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 verylong 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 cos(x)=0 and you press solve, the screen you then see is x= 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 80 as your value your answer will be 90, then when you solve it again and enter 250 as your value your answer will be 270


This allows you to calculate the sum of a series between two terms. 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

In complex mode, the calculator is able to perform all expected calculations involving complex numbers. The ENG button (left of the open brackets button) becomes i. It is therefore possible to enter complex numbers in the form a + bi. Another option is to enter complex numbers in the modulus-argument form. This can be done by first entering the modulus, then pressing Shift and the (-) button (\angle), then entering the argument of the complex number in the correct format (degrees/radians/grads). It is possible to convert between the two forms from the complex menu (Shift-2). This menu also allows you to find the complex conjugate of the complex number, as well as the argument. To find the modulus of a complex number, use the Abs function.

Mode 3: STAT

Statistics mode allows to input data in multiple forms and will give you the procedures required to calculate many values from the data sets (eg. standard deviation).

You can enter data with only one variable or two. After entering data, it is possible to carry out several different functions including variance and regulation calculations (which can be useful for A-level maths).

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 portions of chips for £2.80 A woman buys 1 fish and 4 portions 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:


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 4,3 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

The table mode allows the generation of a table of numbers based on a function of X. This is a quick method of calculating several values for a function of X, as well as helping sketch graphs. Upon entering table mode, you'll be prompted to enter a function of X. Enter the value of X at which the table should begin, end, and the difference between each value and the table will be generated.

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

In contrast to convention the dot product has higher precedence than the vector product so you will sometimes need brackets around a vector product expression.


 |\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

The Casio fx-991ES can convert between different units of measurement, by first inputting the value to be converted, then pressing Shift and 8. A full list of conversions is given on the case of the calculator.

Scientific Constants

By pressing Shift then 7 in any mode except Base-N, it is possible to recall one of 40 constants stored on the calculator by inputting a number from 01 to 40. Although none are necessary for A level exams (they'll be given to you), the following may be helpful (though be aware that examinations may expect you to use specified rounded values):

  • 01 - Mass of proton
  • 02 - Mass of neutron
  • 03 - Mass of electron
  • 06 - Planck's constant
  • 17 - Atomic mass unit
  • 24 - Avogadro's constant
  • 27 - Molar gas constant
  • 28 - Speed of light
  • 35 - Gravitational field strength
  • 38 - Difference between Kelvin and Celsius
  • 39 - Gravitational Constant
  • 40 - Atmospheric pressure