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** These operations are used in Math 099.
Overview of Math 099
Math 099 is intermediate algebra with geometry. It presupposes that you already know how to add, subtract, multiply, divide, raise to powers and extract roots using real numbers and fractions. Chapter one reviews the steps for solving equations with one variable and no powers (linear equations in one unknown). The rest of the book talks about graphing equations with two unknowns, solving simultaneous linear equations, rational expressions, complex numbers, radicals, and the "pièce de résistance", the "Grand Finale", the Big Boom when we set off all the remaining fireworks, which is the quadratic formula for solving second order equations in one unknown. All of this is done with an accompaniment of geometry examples in the background.
Another concept that is important in Mathematics is the inverse of a function. This would be the function that does the opposite of some other function. It can be used to undo what the other function did. For example if your function adds 5 to a number, the inverse function would be to subtract 5. If the original function multiplied by 12 then the inverse would divide by 12 or, equivalently, multiply by the reciprocal of 12. In this course we use the concept of the inverse function in chapter one as well as when we talk about radicals and powers being inverse functions.
My Grading Policy
I do not grade effort, I do not give extra points for good behavior, I do not give a minimum grade for perfect attendance. Worse yet, I do not give extra points for doing your homework, nor do I give partial credit for answers that are almost right. I only care about whether you can do math and get the right answer. Your answers are either right or they are wrong. Period. Either the Mars Lander is on target or it isn't. Either the patient gets the right medicine or he doesn't. If the lander misses the target, it is lost. If the patient is given the wrong amount of medicine he may die. I hope you understand how critical it can be for you to get the right answer. When you add ½ + ¾ I want you to be right every time.
Using calculators
There are so many new things to learn we really don't have time to work on the stuff you are supposed to already know. Some teachers may shrug and say there's nothing they can do about "your problem" and others may take time out from what they are supposed to be teaching and review remedial concepts. I really want to help you pass this course, but I want you to do it on my terms; you need to be able to get the right answers on my tests. I am so hung up on getting right answers that I will even allow you to use calculators. Because every calculator seems to work differently, you should never try using a calculator on a test until you have had a chance to do every problem with it. There are several calculators you might want to try but I am currently very positive on the Casio fx-115MS. It does fractions and complex numbers although it doesn't do radicals. I strongly encourage you to use it or another calculator that has the same fuctionality. In fact, even if you don't think you need a calculator, I strongly encourage you to use one to check your answers. This way you should be able to concentrate on understanding WHY you are doing the math instead of getting hung up on HOW to do it.
Notation
I use curly braces
For example, to add 2 plus 3 Press:
I hope it doesn't confuse you but sometimes I got lazy and combined several keystrokes into one curly bracket, like this:
The
Browsers
Web pages are rendered differently by different browsers. Firefox and Netscape seem to do a better job of displaying the HTML special characters; however, even if you are using MS-InternetExplorer, if you look at my Keyboard layout and compare it to your calculator you should be able to figure out which symbol I am using for each key. If you find anything to be confusing, please let me know sooner rather than later.
The fx-115MS is called the fx-912MS in Japan. It is a general purpose scientific calculator that has approximately 300 built-in functions but no graphing capabilities. It has a two line display and some deceptively sophisticated programmability. The top line is where you put your input, the bottom line is where the calculator displays its answers. The only tricky part to using the calculator is the funny looking circle labeled "copy" and "replay". This button also has four cursor controls for up, down, left and right.
Other keys can be used for various operations depending on the state of the calculator. The state can be set by using the SHIFT and ALPHA keys or by putting the calculator into certain MODES . You can tell the state of the calculator by looking at the little indicators above the first line of the display. Here is how to access the color coded functions associated with each key.
| Color | How to get it |
|---|---|
| Black | Normal |
| RED | |
| Brown | |
| BLUE | SD or REG MODE |
| GREEN | BASE MODE |
Different Calculator Logic Types. Calculators can be radically different in the way they operate. They use different keystroke sequences to enter the same formula. Do not make the mistake of borrowing someone else's calculator for a test unless you have been using it all along to do your homework. Here is a summary of various logic types that I saw at www.rskey.org .
