Casio fx-991MS Tips and Tricks

Casio fx-991MS displaying “133Tne55!”

Casio fx-991MS is a scientific calculator with many powerful features, some of which can greatly reduce work, and some of which can be abused in interesting ways. Its labour-saving features come in handy on tests an exams ‒ and since it’s not a graphing calculator, it is allowed on most tests and exams.

Manuals

These are the official manuals that come with the calculator, in PDF format. Please consult the manuals if you are not familiar with the functions.

Source: http://world.casio.com/calc/download/en/manual/

Standard tricks

Solving polynomials and linear systems

To solve a quadratic or cubic polynomial equation:

  1. Go to EQN mode and scroll over to the Degree? section (MODE MODE MODE 1 →)
  2. Enter the degree (2 or 3)
  3. Enter the 3 or 4 (real) polynomial coefficients, from highest degree downward.
  4. Scroll through the solution set (which may have complex values).

To solve a system of linear equations of 2 or 3 variables:

  1. Go to EQN mode and stay at the Unknowns? section (MODE MODE MODE 1)
  2. Enter the number of variables, which is also the number of equations (2 or 3)
  3. Enter the 6 or 12 coefficients.
  4. Scroll through the solution vector.

Polynomials equations and linear systems come up quite often on tests and contests in mathematics, physics, and chemistry, which makes this feature quite useful.

Solving arbitrary equations

To solve an arbitrary equations of one variable:

  1. Enter the equation on the formula line. (e.g., 3X−8=5)
  2. Press SOLVE (SHIFT CALC).
  3. Give an initial guess for the variable and press the equals button. Try to give a value near a solution, or else solving will be slow or will fail. However, this step is generally not necessary for linear equations due to the nature of Newton’s method.
  4. Press SOLVE again (SHIFT CALC).
  5. Read the result. (e.g., X=4.333333333)

As mentioned in the manual, the calculator internally uses Newton’s method (but with numerical differentiation) to solve equations. The algorithm can easily hang, fail, or give a wrong answer, so beware – it is not the solution to all your problems.

Although linear equations are simple to solve in theory, letting the calculator solve it for you can still save some algebraic manipulation. Here’s an example problem: The 3 angles in a triangle are A, B, and C. B is twice of A. C is triple of B. Find the value of B. The solution: Solve the equation B÷2+B+3B=180.

Tabling functions

It is possible to evaluate an arbitrary function (of one or more variables) at different arguments without re-entering the function’s definition every single time. Procedure:

  1. Enter the expression defining the function. (e.g. X2)
  2. Press CALC.
  3. For each variable in the expression, give it a value.
  4. Read the value of the evaluated expression.
  5. Go to step 2 (not step 1) to evaluate the function at another argument.

This is a tremendous aid when applying the rational root theorem to factor or solve polynomials.

Random number generation

The Ran# variable (SHIFT .) gives a uniformly distributed random number in the range [0, 1) with a step size of 0.001. It takes on a random value for each instance in a formula and for each evaluation of a formula. (e.g. Ran#+Ran# has a different distribution than 2Ran#.)

If you want an integer in the range [0, N), evaluate N×Ran# and mentally discard the fractional part of the result. This can help with pesky questionnaires and surveys that have too many or too personal questions, although you may end up answering some basic facts wrong (e.g. gender, age).

For a finer step size, use Ran#+0.001Ran# (step size 1×10−6) or Ran#+0.001Ran#+0.000001Ran# (step size 1×10−9).

Degree-minute-second calculations

Except for certain of division calculations, decimal fractions are far easier to work with than the sexagesimal (base-60) system we’ve inherited from the Babylonians. But for those times where you need to work in sexagesimal or convert between it and decimal, this calculator supports degree-minute-second input and conversion to and from decimal format. See the manual for details, or just play around with the °′″ button.

This feature is obviously useful for dealing with angles expressed in DMS notation, but I think it’s not well known that it helps time calculations as well. Don’t you remember that there are 60 minutes in an hour and 60 seconds in a minute? This has the same subdivisions as the DMS scheme. Example applications:

Numerical differentiation

The numerical differentiation operation (SHIFT ∫dx) takes 2 or 3 arguments:

  1. The function of X to differentiate
  2. The point where the derivative is evaluated at
  3. The change in X (optional)

For example: d/dx(X^X,0.5) = 0.216976666. (The actual value is about 0.216977710.)

