MIKIPEDIA SCIENCE BLOG: December 2012

Saturday, December 29, 2012

WRITING MATHEMATICAL EXPRESSIONS IN PLAIN TEXT - EXAMPLES AND CAUTIONS

Mathematical expressions can be typed online in a number of ways including plain text, ASCII codes, HTML tags, or using an equation editor (see Writing Mathematical Notation Online for overview). The most common way is to write expressions horizontally in plain text, but you have to format the expressions carefully using appropriately placed parentheses and accurate notation. This document provides examples and important cautions for writing mathematical expressions in plain text.
Section 1. How to Write Exponents
Just as on a graphing calculator, when writing in plain text the caret key ^ (above the 6 on a qwerty keyboard) means that an exponent follows. For example
2x
would be written as x^2.
Example 1a.
234xy
would be written as 4 x^2 y^3 or for added clarity with the multiplication mark as 4*x^2*y^3.
Example 1b. With more than one item in the exponent you must enclose the entire exponent in parentheses to indicate exactly what is in the power.
2nx
must be written as x^(2n) and NOT as x^2n. Writing x^2n means
2xn
Example 1c. When using the quotient rule of exponents you often have to perform subtraction within an exponent. In such cases you must enclose the entire exponent in parentheses to indicate exactly what is in the power. The middle step of
55232xxxx
must be written as x^(5-2) and NOT as x^5-2 which means
52x
Example 1d. You must enclose fractional exponents in parentheses to indicate that the entire fraction is in the power.
23x
must be written as x^(2/3) and NOT as x^2/3. Writing x^2/3 means
23x
Section 2. How to Write Repeating Decimals
There is no over-strike bar on your keyboard, so the only way to write a repeating decimal in plain text is to write several repetitions of the repeating numbers followed by an ellipsis (dot dot dot).
Example 2.
1.23
would be written as 1.2323232323…
Section 3. How to Write Absolute Value Bars
On the right of your keyboard, usually above the backslash, is a mark that looks like a vertical dash above another vertical dash. It produces a single vertical bar |. Use it to produce absolute value bars.
Example 3.
23x
could be written as 2-|x-3|.
Section 4. How to Write Inequalities
The less than and greater than keys are on your keyboard. To type “less than or equal to” or “greater than or equal to” type the original inequality, select it, and underline it to produce < or >. Alternatively use an ASCII code to produce these symbols (ALT+243 and ALT+242). On a MAC computer hold down ALT while typing the original inequality and it will automatically create ≤ or ≥.
Example 4. 30 is less than or equal to x could be written as 30 ≤ x.
Section 5. How to Write Fractions
When writing rational expressions (fractions) horizontally USE PARENTHESES. When the numerator contains more than one item, you MUST have parentheses around the entire numerator. When the denominator contains more than one item, you MUST have parentheses around the entire denominator.
Example 5a.
23xx
would be written horizontally as (2x)/(x-3).
Parentheses must enclose the entire numerator and parentheses must enclose the entire denominator to indicate that the entire 2x is divided by the entire x-3. I can’t stress enough how important this is!
If you neglect the parentheses and write 2x/x-3 then, by order of operations, this means
23xx
, a completely different expression.
Example 5b.
32x
must be written as 3/(x-2) and NOT as 3/x+2. Writing 3/x+2 means
32x.
Example 5c.
32x
must be written as (x+3)/2 and NOT as x+3/2. Writing x+3/2 means
32x
Example 5d.
(1)(2)(3)(4)xxxx
must be written as [(x-1)(x-2)] / [(x-3)(x-4)] and NOT as (x-1)(x-2)/(x-3)(x-4). Note the extra set of grouping symbols (I like to use square brackets) around the entire numerator and around the entire denominator. Believe it or not, by order of operations, writing (x-1)(x-2)/(x-3)(x-4) means
(1)(2)(4)(3)xxxx
, ugh! See how messed up things can get if you neglect to include the parentheses to clearly indicate what expression is in the numerator and what expression is in the denominator?
Example 5e. This common “neglected parentheses” notation error only occurs when you are typing the problem horizontally. Instead you could write the fraction vertically. For example, instead of writing
23xx
horizontally as (2x)/(x-3) you could write the fraction vertically (see below) and then you don't need the parentheses because it is clear exactly what is being divided by what. 2x_ x – 3 To produce this fraction type the numerator, type the denominator in the next line down (use SHIFT+ENTER), highlight the numerator, and click the underline key U to form the fraction bar. Voila!
Section 6. How to Write Radicals
You can use “sqrt” for square root, “cubert” or “3RT” for cube root, “fourthrt” or “4RT” for fourth root, etc. No matter what the index of the radical is, be sure to enclose the entire radicand in parentheses to indicate exactly what is in the radical.
Example 6a.
3x
would be written as sqrt(x+3).
Don’t forget the parentheses! Do NOT write sqrt x+3 as this means
3x
