Making the Most of your Casio fx-991ES Calculator

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Making the Most of your Casio fx-991ES Calculator

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TSR Wiki > Study Help > Subjects and Revision > Subject Guides > Mathematics > Making the Most of your Casio fx-991ES Calculator

A lot of people own an fx-991ES, but I've found that very few of them realise how powerful this unassuming little calculator actually is. Here's a (not entirely comprehensive) guide to getting the best out of the Casio fx-991ES. The guide assumes basic knowledge of how to get around the calculator (changing mode, changing from degrees to radians, memory etc). For more information, consult the manual.

Mode 1: COMP

This is the mode you'll be using most of the time. It's probably the most versatile mode, giving you the options of definite calculus, calculation of functions and a rudimentary equation solver. Unless otherwise specified, this guide will be using MathIO, (Shift/MODE/1), Norm1 (Shift/MODE/8/1).

Definite Integration

It must be noted that integration can only be performed with respect to x. To perform definite integration, first press the $\displaystyle \int _{[]} ^{[]} []$ button. This will bring up an integral sign, with blanks for the upper and lower limits and the integrand, and a 'dx' at the end. The cursor will be flashing for you to enter the integrand (in terms of x), using ALPHA - for example, X^2 + 2X . Once you have entered your integrand, input your upper and lower limits by scrolling up and down using the REPLAY/directional arrow buttons. Press [=], and it'll perform the integration, giving the answer as a fraction if appropriate.

Numerical Differentiation

Again, this can only be performed with respect to x. The calculator will, given f(x), calculate f'(x) for a particular value of x. Press SHIFT/ $\displaystyle \int _{[]} ^{[]} []$ (that is, $\frac{d}{dx}[]$ ). This will bring up something that looks a bit like this: $\displaystyle \frac{d}{dx}\left( [] \right) | _{x = []}$ . Enter f(x), and scroll right to enter the value of x at which you want to calculate the gradient. Press [=].

The CALC Button

This allows you to calculate the value of a function (of up to 7 variables) at a particular value of each variable. At a blank screen, enter your function in terms of the variables (using ALPHA) - e.g. (X + Y)^2 . Press CALC. Prompts will appear for you to enter values of your variables. Don't worry if there's already a value displayed - this is just the most recent value that has been stored as that letter. Simply enter your value (in terms of pi, ln, log, fractions...) and press [=]. You will have to repeat this for each variable. The final press of [=] will calculate the value of the function.

The SOLVE Operation

This will solve most equations that you throw at it - with a few catches. The first major catch is that it only ever returns one solution - so if you're trying to find x with an equation in x^2, you're better off in equation mode, however multiple values can be found. The second catch is that you can only ever solve for one variable at a time - entering multiple variables will cause the calculator to prompt you to enter specific values for each of those variables save for x (or the one you've specified (by adding, say, ",Y" to the end of the equation). The third catch is that if your equation is rather complicated, it can take a very long time to solve accurately. The fourth catch is that it will only ever give you a solution as an integer or decimal.

I'll deal only with equations in terms of x, as doing it this way saves a few keypresses, and potentially a lot of valuable time. Enter the equation you want to solve, using ALPHA. It is not necessary for this to be in the form f(x) = 0. For example, 3X + 2 = 6X . Then press SHIFT/CALC (SOLVE). The calculator will then display "Solve for X", along with the current value of X in memory (this can generally be ignored - see below). Press [=] again, and it'll come up with this:

3X+2=6X X= 0.6666666667 $L-R= \, \, \, \, 0$

What this means is that for $X \approx \frac{2}{3}$ , 3X + 2 - 6X = 0. "L - R" gives you an idea of the accuracy of the calculator's solution. In most cases , it'll be exact (assuming you have the sense to change recurring decimals into fractions).

If your equation has more than one solution there is a way in finding them take 2x^2+3x-5=0 and you press solve, the screen you then see is if you type in a value the calculator will solve your equation giving you an answer nearest to the value you entered so for the example above, if you entered 5 as your value your answer will be 1, then when you solve it again and enter -5 as your value your answer will be -2.5

Summation

This allows you to calculate the sum of a series between two values. The method for this is essentially the same as that for integration. To access the summation function, press SHIFT/ $\log _{[]}[]$ ( $\displaystyle \sum _{x=[]} ^{[]} ([])$ ).

