": 2 x 3 = 36 / 9 = 4 The underline ("36") indicates that the number underlined is the result of the operation to its immediate left. One might verbalize this as "twelve times three is thirty-six, divided by nine is four." You only use continuation notation for "intermediate results," not for what's on either end. Obviously 2 x 3 is not equal to 4. You have to read through the entire line to understand the result, but that's the whole point!

6 6 Conclusion Really, this is all you need to understand the universe. Bangileto has one more thing to say, funny monkey. He tells me that 2 x 2 =. Oh Bangileto, will you never learn?? Cheers all!!