Arithmetic. This logic type is typically associated with desk-top adding-machine type calculators. It's fine for accountants and bookkeepers, don't even think of using them in a Math or Science course.
Simple Algebraic. This is the algebraic logic used on most so-called four-function calculators. Calculators using this logic method will have an equal key but no parentheses keys. Slightly better than an arithmetic calculator but not really.
RPN stands for Reverse Polish Notation. RPN is characterized by an "Enter" key, and the absence of equals or parentheses keys. Hewlett-Packard has marketed RPN calculators for nearly 30 years. Today they still offer several RPN calculators, including a few dual mode RPN-algebraic calculators. RPN calculators are relatively expensive, with the cheapest going for around $60. They can do anything but they require you to totally rethink your formula before you enter it into the calculator.
Traditional Algebraic. This is the algebraic logic type used on many electronic calculators since the '70s. This logic type utilizes parentheses and an equal key. Unary operations (e.g. square root) are performed on a number already existing in the calculator's display register. Texas Instruments refers to their version of the traditional algebraic logic by the trademarked name AOS (Algebraic Operating System).
Formula Algebraic. This logic type is referred to by various trademarks: Casio's V.P.A.M. (Visually Perfect Algebraic Method), Sharp's D.A.L. (Direct Algebraic Logic), and Texas Instruments' EOS (Equation Operating System). It allows expressions to be entered in the same way as a mathematician would write them in an equation. For example, square root is entered before the expression on which it operates while square is entered after the expression. It is different from traditional algebraic calculators like Texas Instruments' AOS which use parentheses and an equal key but all unary operations are performed on the number in the calculator's display register.
S-V.P.A.M. stands for Super Visually Perfect Algebraic Method. This is the algebraic logic used on your fx-115MS. It is VPAM enhanced with a two line screen that lets you see your input together with the result. The calculator keeps a history of previous calculations which you can recall with the Replay feature, make any changes that you want, and then recalculate.
I have found instructions for the calculator in a PDF file on the CASIO web site here .
|
| Shift | [S] | Execute |
|---|---|---|
| Alpha | [A] | Execute |
| Hyperbolic | hyp | Execute |
| Memory | M | Non zero value in Memory location |
| Store | STO | |
| Recall | RCL | |
| Standard Deviation | SD | Standard Deviation and other single-variable statistical calculations. Execute |
| Regression | REG | Regression and Paired-variable statistical calculations. Execute |
| Complex Numbers | CMPLX | Complex numbers |
| Matrix | MAT | Doesn't seem to be used on this model |
| Vector | VCT | Doesn't seem to be used on this model |
| Equation | EQN | Solutions of Simultaneous Linear Equations or Polynomial Equations |
| Degree | [D] | Angles are measured in degrees |
| Radian | [R] | Angles are measured in radians |
| Grad | [G] | Angles are measured in grads |
| Fixed | FIX | Numbers are displayed in Fixed Mode |
| Scientific | SCI | Numbers are displayed in Scientific Mode |
| r θ | r∠θ | complex numbers are displayed in |
| Engineering Units | Eng | Numbers are displayed in |
| Complex Number | R⇔I | the result is a complex number, you need to use the |
Addtional indicators on the screen are:
| Up arrow | ↑ | There are prior screens in |
|---|---|---|
| Down arrow | ↓ | There are subsequent screens in |
| Right Arrow | → | there are additional menu choices you can get with the |
| Exponent Base | ×1088
d H b o | Used in Scientific Mode to indicate the exponent of 10
This same indicator is used in Base mode to indicate whether the displayed units are: d-Decimal, H-hexadecimal, b-Binary, o-octal and to execute |
| Multi-Statement Mode | Disp | Indicates that the calculator is executing a Multi-statement command. Special thanks to Adam Sundor for telling me what this does. |
| Imaginary | i | The imaginary component of a Complex number number in a+bi form. |
| Angle | ∠ | The angle argument of a Complex number number in r∠θ form |
| Left Arrow | ← | there are additional menu choices you can get with the |
How to test all pixels on the screen.