, a very different expression. The same caution holds if you use ASCII codes to create the radical symbol (ALT+251 will create √). You must write √(x+3) not √x+3. Writing √(x+3) means the square root of the entire x+3, i.e. x  3 , but √x+3 means only the x is under the radical, i.e. x  3 . Similarly √5x means
5x
with only the 5 under the radical, but if you want
5x
you must instead write √(5x).
Example 6b.
23x
could be written as cubert(x^2) or as 3RT(x^2) or using fractional exponents as x^(2/3).
Do NOT write as 3sqrt(x^2) nor as 3√x^2 as these both mean “3 times a square root” not “cube root”. The
symbol can only be used for square roots, not higher roots.
It is a bit of a cheat, but here is a shortcut for higher index radicals. Using ASCII codes to create the radical sign type 3√(x²) then select the 3 and hit the superscript key in the discussion board message menu (the icon looks like x²) so that the 3 turns into a little 3 index as in ³√x which is recognizable as the cube root of x.
Example 6c.
321x
could be written as cubert(2x-1) or as 3RT(2x-1) or using fractional exponents as (2x-1)^(1/3).
Example 6d.
45x
could be written as fourthrt(x/5) or as 4RT(x/5) or as (x/5)^(1/4).
Example 6e.
52xy
could be written as 2x*fifthrt(y) or as 2x*5RT(y) or as 2xy^(1/5).
Example 6f. The quadratic formula
242bbacxa
could be written as x = [-b +/- sqrt(b^2 – 4ac)] / [2a] I know it’s ugly, but accurate! Using ASCII codes you could clean it up a bit: x = [-b ± √(b^2 – 4ac)] / [2a].
Section 7. How to Write Logarithms
Natural and common logarithms can easily be typed as is. It is only logarithms with a specific base displayed that require special formatting. No matter what type of logarithmic function, be sure to enclose the entire argument in parentheses to indicate exactly what is in the logarithm.
Example 7.
3log(2)x
could be typed initially as LOG3(x-2), then select the 3 and click the subscript key in the discussion board message menu (the icon looks like2x) to create LOG3(x-2). Using capital letters for the logarithm isn’t necessary but does intensify the smallness of the base.
Section 8. How to Write Trigonometric Expressions
Example 8a. Some Greek letters can be typed using ASCII codes (or HTML tags if you are familiar with HTML programming). Otherwise, I recommend that you use acceptable alternatives. For instance,
cos
can be written as cos(theta), cos(t), or cosT.
sin(2)
can be written as sin(2 alpha), sin(2a), or sin2A.
Example 8b.
2cossin
could be written as 2cos(t)sin(t) or as 2cosTsinT.
Example 8c.
sincos
could be written as sin(t)/cos(t) or as sinT/cosT.
Example 8d.
2sin
could be written as (sin(t))^2 or as (sinT)^2.
Do NOT write sinT^2 which might be interpreted
2sinT
, nor sin^2T which looks like the nonsensical
2sinT
Example 8e.
1cosx
is best written as arccos(x).
Do NOT write cos^-1(x) because the -1 is not an exponent.
Example 8f. Common trig symbols include the angle symbol and the degree mark. For the angle symbol personally I use HTML tags, but in plain text you can just use words such as writing ∠B as “angle B”. To produce the degree mark (which is vitally important to include on every angle that is in degrees so as to not confuse it with radians) using an ASCII code is an easy way (ALT+0176 will create ° ). Or you can click the superscript key in the discussion board message menu (the icon looks like x2) then put a little oh for the power. Or you can just use words such as writing 30° as “30 degs”.
Section 9. How to Write Calculus (Limits, Derivatives, and Integrals)
Using an equation editor to type calculus expressions online is preferred, but occasionally a quick plain text message is appropriate so here are examples of typing limits, derivatives, and integrals.
Example 9a.
0sinlimxxx
can be written in words as “the limit of (sinx)/x as x->0” or by writing “lim x->0 (sinx))/x” preferably with x->0 as a subscript so it looks like “limx->0(sinx)/x”.
Example 9b.
2lndxdx
can be written as d/dx (ln x^2).
Example 9c. If evaluating the derivative at a specific value such as
21lnedxdx
you can use the horizontal “evaluate at” bar which is, on a standard keyboard, on the right side above the \ symbol and looks like two vertical lines but types as |. So this expression could be written as “d/dx (ln x^2)|x=1 to x=e” or use more words such as “d/dx (ln x^2) evaluated from x=1 to x=e”.
Example 9d.
sinxexdx
can be written using INT for “the integral of” as in “INT(e^x*sin(x))dx”. Note that the integral symbol can be typed using HTML tags if you are familiar with HTML programming.
Example 9e. If evaluating a definite integral, more words must be used or again the horizontal “evaluate at” bar. For instance
1sinxexdx
could be written as “INT(e^x*sin(x))dx|x=1 to x=pi” or “INT(e^x*sin(x))dx evaluated from x=1 to x=pi”.
Example 9f.
512kk
can be written as “SUM(1/(k-2))|k=5 to k=infinity” or “SUM(1/(k-2)) from k=5 to k=infinity”. Note that the summation sign and the infinity sign can be written using ASCII codes or HTML if you prefer.
Copyright © 2009 Sally J. Keely. Writing Math Plain Text.docx
All Rights Reserved. revised 2012-10-22
http://www.integreat.ca/OL/docs/WritingMathPlainText.pdf

