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MATLAB Programming/Portable Functions

Functions in MATLAB
A function is a group of sequential expression statements (a.k.a pseudo-algorithm) that are formed together perform a task. In MATLAB, functions are defined in separate files. The name of the file and of the function MUST be the same. Functions operate on variables within their own workspace, which is also called the local workspace, separate from the workspace you access at the MATLAB command prompt which is called the base workspace.

Functions can accept more than one input arguments and may return more than one output arguments.

The sntax of functions as followed

function [y1,...,yN] = myfunc(x1,...,xM)

where syntax declares a function named myfunc that accepts inputs x1,...,xM and returns outputs y1,...,yN. This declaration statement must be the first executable line of the function. Valid function names begin with an alphabetic character, and can contain letters, numbers, or underscores.

Create Functions in Separate Files
To create a new function, at "Home" tab, click on "New" ---> "Functions" It will create a template looks like the following:

Declare a function
To demonstrate a function usefulness, we are going to take a hypothetical situations , ''Recently, in your class there are some exchange students from America whom are consistently complaining the weather is hot outside, when they check on their weather app, they exclaimed the weather is 100 degrees outside. But as to the people who never uses the imperial units before, you might wonder, what is the temperature outside right now ?''

So, to solve the question, it might be a good time to create a custom functions to convert from Fahrenheit to Celsius. First of all, we must know the conversion formula from Fahrenheit to Celsius which is given as: $$C^o = (F^o-32)*\frac{5}{9}$$

At "Home" tab, click on "New" ---> "Functions"

Click on save button to save the functions as convertTemp.m file Note: The filename MUST be same as function name or else it won't work !

Later, on the Current Folder Panel, select the directory where you save the function files just now.

Calling a function
To call the newly created function, do the following: On the Command Window Panel, type this command as shown

Hence, the weather outside is almost 38 Celsius degrees.

Benefits of functions
Why this is beneficial, you might ask ?

Well, from this exercise, you can see that you can just type function name to execute the calculation especially for repetitive tasks. Just imagine for example above, it will be a real time saver to just type function convertTemp instead of manually typing formula each tims.

Type of functions
The type of functions includes such as:
 * Anonymous Functions
 * Local Functions
 * Nested Functions
 * Private Functions
 * Inline function (Phased out in future version)

Anonymous Functions
Anonymous functions allow you to define a function without creating a program file, as long as the function consists of a single statement. A common application of anonymous functions is to define a mathematical expression without creating a separate files.

Here is how the anonymous syntax looks like output = @(arguments) expression where, output = output to be returned arguments = required inputs to be passed expression = a single formula/logic

It can accept multiple inputs and return one output. They can contain only a single executable statement.

Here, we have an examples of converting Fahrenheit to Celsius.

Another example here is using anonymous function to convert minute to seconds (NOTE: 1 minute is equal to 60 seconds)

Local functions
Sometimes in mathematics, you might be encountering an issue where you might find a use case where function that are better to be calculated within it's own function. Rather than running separate functions to get one output, you just need to run just one function to get the multiple output. This is where local functions comes in. To clarify further, in a function file, the first function in the file is called the main or parent function. Additional functions within the file are called local functions, and they can occur in any order after the main function. Local functions are only visible to other functions in the same file. These are equivalent to subroutines in other programming languages, and are sometimes called sub-functions.

The application for this local functions is in a complex function, it would be easier to be broken down to smaller piece of functions and we are certain that the functions are going to be used in same functions and not at other functions.

Examples of local functions
To illustrate an examples, we will create a function statistik which will take in a series of number and return an output of basic function of statistic such as max, min, average and standard deviation. We will give it just a sting of random numbers and it will produce all the usual statistic results.

First of all, we need to create a function named statistik, follow the instruction here at Create Functions in Separate Files

After creating function statistik in a separate file, we generate a random set of number inside the Command Window using randi function

We call the function statistik as followed inside the Command Window: [maximum,minimum,average,stdeviation]=statistik(V)

Although you cannot call a local function from the command line or from functions in other files, you can access its help using the help function. Specify names of both the file and the local function, separating them with a > character.

Nested Function
Nested functions are functions that are nested entirely inside the main functions. The primary difference between nested functions and local functions is that nested functions can use variables defined in parent functions without explicitly passing those variables as arguments.