Each digit on the second line is made up of 7 line segments. If any of them cease to operate correctly you could be misreading the data and putting the wrong answers on your homework and tests. I think it is important to check your display from time to time. To do this, follow these instructions.
Press and Hold
Press and Hold
Press
Release all three keys.
Press
Pressing
Press
I've been told that on earlier versions of the fx-115MS you only need to press the
The
These two keys change the
When you press the
When the calculator is in the [S] state, pressing any key that has a BROWN inscription will invoke the corresponding function. In the [A] state the RED functions are executed.
Pressing any key will cause a state to be cleared whether or not there is a function associated with that state and key combination.
The
This key is used all over the place to review input data as well as to scroll through output results.
The Copy Key allows you to several combine lines from
The
This key is used when editing the top line of the display as well as in switching between alternate sets of menus. You can tell if there are additional menu choices by checking the → or the ←
The
This key is used when editing the top line of the display as well as in switching between alternate sets of menus. You can tell if there are additional menu choices by checking the → or the ←
The
The replay key allows you to call up SOME previous calculations, optionally change them, and recalculate them. I say SOME previous calculations since there seem to be a lot of situations which clear out the replay memory. If there is anything in replay memory you should see the either the ↑ or ↓
The
This key is used all over the place to review input data as well as to scroll through output results.
The
The Mode key allows you to switch between different calculator modes and to change settings having to do with how information is displayed.
Press the
The CLR key gives you the following options: 1-Mcl 2-Mode 3-All
In SD MODE this changes to : 1-Scl 2-Mode 3-All
The
This key turns the calculator on. The calculator has an Auto-Off function which turns the calculator off after six minutes of inactivity. If you want to turn the calculator off manually you can do this with the
To use this key you need to enter an equation using the
For example, to solve the equation A = B² + C given A = 10 and C = 1 you would enter:
WAIT a long time... the answer
TIP: People who had trouble with this function were using the wrong key to enter the
Note: this key is totally different from the main
Use this key when you want to enter an equation to be solved by the equation solver using the
The
The CALC key enables you to enter a formula or an expression and evaluate it by substituting values for the variables.
Example:
B?
A?
C?
If you press the
uses the difference quotient ( ( f(x+h) - f(x) ) ÷ h ) to approximate the derivative. The function requires three inputs:
This key allows you to enter a "multi-statement" which is the concatenation of more than one instruction in the same line. Another way to create a multi-statement is by using the Copy key. I believe it can probably be very useful. It seems that if you use the up arrow key after the execution of a multi-statement you get the individual statements that had been executed but if you use the left arrow key you get a chance to edit the multi-statement. There are some other idiosyncrasies but I haven't cared enough to figure them out.
Example: Here is a neat trick for generating pairs of data points automatically. Graph the function f(x)=x² - 3x + 2 in the interval from x = -1 to x = +3. Generate data points for values of x 0.5 units apart.
We need to do two things at each step, calculate a y-value and get the next x-value.
We will enter both of these statements into the calculator on one line and use a colon to separate them.
The finished program will look like this:
Here is the process:
| X | Y |
|---|---|
| | |
| | |
| | |
| |
The
Use Simpson's rule to calculate an approximation to the integral. This key requires 4 input arguments:
x=2 ∫ ln(x)dx =Example 1:[ 0.386294361 ] x=1
calculates the factorial function. The factorial can only be calculated for integer values between 0 and 69. The
factorial function is usually defined recursively as follows:
0! = 1
1! = 1
2! = 2 × 1!
n! = n × (n-1)!
Example:
The
This key is only available in Base MODE. Each time you press it, you will get a different menu:
The
x-1 calculates the reciprocal function which is the same as 1 ÷ x. Note that in this context the -1 is an actual exponent and not an inverse function. In other words this works like x² and gives the same result as pressing:
Example:
calculates the cube root of the value.
Example:
Be sure to use parentheses if needed:
The
calculates x × x × x
Can also be calculated with the exponent key by entering the key sequence:
Example:
After you have displayed an answer you can use this key to convert it to an improper fraction, or back to a proper fraction format.