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THE HARVARD YARD

JOHANNES KEPLER-NEW ASTRONOMY-PART 1, ON RELATIONSHIPS OF HYPOTHESES-CHAPTER ONE, OBSERVING THE HEAVENS

The testimony of the ages confirms that the motions of the planets are orbicular. It is an immediate presumption of reason, reflected in experience, that their gyrations are perfect circles. For among figures it is circles, and among bodies the heavens, that are considered the most perfect. However, when experience is seen to teach something different to those who pay careful attention, namely, that the planets deviate from a simple circular path, it gives rise to a powerful sense of wonder, which at length drives men to look into causes. (p.115)

The Starry Vault

Our earth, which we are standing on, is the huge blue sphere, and the blue circle around it is the horizon, the limits of our sight. Imagine you are on your back, looking up. This is a live view of the night sky from an undisclosed location. Watch it carefully!

Actually, this picture doesn't move at all, but that's what stargazing is like! You don't see a particular star move any more than you can see the hour-hand on a watch moving, yet the change happens. Let's say you look up again in an hour. Here's what you'd see:

What happened now? Take another look a while later, and you'll see this:

The stars all seem to move in the sky. Doing this for an entire night until the sun comes up and ruins your view, you observe this (noting one particularly bright, redish star that stands out to you):

Do certain stars move faster than others? It seems that each star moves in its own particular circle, and there seems to be a spot where the star doesn't move at all. This spot that doesn't seem to move during the night, but around which the other stars move, acts like the end of a large pole, so we can call the star located there the pole-star. Now we have some directions! It also seems that the sun rises and sets not in the pole direction, or opposite it, but in between its direction and the opposite. We can name these directions: the pole-star direction is North (unless you're in Australia and see a certain cross that seems to rotate while maintaining the same position and call this pole South), its opposite is South, and the directions of the rising and setting of the sun East and West, respectively.