Requirements of nested functions
(a) To nest any function in a program file, all functions in that file must use an end statement. (b) A nested function can't be defined inside any of the program control statements, such as if/elseif/else, switch/case, for, while, or try/catch. (c) A nested function either directly by function name, or using a function handle that you created using the @ operator (refer to anonymous function). (d) All of the variables in nested functions or the functions that contain them must be explicitly defined. That is, you cannot call a function or script that assigns values to variables unless those variables already exist in the function workspace.

Examples of nested functions
We have a hypothetical situations where we need to estimate the weight of metal ingots which are in cylindrical shape. First, before we started on figuring out the metal ingots weight, we need to break down the problems into smaller piece. To figure out the weight of the metal, we need to figure out the density and before figuring out density we need to figure out volume of the metal.

To start off, before we start to know the volume of ingot, we first calculate the surface area of the cylinder base which is circle. Area of circles is defined as $$ A = \Pi r^{2} $$ ,r is radius of base circle

We can use the area of circle to calculate the volume of cylinder $$ V = A*h $$ ,h is height of cylinder. $$ V = ( \Pi r^{2} ) h $$

Lastly, we use the volume and density to calculate the weight $$ W = V * D $$ where D is density.

In MATLAB, we formulate the formula such as these. Note: For the nested functions beside main funtions, we need to indent the functions. Editor -> Indent.

Note:

When we calling ingot_calc function in the command window, it should display the values

Inline Functions
Inline functions are currently being phased out. Anonymous functions should be used instead. Inline functions are included here for information purposes.

>> convert_s_to_ms = inline('x*1000','x'); >> convert_s_to_ms(20) ans = 20000

Function Handles
Function handles are doubles serving as an abstract reference to the function. Handles allow an equation to be passed to another function for direct evaluation. Anonymous functions are useful for command-line evaluation or for multiple evaluations in the same m-file.

The ampersat returns the handle of a function either built into Matlab or defined in an M-file.

To illustrate example, we need to build a tent as shown on the picture but we need to figure out what is the amount of tarp that are needed to build the carnival tent. Therefore, we will create two separate functions which take the radius and slant height of cone (cone) plus radius and height of cylinder (cylinder).

To refresh the your memory, formula for cone surface (remember we focus slanting area and ignore the base area) is $$ \Pi r l $$ and cylinder surface (remember that one side of circle of cylinder is not calculated) is $$ 2 \Pi r h + \Pi r ^{2} $$ , where the r is the shared radius of cylinder and cone, l is the length of slanting height of cone and finally h is height of cylinder.

Follow steps in Create Functions in Separate Files to start create totalsurftent

We can know the surface area by typing following commands:

Function Handles in m-files
If you are not familiar with m-files, then skip this and come back.

A function handle passes an m-file function into another function. This of course lets you have more control over what's passed there, and makes your program more general as it lets you pass any m-file (as long as it meets other requirements like having the right number of input arguments and so on). The functionality is similar to that of function pointers in C++.

To pass an m-file to a function, you must first write the m-file, say something like this:

function xprime = f(t,x) xprime = x;

Save it as myfunc.m. To pass this to another function, say an ODE integrator, use the @ symbol as follows:

>> ode45(@myfunc, [0:15], 1)

One advantage of using function handles over anonymous functions is that you can evaluate more than one equation in the m-file, thus allowing you to do things like solve systems of ODEs rather than only one. Anonymous functions limit you to one equation.

How to write a function that accepts a function handle
Functions can accept function handles. To do this define them as variables in your header, and then call the handles as if they were functions:

% myadd adds two variables together function result = myfunc(func, a, b); result = func(a, b); [in a separate m-file] function sum = myadd(a, b) sum = a+b;

The command you send to myfunc looks like this:

>> result = myfunc(@myadd, 1, 2); result = 3

title: MATLAB Programming/Portable Functions


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Bagpipe Maintenance

The bagpipes are a woodwind instrument, and with all woodwinds, they are affected by the environment in which they are stored and played in. This can result in problems with the care and maintenance of the bagpipes.

Wood maintenance
A woodwind instrument, being made of wood, expands and contracts slightly with the rising and lowering of humidity. If the wood is allowed to dry out, the wood can crack.