Example 1:
Example 2:
Hint: You can use the
The
This key can be used in two different contexts.
The
Calculates the square root of a value. In Computational Mode you can not take the
square root of a negative number. In Complex Number Mode you can get a complex result.
Can also be calculated with the exponent key by entering the key sequence:
Example:
The
This key is only available in Base MODE.
It converts the value shown in the display to Decimal (Base 10)and sets the "d"
The
calculate the square or x × x.
Can also be calculated with the exponent key by entering the key sequence:
Example:
calculate the x root. this is the inverse of the
Example:
In this course you need to know that rational exponents and that this is the same as the (x-1) or the 1/x power.
So
The
This key is only available in Base MODE. It converts the value shown in the display to Hexadecimal (Base 16) and sets the "H"
The
Raises a value to a specified power which can be an integer, a fraction or it can even be negative. The inverse of this function is the
Example:
Note: in Complex Number Mode this key is restricted. You can not use it raise a complex number to a power.
Example:
allows you to calculate any power of ten. This function is also known as the antilogarithm since its inverse is the function
Example:
The
This key is only available in Base MODE. It converts the value shown in the display to Binary (Base 2) and sets the "b"
The
Calculates the base 10 logarithm. The inverse of this function is the
Example:
Note: To calculate a logarithm to some other base Use the formula:
loga(b) = log(b) ÷ log(a)
log2(3) is
You can check this by entering
calculates a power of e . This is the inverse of the natural logarithm function
Example: To calculate e1.5 press:
The
This key is only available in Base MODE. It converts the value shown in the display to Octal (Base 8) and sets the "o"
e is a constant like π which can not be represented with decimal digits. Its approximate value is 2.71828182845904523536028747135266249775724709369995957496696762772407663035354
For a more precise value you can click here
The
The ln key is the inverse of the
Example:
This key is only available in Complex Number Mode and is used to enter a complex value in polar coordinates.
Example:
This key is used to refer to the variable A.
Example: To add the contents of variable A to 5
Note: In Base 16 - HEX mode use this key WITHOUT the
The
This key is used for entering negative quantities.
This key allows you to convert decimal values to sexagesimal values. Sexagesimal values can correspond to hours° minutes° seconds or degrees° minutes° seconds or any other base 60 numbering system.
This key is used to refer to the variable B.
The This key can be used in two different contexts. This key is used to refer to the variable C.
The This key is used to call the Hyperbolic and the Inverse Hyperbolic functions. When you press this key it turns on the hyperbolic To change the default angle unit (degrees, radians, grads) see the This key is used to refer to the variable D.
The To change the default angle unit (degrees, radians, grads) see the To change the default angle unit (degrees, radians, grads) see the This key is used to refer to the variable E.
The To change the default angle unit (degrees, radians, grads) see the To change the default angle unit (degrees, radians, grads) see the This key is used to refer to the variable F.
The To change the default angle unit (degrees, radians, grads) see the This key allows you to store a value into one of the variables A, B, C, D, E, F, M, X, and Y. In Note 2: Special thanks to Paul Bonarrigo, P. E., for pointing out a strange side effect of this key. Most of the time if you unnecessarily press the Let me paraphrase Paul's example:
The Pressing this key sets the "RCL" This key takes a result and displays it as a value in the range (.010-9.99) multiplied by a power of ten that is divisible by three. Look at the Actually the whole thing makes a lot more sense if you turn engineering units The This key is only available in Complex Number Mode and is used for entering the imaginary number i.
The This key takes a result and displays it as a value in the range (1-999) multiplied by a power of ten that is divisible by 3. Look at the Each additional press of the key multiplies the range by 1000 until the result would exceed ten digits.
Actually the whole thing makes a lot more sense if you turn engineering units Parentheses serve to group a portion of an expression into a separate basket which is evaluated before things that are outside the basket. In this way they serve to change the natural order of operations.
This key is only available in Complex Number Mode. It calculates the argument of the complex number - that is it calculates the principle angle θ in the polar coordinates representation of a complex number.