Mode 2: CMPLX

Numbers can be displayed in "Modulus Argument" (r theter) and also the normal real / imaginary form (it is possible to convert between the two as well)

Mode 3: STAT

Mode 4: BASE-N

Modes: Binary, Decimal, Hex, and Oct. This allows you to add / subtract / divide and multiply in different number bases. It goes up to 16 bit and does "Twos Complement" as well (which is used to store negative numbers).

Mode 5: EQN

This is possibly the most useful of the fx-991ES's 'extra' modes. You can use it to solve simultaneous equations of two or three variables, quadratics in the form ax^2 + bx + c = 0 , and cubics in the form ax^3 + bx^2 + cx + d = 0 .

Upon entering equation mode, you will be presented with the following options:

1: a_nX + b_nY = c_n
2: a_nX + b_nY + c_nZ = d_n
3: aX^2 + bX + c = 0
4: aX^3 + bX^2 + cX + d = 0

Simultaneous Equations

Options 1 and 2 are the simultaneous equation modes. Select option 1 if you have two unknown variables, or option 2 if you have three. Here I'll deal with two unknowns - the procedure for three is exactly the same. I'll use the simultaneous equations obtained from the solution to the following question:

"A man buys 3 fish and 2 chips for £2.80 A woman buys 1 fish and 4 chips for £2.60 How much are the fish and how much are the chips?"

3x + 2y = 2.8 (equation 1),

x + 4y = 2.6 (equation 2),

where x is the price of 1 fish, and y is the price of 1 chip. The more mathematically able among you may already have seen that a fish costs £0.60, and a chip costs £0.50.

Upon selecting option 1, you'll be faced with a 3*2 matrix - the columns of which are labelled a, b, and c, the rows of which are named 1 and 2.

Enter equation 1's x coefficient, then press [=]. Do the same for the y coefficient and the constant on the RHS. Press [=] again, and do the same for equation 2. Pressing [=] again will show you first the solution for x, and another press shows the solution for y. So the keypresses are:

[3][=][2][=][2][.][8][=][1][=][4][=][2][.][6][=][=]

Quadratic and Cubic Equations

Options 3 and 4 are the polynomial modes - choose option 3 if you have a quadratic equation, and option 4 if you have a cubic equation. Note that they will provide you with all roots, both real and complex. Here I'll be dealing with a quadratic, 2x^2 - 3x + 1 = 0 , the roots of which are x = 1 and x = 0.5.

Enter the coefficients of x^2, x, and 1 in exactly the same way as has been outlined for simultaneous equations. Press [=] again and you will be shown the two roots of the equation, X_1 and X_2 . In the case of a repeated root, simply "X = " will be displayed.

Mode 6: MATRIX

Adding Matrices

$\begin{bmatrix}1 & 3 & 1 \\1 & 0 & 0 \\1 & 2 & 2\end{bmatrix}+\begin{bmatrix}0 & 0 & 5 \\7 & 5 & 0 \\2 & 1 & 1 \end{bmatrix}=\begin{bmatrix}1+0 & 3+0 & 1+5 \\1+7 & 0+5 & 0+0 \\1+2 & 2+1 & 2+1\end{bmatrix}=\begin{bmatrix}1 & 3 & 6 \\8 & 5 & 0 \\3 & 3 & 3\end{bmatrix}$

To solve that press mode,6,1,1* and then enter values for your first matrix, then press on, then shift,4,2,2,1* then enter values again for your second matrix, then on again then press shift 4,3 then press add, then press shift 4,4, then press equals

for 3x3

Scalar Multiplication

$2 \cdot\begin{bmatrix}1 & 8 & -3 \\4 & -2 & 5\end{bmatrix}=\begin{bmatrix}2 \cdot 1 & 2\cdot 8 & 2\cdot -3 \\2\cdot 4 & 2\cdot -2 & 2\cdot 5\end{bmatrix}=\begin{bmatrix}2 & 16 & -6 \\8 & -4 & 10\end{bmatrix}$

To do a scalar multiple of a matrix press mode 6,1,4* then enter your values of your matrix, then on, then press the number your multiplying by and then times, then press shift 3 and then equals

for 2x3

Matrix multiplication

$\begin{bmatrix}1 & 0 & 2 \\-1 & 3 & 1 \\\end{bmatrix}\times\begin{bmatrix}3 & 1 \\2 & 1 \\1 & 0 \\\end{bmatrix}=\begin{bmatrix} ( 1 \times 3 + 0 \times 2 + 2 \times 1)& ( 1 \times 1 + 0 \times 1 + 2 \times 0) \\(-1 \times 3 + 3 \times 2 + 1 \times 1)& (-1 \times 1 + 3 \times 1 + 1 \times 0) \\\end{bmatrix}$