On the subject of oiling, there are different schools of thought. Certainly blackwood is naturally an extremely oily wood, even when it appears to be bone dry. Many pipers never oil sets made of blackwood to no apparent detriment. Others oil only the exterior. If oil is applied, it should be done lightly, and not too often. Woodwind instrument dealers or bagpipe will sell suitable oils; other substitutes, such as olive oil, vegetable oil, mineral oil, engine oil, etc., are not substitutes!

Hemping
Hemping is the process of coating a joint with hemp. In order for bagpipes to perform well, joints must be properly hemped so they are airtight but moveable. Overhemping a joint should be avoided. Various types of hemp are available - it is actually made for cobblers, but are ideal for bagpipes - but the most convenient kind is the pre-waxed yellow hemp. The black hemp is coarser, but more moisture resistant, and can be useful for joints such as the blowpipe which get very wet. If unwaxed hemp is used, it needs to be coated before use either with beeswax or cobblers wax.

Hemp should be wound onto the join evenly and under firm tension, building the layers up one by one. A piece of timber can be used to roll the joint on a flat surface: this compresses the hemp down and speeds up the process of "bedding in".

In general, PTFE (plumber's tape) should not be used as a substitute for hemp, though some players use a very fine layer on tuning slides to assist in fine tuning. There is a school of thought which counsels against this, claiming that PTFE can become embedded in cracks and force them open. Whether this is true or not is not clear.

Seasoning the bag
Natural leather bags usually require a dressing which renders the inside of the bag airtight yet moisture absorbent; this dressing, and the process of applying it, is called seasoning. It is important to note that some bags are treated in such a way that they do not require seasoning, so it is important to check! Synthetic bags, made of nylon or gore-tex, do not require any seasoning (though they do need to be checked regularly for airtightness), and should be kept as clean and dry as is practicable.

Historically, seasonings were concocted from various widely available substances - honey and treacle were popular - but nowadays commercial preparations, such as Hardie's and Robertsons are available from bagpipe dealers. The recipes are trade secrets, but are widely presumed to be largely glycerine. Directions are unique to each product, but in general the best approach is to use generously - at least a cup or more - and once worked into the bag fully, allowed to drain out back into the container for re-use. Metal containers for seasoning are now mostly a thing of the past, but if encountered these should be transferred into a plastic container, as once opened the seasoning reacts with air and the tin and will go bad.

Natural bags do have a limited lifespan: sheepskin bags may last only a year (or even less if used in wet weather or if neglected), and hide bags may last four or five years (though they are now uncommon as they are very poor at controlling moisture).

Reed Manipulation
Only a certain amount can be said about reed manipulation on the printed page, perhaps the best of which is: don't! Nowadays, we have many excellent reed makers and it should not be hard to find reeds that work well at a comfortable pressure for anybody. There are individuals who specialise in picking reeds for mail order customers who cannot visit a reedmaker personally.

Before altering a reed, it is very important to look for other causes of a problem, such as dirt and gunk built up inside the chanter. In general, a new reed can be gently squeezed at the top of the reed (too much will damage the cane: no more than is required to close the lips together is about right), to ease it slightly; if it is still too hard after squeezing, the best approach is to discard it in favour of another. However, it is possible to remove a little cane in order to ease a reed if needed; this should be done with a very sharp knife or fine sandpaper, and be done a very little at a time: every cut or scrape will make an audible difference, and cane cannot be put back on! This approach will also alter the pitch of the reed, and so should be done with care.

Reed Too Hard

 * Gently squeeze lips of blade
 * Lightly sandpaper or scrape blades above sound box
 * With a pair of pliers gently close staple - this is an extreme measure and is likely to destroy the reed.

Reed Too Weak

 * Open reed with a mandrel
 * Snip a tiny amount off the lips of the blades
 * Dispose of the reed and get a new one

Top Hand Too Sharp

 * Raise reed in reed seat
 * Sandpaper or scrape the upper part of the blades (not the lips)

Top Hand Too Flat

 * Seat reed further into chanter
 * As a last resort, snip a tiny amount off the lips of the blades.

Individual Note Flat

 * Use a small file or very sharp scalpel to undercut the top of the hole. DO NOT MANIPULATE THE CHANTER UNTIL YOU HAVE EXHAUSTED EVERY OTHER POSSIBLE OPTION

Individual Note Sharp

 * Apply tape to the top of hole
 * Occasionally, single notes can be brought into tune with small movements of the reed - this is particularly true of D, F, and high G.