First and Second Motions

Around a dozen or so days later, you see the moon every night as well. Let's take a few nights and spend them watching the sky. Although you may find it easier to look at two at once to compare them, how would you do so without a computer animation? The motion would only be in your mind, or, expressed by your mind in a drawing.

First night

Next night

Last night
What did you discover? How high was the moon in the sky at sun-down on the three days? Among which stars is it found? Does everything in the sky move at the same speed? Which moves faster: the stars or the moon? You have now discovered the difference between the First Motion and the Second Motion: the first motion presenting itself to the mind of all the stars, including the moon, and the second motion, seen only through thought and comparison, of the particular heavenly bodies moving against the stars:

The first motion is that of the whole heaven and of all its stars from east past the meridian to the west... The second motions are those of the individual planets from the west to the east. (p.115)

Taking more care in the observations, it is noticed that the sun, too, moves more slowly than the stars. The stars are more advanced at each sunset. Here are a week's worth of sunsets, as they appear to the animating mind (not the eyes):
[a week of sunsets]

A week of sunsets

This first adumbration of astronomy explains no causes, but consists solely of the experience of the eyes, extremely slowly acquired. (p.117)

At this same time of year, while the red star rises near sunset, and is particularly bright (do other stars change their brightess during the year?), we can observe it at its peak, around the middle of the night. By using our mind to remove the daily motion of the heavens, we can focus our powers of reason upon the single star's second motion: this brings out more problems! Let's combine a couple of weeks of observations, creating the following animation in our minds:
[Mars

This bright star, which, like the sun, usually was slower than the stars, being left behind them every night in the westward motion, reaches a time of staying even with them, and then, contrary to its usual motion, moves faster to the west than the fixed stars, before again stopping and resuming its eastward motion... This

"...gives rise to a powerful sense of wonder, which at length drives men to look into causes."

Retrograde

A paradox:

It was apparent that the three superior planets, Saturn, Jupiter, and Mars, attune their motions to their proximity to the sun. For when the sun approaches them they move forward and are swifter than usual, and when the sun comes to the sign opposite the planets they retrace with crablike steps the road they have just covered. (p.118)

(Click here for a visualization of how Ptolemy and Brahe conceived this occuring.) That this retrogression, or backwards-motion, occurs always when the sun is opposite these planets ("wanderers" in Greek), is peculiar, for they do not make their path through the stars in the same time as the sun. Our bright red star Mars, for example, moves through the stars and regains its original position in a little less than two years, although the exact amount of time changes from cycle to cycle, owing to the influence of its proximity to the sun on its path. These two, different, but interacting tendencies among the second motion of a particular planet, are known as the first inequality (its motion through the stars) and the second inequality (the relationship of its proximity to the sun on its motion).
In order to study the first inequality by itself, we'll have to remove the influence of the second inequality.

But in order to separate the second inequality from this first one, they could proceed no otherwise than by considering the planets on those nights at whose beginning they rise while the sun is setting, which thence were called akronychioi, or night rising... [A]t the very moments of conjuunction with and opposition to the sun they are traversing their own true and proper positions. But since they cannot be seen when in conjunction with the sun, only the opposition to the sun remains as suitable for this purpose. (p.120)

Retrograde occuring at opposition. The light blue spot is the location opposite the sun -- the Earth's zodiacal position. (Note the extreme latitude at opposition. The blue line at the bottom of the animation is the ecliptic -- the path that the sun traces against the stars, and the average latitude of Mars.)
Keep reading for mean and apparent oppositions!

You can click here for a draft, 3D version of the spherical animation.
http://science.larouchepac.com/kepler/newastronomy/