= $\begin{bmatrix}5 & 1 \\4 & 2 \\\end{bmatrix}$

To multiply the above example press mode,6,1,4* and then enter values, then press on, then shift,4,2,2,2** then enter values again then on again then press shift 4,3 then press times, then press shift 4,4, then press equals

for 2x3
- for 3x2

Transpose Matrix

$\begin{bmatrix}1 & 2 \\3 & 4 \\5 & 6 \end{bmatrix}^{\mathrm{T}}= \,\begin{bmatrix}1 & 3 & 5\\2 & 4 & 6 \end{bmatrix}$

To transpose a matrix press mode 6,1,2* then enter your values and press on, press shift 4,8 then shift 4,3 and then equals

for 3x2

Inverse of Matrix

$\begin{bmatrix} 1 & 2 \\3 & 4 \end{bmatrix}^{-1}=\begin{bmatrix} -2 & 1 \\1.5 & -0.5 \end{bmatrix}$

To find the inverse press mode 6,1,5* then enter your values and press on, then press shift 4,3, then press the $x^{-1}$ button underneath the mode, then press equals

for 2x2

Mode 7: TABLE

Mode 8: VECTOR

Throughout this section, I'll be using two 3D vectors, VctA and VctB - $\begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix}$ and $\begin{pmatrix} 6 \\ 8 \\ 10 \end{pmatrix}$ respectively.

This mode allows you to perform calculations on 3D and 2D vectors - up to three at a time. Upon selecting vector mode, you'll be prompted to choose a vector memory slot to enter (VctA, VctB, or VctC). After choosing which memory slot you're going to use, you will be prompted to choose the dimensions of the vector (either 2 or 3). Now enter the values of your vector. Rather than using one column and three rows to indicate i, j and k, the fx-991ES uses one row and three columns. Enter the i value (in the leftmost box), then press [=], which will prompt the calculator to scroll to the centre box, in which you enter the j value. Press [=] again, and it'll scroll to the right-hand box, in which you enter the k value. Press AC, and you'll return to a blank screen.

To enter data for another vector, press SHIFT/5 (VECTOR) (which we'll now call 'Vector Options') and choose option 2 (Data). This will return you to the screen you were presented with when you first entered vector mode. Proceed from there as before, choosing a different memory slot.

Adding Vectors

We are going to add together vectors A and B. So we should be looking for an answer of $\begin{pmatrix} 3 + 6 \\ 4+8 \\ 5+10 \end{pmatrix} - \begin{pmatrix} 9 \\ 12 \\ 15 \end{pmatrix}$ . At a blank screen (AC), go into Vector Options and choose VctA (option 3). This will cause "VctA" to show up on the main screen. Now press [+]. Go into Vector Options again, and choose VctB (option 4). Press [=]. This will take you to VctAns (a fourth memory slot, uneditable by the user) - which indeed shows $\begin{pmatrix} 9 \\ 12 \\ 15 \end{pmatrix}$ .

The Cross Product

This procedure is basically the same as that of adding vectors - only you press [ $\times$ ] rather than [+]. Performing this for $\mathbf{A} \times \mathbf{B}$ should yield $\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ .

The Dot Product

You might have been wondering what option 7 in Vector Options is for. It's for calculating the dot product of vectors.

$\mathbf{A} \bullet \mathbf{B} = \begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} \bullet \begin{pmatrix} 6 \\ 8 \\ 10 \end{pmatrix} = 18 + 32 + 50 = 100$ .

Select vector A, re-enter Vector Options, select Dot (option 7), go into Vector Options one more time and select vector B. Press [=]. You should see the following:

$\mathrm{VctA} \bullet \mathrm{VctB}$

$\, \, \, \, \, \, \, \, \, \, 100$

The Modulus of a Vector

$|\mathbf{A}| = \sqrt{3^2 + 4^2 + 5^2} = 5\sqrt{2} \approx 7.071$

Press SHIFT/hyp (Abs), go into Vector Options, choose your vector, and press [=].

Unit Conversion

Scientific Constants

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