Dull Sounding Reed

 * Reed may have become dry, so moisten either by blowing into it for a period of time for a slightly dry reed, or soak the reed in water and allow it to dry naturally
 * Lightly sand the lips of the blades (This will sharpen the top notes)
 * Reed may just be bad, toss

Squeaky Reed

 * Try remedy for weak reed
 * Reed may just be bad, toss

Reed Choking

 * Reed is probably be too strong, or bad blowing
 * Check for possible leaks
 * Could be a bad chanter or incompatible reed

Double Toning F

 * Carefully squeeze sound box until you feel the reed give a little. You may need to perform the squeeze a few times but the problem should disappear as the reed comes down in strength
 * If it appears in a played in reed, bin it. You might be able to fix it by cutting and scraping, but the reed will never be reliable.

Scratchy High A

 * Wait until you have played the reed for a couple of hours, and if the problem continues, try using very fine sandpaper to sand the tops of the lips

Drone Reed Too Weak

 * Move the drone bridle away from the tongue seat

Drone Reed Too Strong

 * Move the drone bridle toward the tongue seat

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Bagpipe Maintenance


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Talk:Bagpipe Maintenance

Perhaps, the article should be moved to Wikibooks how to section

 * Good idea. I've never done a transwiki before or else I would.  Anyone...? - Lucky 6.9 06:59, 6 November 2006 (UTC)

 * Before this is moved or merged anywhere, there is a lot of bad or misleading or only vaguely correct information. Chanters usually fall out because the hemp dries out, not the wood.  Whether or not you should oil depends on the type of wood: African Blackwood should not be oiled, and olive oil and neatsfoot are not suitable.  When hemping, the whole piece of hemp should be waxed; cobbler's wax is useful in providing a firm grip on the wood for the first turns.  There is no need to rehemp unless the hemp is rotten or poorly applied to begin with (most new sets of pipes are badly hemped, though).  Natural bags can be cured so as not to require seasoning, so you need to know how your bag was cured.  A bag needs seasoned whenever the insides of the bag do not slip against one another smoothly.  Clean the seasoning from the stocks before leaving the bag overnight (or it will cool into solid lumps that are a pain to remove).  After a months regular practice, any leather bag is going to be as unhygienic as it is going to ever be.  A regularly played sheepskin bag will only last a year anyway, and an unregularly played sheepskin bag is worthless.  Squeezing a staple with pliers leads to non-functioning reeds - pick ones you can play in the first place.  Don't throw away easy reeds, keep them for beginners.  Don't soak a reed in water unless you are going to chuck it anyway.

 * I can see how we could in theory have an encyclopedia article on bagpipe maintenance, but this is not it. And I don't think it is worth keeping, here or elsewhere, either. Calum 23:01, 6 November 2006 (UTC)

 * I also think this would be a good topic for Wikibooks and not Wikipedia. This is a howto article. Chapium 22:54, 12 November 2006 (UTC)

title: Talk:Bagpipe Maintenance


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Real Analysis/Section 1 Exercises

These are a list of problems for The Real Numbers section of the wikibook. Most of these problems can be described as algebraic problems, although this problem set also includes theorems and concepts from Number Theory. Number Theory is not a primary topic in this wikibook, but there is an appendix section devoted to formalizing several concepts from it. It is recommended to do some problems from the Number Theory heading since its scope of discussion—theorems related to the natural numbers and its supersets integers and rationals—are seldom discussed clearly and usually left to intuition.

Unsorted

 * 1) Show that $$-1<0\ $$
 * 2) Show that $$\forall z\in\mathbb{Z},\ z^2\ge0$$
 * 3) Complete the proofs of the simple results given above.
 * 4) Show that the complex numbers $$\mathbb C$$ cannot be made into an ordered field.
 * 5) Complete the proof of the square roots theorem by giving details for the case $$x<1$$.
 * 6) Suppose A is a non-empty set of real numbers that is bounded above and let s = sup A.  Show that if s is not in A, then for any ε > 0,  there exists an element a in A such that s − ε < a < s.

Algebra
The following questions are meant to formalize a good chunk of algebraic rules one may have simply memorized in elementary mathematics as axiomatically true. However, provided with even the first few laws established in the section The Real Numbers such as the commutative law and algebraic manipulations like moving variables around the equal sign, the following questions should be an easy way to get used to applying theorems to prove one's claim—a very important skill in mathematics.