Numerical integration

The numerical integration operation (∫dx) takes 3 or 4 arguments:

  1. The function of X to integrate
  2. The lower bound
  3. The upper bound
  4. The amount of partitioning (optional)

If the amount of partitioning is n, then the number of partitions is 2n. See User’s Guide 2 for more details.

Example of usage and amount of partitioning:

The actual value is ln 2, or approximately 0.693147181.

Note: Most functions take much longer to integrate and yield much more pathological results than this example. On this calculator, it’s dangerous to rely on numerical integration.

Display and key test

Press and hold SHIFT and 7, then ON to get the display and key test. Pressing ON at any time exits the test. (Pressing 7 is not necessary on some older models; activating this test by accident happens sometimes.)

In the display test, press SHIFT to step through the sequence:

  1. All segments on
  2. All segments off
  3. Half of segments on
  4. The other half of segments on
  5. Lots of copies of a single digit, from 0 to 9

The last display test shows 999999999999; pressing SHIFT starts the key test. From left to right, top to bottom, press every key in sequence. If the number displayed increments, you’ve pressed the right key; otherwise, try another key. Do not press ON unless you intend to quit!

Iterating formulas

A major difference between this calculator and classic scientific calculators is that it does not evaluate the expression while you input it; it evaluates it all at once when you are done.

The most recent expression evaluated can be re-evaluated by pressing the equals key (=). The result of each evaluation is always saved in the answer variable (Ans). For single-statement iterations, it is most convenient to use Ans as the iterated variable. For example, Ans+1 is functionally equivalent but easier to type than X=X+1.

Multi-statement iterations are possible: use colon (ALPHA ∫dx) to separate the statements. Press the equals key once per statement, which evaluates them in sequence. When all the statements in the line have been evaluated, pressing the equals key will go back to evaluating the first statement. For example, A=A+2:B=B−3 is a multi-statement line.

Remember to give initial values to the variables used in the iterations. Also remember that for most formulas, you iterate them as many times as you need until you’re bored or satisfied.

Simple iterations

Arithmetic progression

For example, starting at 0 and counting up by 1:

  1. Initialize: Evaluate 0 (sets Ans to 0).
  2. Iterate: Evaluate Ans+1.
Geometric progression

For example, starting at 1 and doubling:

  1. Initialize: Evaluate 1 (sets Ans to 1).
  2. Iterate 2Ans.
Iterated squaring

For example, starting at 1.000000001 and squaring:

  1. Initialize: Evaluate 1.000000001 (sets Ans to 1.000000001).
  2. Iterate Ans2.

(It’ll take 38 iterations to overflow.)

Logistic map

For example, the chaos at r = 4:

  1. Initialize: Evaluate 0.2 (sets Ans to 0.2).
  2. Iterate 4Ans(1−Ans) and watch the randomness.

Newton’s method iteration

To solve an equation of the form f(x) = 0:

  1. Set Ans to an initial value close to a root (solution).
  2. Iterate the expression Ans − f(Ans)/f′(Ans), where f′ is the derivative of f.

For example, to solve x2−3 = 0:

  1. Initialize: Evaluate 1000(Ran#−0.5).
  2. Iterate Ans−(Ans2−3)/(2Ans).

After a number of iterations, the result should converge to 1.732050808 or −1.732050808, depending on the initial value.

Fixed point iteration

It is possible to solve equations of the form f(x) = x by simply iterating x′ = f(x), where x′ means the next value of x.

Examples:

Fixed point iteration is generally not as fast or as reliable as Newton’s method, though.

Taylor series iteration

Examples:

Exponential function
  1. Initialize X with an argument of your choice.
  2. Initialize: Y=1, A=0, C=0
  3. Iterate: A=A+Y : Y=YX÷(C+1) : C=C+1
  4. Read the answer from A.
Cosine function
  1. Initialize X with an argument of your choice.
  2. Initialize: Y=1, A=0, C=0
  3. Iterate: A=A+Y×(−1)^(C÷2) : Y=YX2÷(C+1)÷(C+2) : C=C+2
  4. Read the answer from A.
Sine function
  1. Initialize X with an argument of your choice.
  2. Initialize: Y=X, A=0, C=1
  3. Iterate: A=A+Y×(−1)^((C−1)÷2) : Y=YX2÷(C+1)÷(C+2) : C=C+2
  4. Read the answer from A.