This key is used to refer to the variable X.
This key is only available in Complex Number Mode and calculates the polar r coordinate for a complex number. The coordinate r is the absolute value of a + bi which is given by r = √(a² + b²)
This key allows you to enter multiple occurrences of the same data point in Standard Deviation and Regression calculations.
This key is used to refer to the variable Y
The This key separates the parameters in a multi parameter function.
This key is only available in Complex Number Mode. It calculates the complex conjugate of a value.
This Key allows you to subtract from memory M.
This key is used to refer to the variable M.
Example 1: Add negative 3 to negative 4:
You can not use this key to subtract two numbers. For that you must use the
Example 2:
The correct way to do this is
Warning: Unlike Excel and some other calculators, this calculator assigns a lower precedence to the negative operator than other operations such as raising a value to a power. For example
Note: in Base 16 (hexadecimal) Mode this key is used to enter the Hex digit A16 (value 10)
Example: To convert 4.085° (degrees to degrees° minutes° seconds)
Example 2: To convert three and a half hours (hours to hours° minutes° seconds)
Note: The key is actually a toggle key - if you press it a second time it will convert the data back to decimal format. As far as I can see this is a totally useless key since you can use the
Example: To add the contents of variable B to 5
Note: In Base 16 - HEX mode use this key WITHOUT the
Example:
Note 1: There is a discrepancy between how the data appears when you enter it and how it appears in the result screen. Every time you press the
Note 2: Degrees° Minutes° seconds can also be read as Hours° Minutes° Seconds. This almost enables you to do time calculations except that instead of a 12 hour day or 24 hour day, there are no adjustments made for the end of the day. So 10:30 plus 4 hours is 14:30 and 21:30 plus 4 hours is 25:30.
Example: If you add 3 hours and 45 minutes to 2:40 the answer is 6:25
Example:
Note: You can also use the
Side Note: in Base 16 (hexadecimal) Mode this key is used to enter the Hex digit B16 (value 11)
Example: To add the contents of variable C to 5
Note: In Base 16 - HEX mode use this key WITHOUT the
To call inverse hyperbolic functions it doesn't matter in which order you press the
Example 1: sinh(3.6) =
Example 2: sinh-1(30) =
or also
These are the definitions of the hyperbolic functions:
Note: in Base 16 (hexadecimal) Mode this key is used to enter the Hex digit C16 (value 12)
To override the default angle unit see the
Example:
Example: To add the contents of variable D to 5
Note: In Base 16 - HEX mode use this key WITHOUT the
To override the default angle unit see the
Example (in degree mode): sin 63 ° 52 ’ 41 ”
Note: in Base 16 (hexadecimal) Mode this key is used to enter the Hex digit D16 (value 13)
To override the default angle unit see the
Example:
Example: To add the contents of variable E to 5
Note: In Base 16 - HEX mode use this key WITHOUT the
To override the default angle unit see the
Example:
Note: in Base 16 (hexadecimal) Mode this key is used to enter the Hex digit E16 (value 14)
To override the default angle unit see the
Example:
Example: To add the contents of variable F to 5
Note: In Base 16 - HEX mode use this key WITHOUT the
To override the default angle unit see the
Example:
Note: in Base 16 (hexadecimal) Mode this key is used to enter the Hex digit F16 (value 15)
Note: when you press the
Example: To store the value of 2 plus 3 into A Press:
Step one: Calculate 1+1 and leave the answer in Answer Memory:
Step two: Multiply the answer by 3 and put the result into Variable Memory A:
HOWEVER, if you get into the habit of (unnecessarily) pressing the
Here's how it works. Let's say that in step two you use this sequence of keystrokes:
In other words, if you are using the Ans value in your expression you have to remember that every time you press any of the
Example:
Example:
Pressing the
Note: This key looks like it would use the
Example:
Example 1:
Example 2:
Example 3:
Example:
Example:
Example:
To see how this key might be used, see the Example that follows the
Memory M is a special location. To see how it is used, see the Example that follows the
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