1. Prove the following theorems on inequalities (Assume that the variables, unless explicitly restricted, can be any number in its assumed domain)  If 0 ≤ x, then -x ≤ 0 If a < b, then -b < -a Given x < 0, if y < z, then xy > xz If a < b and c < d, then a + c < b + d If a d, then a - c < b - d If 0 ≤ a < b and 0 ≤ c < d, then ac < bd

2. Prove the following inequalities (Assume that the variables, unless explicitly restricted, can be any number in its assumed domain)  If 1 ≤ x, then x ≤ x2

If 1 ≤ x, then 1 ≤ x2

If 0 < x < 1, then x2 < x

If 0 ≤ x < y, then x2 < y2

Given x, y such that 0 ≤ x, y, if x2 < y2, then x < y

Given an odd number n, if x < y, then xn < yn

Given a natural number n, if 0 ≤ x < y, then xn < yn

3. Prove the following consequential theorems related to the laws provided in this chapter

If there exists the number 0, then $$0 = -0$$

$$\forall x,y\in\mathbb{Z},\ -(xy) = (-x)y = x(-y)$$

4. Prove the following theorems on rational numbers

Given $$a \ne 0, b \ne 0$$, $$\frac{1}{ab} = \frac{1}{a} \cdot \frac{1}{b}$$

Given $$b \ne 0, c \ne 0$$, $$\frac{a}{b} = \frac{ac}{bc}$$

Given $$b \ne 0, d \ne 0$$, $$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$

Given $$b \ne 0, d \ne 0$$, $${\frac{a}{b}}\div{\frac{c}{d}} = \frac{ad}{bc}$$

Answers

Absolutes
1. Prove the following inequalities (Assume that the variables, unless restricted, can be any number)

|a| + |b| ≤ |a + b|

Answers

Number Theory

 Prove the following properties on even and odd numbers:

If you add two even numbers, then the sum is even.

If you add two odd numbers, then the sum is even.

If you multiply an odd number with an even number, then the product is even.

If you multiply two odd numbers, then the product is odd.

 Prove that no consecutive number of a perfect square is also a perfect square for all natural numbers. You don't have to factor in 0 for this problem.

 Prove that there exists no primitive Pythagorean triple $$a, b, c$$ such that either a and b are even or a and b are odd.

 Given that $$\forall a \in \mathbb{Z} : [\exists b \in \mathbb{N} : \exists r, q \in \mathbb{Z} : a = bq + r]$$, prove that if the given holds, then the remainder r has the following property $$0 \le r < b$$.

 Prove that $$\sqrt{2}$$ is irrational.

 Prove that any square root of a prime number is irrational.

 Given the equation $$x - c = A(x - d) + B(x - e)$$ such that $$A, B, c, d, e$$ are constants, prove that if $$x$$ is any number except for $$d$$ or $$e$$ then both $$A$$ and $$B$$ cannot be defined.

Bridge Questions
The following questions can be solved more easily and quickly with more advanced tools. However, solving these questions with restrictions in your mathematical tools provides an excellent understanding of how mathematics as a whole interacts. As a general rule, the answers to these problems should be longer and rely on a lot more properties.

title: Real Analysis/Section 1 Exercises


text:
Real Analysis/Section 1 Exercises/Hints

 * 1) No Hint
 * 2) No Hint
 * 3) No Hint
 * 4) No Hint
 * 5) No Hint
 * 6) No Hint
 * 7) No Hint
 * 8) The only fact you need use about $$\mathbb C$$ is that it contains a square root of $$-1$$.
 * 9) In the general case you will probably want to divide by your prospective square root, as in the part of the proof which was given, so you might want to treat the case $$x=0$$ separately.
 * 10) No Hint

title: Real Analysis/Section 1 Exercises/Hints


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Real Analysis/Section 1 Exercises/Answers

Answers in this wikibook work on the didactic that you should still be able to logically piece together a proof that can sufficiently answer a question. Thus, no copy-paste answer is available on this page. Instead, the answer section simply lays out all the tools to prove the problem sufficiently. Many answers will provide hidden questions since some assumptions may not be known to you. Luckily, some of these assertions will have answers elsewhere in the wikibook or can be solved independently.