Fibonacci sequence iteration

Procedure:

  1. Initialize: Evaluate A=0 and B=1.
  2. Iterate C=A+B : A=B : B=C.

Greatest common divisor iteration

This is based on the original Euclidean algorithm, which uses repeated subtraction rather than the modulus (remainder). The procedure:

  1. Set the mode to complex numbers (MODE 2).
  2. Initialize: Set A and B to be the natural numbers whose GCD will be computed.
  3. Iterate: A = A − B (tanh(20A−20B) + Abs tanh(20A−20B))÷2 : B = B − A (tanh(20B−20A) + Abs tanh(20B−20A))÷2 (Quite a mouthful, isn’t it?)
  4. When A and B converge to the same number, that is the GCD answer.

      5. Note that the function f(x) = (tanh(20x) − |tanh(20x)|) / 2 hackily emulates a step function, with f(x) = 0 for x ≤ 0 and f(x) = 1 for x ≥ 1. Also, on this calculator, the absolute value function is only available in the complex numbers mode.

      Thanks to Bojan Petrovic for suggesting this trick!

      Questionable tricks

      Storing 80 numbers

      In the SD mode, you can store a sequence of up to 80 numbers, which persist even after a power cycle. Enter a number by entering its literal or expression and then pressing DT (M+). Scroll through the sequence of stored numbers by pressing up and down (↑, ↓).

      Any information can be stored as numbers, and digital computers is living proof of this fact. If you want to store text on the calculator, for example, just come up with a coding scheme to convert between letters and numbers. (Note that each number can hold up to about 40.7 bits of information.)

      l33tsp34k

      The calculator contains a palette of symbols, which can be used to spell out words and phrases. The complete (Latin) alphabet cannot be spelled out, though.

      Letter Character Meaning Key sequence
      AAVariable AALPHA (−)
      4Digit 44
      αFine structure constantCONST 10
      BBVariable BALPHA °′″
      8Digit 88
      CCVariable CALPHA hyp
      CCombinationsSHIFT +
      DDVariable DALPHA sin
      EEVariable EALPHA cos
      EE notation separatorEXP
      eExponential functionSHIFT ln
      eElementary chargeCONST 23
      eEuler’s numberALPHA ln
      3Digit 33
      FFVariable FALPHA tan
      FFaraday constantCONST 22
      fSI prefix femtoSHIFT 1
      GGSI prefix gigaSHIFT 8
      GNewtonian constant of gravitationCONST 39
      gStandard acceleration of gravityCONST 35
      HhPlanck constantCONST 06
      IiImaginary unitENG (only in complex mode)
      1Digit 11
      KkSI prefix kiloSHIFT 6
      kBoltzmann constantCONST 25
      L1Digit 11
      MMVariable MALPHA M+
      MSI prefix megaSHIFT 7
      mSI prefix milliSHIFT 5
      NnSI prefix nanoSHIFT 3
      O0Digit 00
      PPPermutationsSHIFT ×
      pSI prefix picoSHIFT 2
      RRMolar gas constantCONST 27
      S5Digit 55
      TTSI prefix teraSHIFT 9
      tCelsius temperatureCONST 38
      7Digit 77
      UuAtomic mass unitCONST 17
      XXVariable XALPHA )
      ×Times×
      YYVariable YALPHA ,

      Discovering the punctuation that can be entered is left as an exercise for the reader.

      The following are the letters that cannot be represented: J, Q, V, W, Z.

      Wish list

      The Casio fx-991MS calculator does not have these features, but it would be nice if it did:

      • Big integers
      • Modular arithmetic operations (e.g. mod, pow mod, reciprocal mod)
      • Number theory functions (e.g. GCD, primality testing)
      • Conditional execution and loops
      • More memory for formulas

      Links

      Last modified: 2008-07-15-Tue
      Created: 2007-09-22-Sat