Note that all solutions are suggestions. There can be multiple ways to solve a question, and as this is a collaborative wikibook, you are free to put down your answer (as long as it follows out guidelines).

Problem 4: Prove that $$0 = -0$$

 * 1) One algebraic property of zero is $$\forall x \in \mathbb{R} : x - x = 0$$
 * 2) And we can use either valid theorem
 * 3) * Subtracting both sides of the equation by the same variable is valid.
 * 4) * Multiplying both sides by negative 1 is valid.
 * 5) And finally, substitution of whole terms is valid.

Algebra
  The following are answers.  As a side note, this is why you do the "inequality sign flip" when you "multiply by -1" $$\begin{align}a &< b \\ a - a &< b - a \\ 0 &< b - a \\ -b &< b - a - b \\ -b &< (b - b) - a \\ -b &< 0 - a \\ -b &< -a \\ \square \end{align}$$  Intuition solved! $$\begin{align}a &< b \\ a + c &< b + c \\ a + c &< b + c < b + d \\ a + c & d \\ -c &< -d\end{align}$$ Part 2: Let's solve it now $$\begin{align}a &< b \\ a + (-c) &< b + (-c) \\ a - c &< b + (-d) \\ a - c &< b - d \\ \square\end{align}$$  Answer me Answer me Answer me Answer me

Answer me

Answer me

Answer me

Answer me

Answer me

The following answers only outline a method to answer the question

The following answers only outline a method to answer the question

The largest hurdle is thinking of ways to prove this since distribution of the power term has not been established yet. However, there is a method:
Start off with $$1=1$$

Apply the existence of an inverse as such: $$(a \cdot 1/a) \cdot (b \cdot 1/b) = 1$$

Redistribute and swap over variables to the other side until the equivalency shows up. As a last resort, consult the following whited out hint in text format:
(ab) × 1/a 1/b = 1 → 1/a × 1/b = 1/ab

Assuming problem 4I has been solved, this question is relatively easy.

Assuming problem 4I has been solved, this question is relatively easy.

Answer me

Absolutes

 The following are answers.

As a side note, you should generally not use squares in your proof! However, it works here just because we're dealing with absolute values, the resulting number when you reverse the squaring operation. $$\begin{align}|a + b| &\le |a| + |b| \\ |a + b|^2 &\le (|a| + |b|)^2 \\ a^2 + 2ab + b^2 &\le |a|^2 + 2|a||b| + |b|^2 \\ a^2 + 2ab + b^2 &\le a^2 + 2|ab| + b^2 \\ 2ab &\le 2|ab| \\ ab &\le |ab| \\ \square \end{align}$$

Problem 1: Prove the following properties on even and odd numbers

 * 1) These definitions can be substituted in for each of the following problems.
 * 2) * The definition of an even number e is $$\exists k \in \mathbb{Z} : e = 2k$$
 * 3) * The definition of an odd number o is $$\exists j \in \mathbb{Z} : o = 2j + 1$$
 * 4) Multiplication and addition are closed under the integers.
 * 5) The distributive law extends to integers, and this axiom is probably the most difficult theorem that is used to solve these problems.
 * 6) Any necessary factor can be expressed as a variable to clean up interpretation.

Problem 2: Prove that no consecutive number of a perfect square is also a perfect square

 * 1) In math notation, the question asks
 * 2) Some valid theorems are the distributive law for natural numbers and adding the same number over both sides of an equation.

Problem 6: Prove that any square root of a prime number is irrational

 * 1) A prime number does not include 1 and should only have itself as a factor.
 * 2) If $$\sqrt{p}$$ is rational, then $$\sqrt{p}=r/s$$ for some coprime integers $$s$$ and $$r$$; $$s$$ and $$r$$ are in its lowest terms.
 * 3) Types of ways to reach a contradiction:
 * 4) * Both $$s$$ and $$r$$ can be shown to be divisible by the prime number $$p$$ after exposing the prime number using algebraic manipulation, thereby breaking the coprime definition.
 * 5) *# "Any natural number has a prime number as a factor" is valid.
 * 6) * The prime number $$p$$, after exposing the prime number using algebraic manipulation, can be shown to have a factor of two of the same natural number squared, which breaks the prime number definition.
 * 7) *# "Any natural number cannot be expressed as a fraction in its lowest common factor form and have its denominator anything other than 1" is valid.