An electronic calculator with brand name CASIO fx-991MS is a non-programmable scientific calculator which is used to perform different calculations. This calculator is common due to its affordability and availability in the market. Most of these calculators have multi-line displays, which enable the user to see both the expression under operations and the result.

The REPLAY button, marked with directional arrows (↑ ↓ ← →), allows the user to scroll through previous calculations. It enhances usability by enabling editing of prior expressions without re-entering them from scratch.

• Some models include a secondary function labeled COPY which is activated using the SHIFT key.

Non-programmable calculators use preceding keys to access secondary or tertiary functions labeled in different colors.

• SHIFT key: Accesses yellow-labeled functions.
• ALPHA key: Accesses pink-labeled functions.

Usage steps:

1. Press SHIFT or ALPHA key.
2. Press the key corresponding to the color-coded label you wish to activate.

Exponentiation

To compute $x^n$:

1. Enter the base number $x$.
2. Press the exponentiation key ∧.
3. Enter the exponent $n$. Use parentheses for fractional exponents.
4. Press = to compute the result.

Xth root of a number

To compute $\sqrt[n]{x}$:

1. Enter the root index $n$ (e.g., 2 for square root, 3 for cube root).
2. Press SHIFT, then ∧ to input the root symbol $\sqrt[x]{\,}$.
3. Enter the number $x$.
4. Press = to compute.

Example

Calculate $\sqrt[3]{5243}$: Answer: 17.00974

The memory function enables temporary storage of values. The keys used include:

• STO (Store)
• RCL (Recall)
• M+ (Memory location)

To store a number

1. Enter the number.
2. Press SHIFT → STO → M+.

To recall a number

1. Press RCL → M+.
2. Press =.

Example

Evaluate: $\frac{200}{31.5 + 2.5}$ Stored sum: 34.0, Result: $\frac{200}{34.0} = 5.882352941$

Scientific notation expresses a number in the form $m \times 10^n$, where:

• $1 \leq m < 10$
• $n \in \mathbb{Z}$ (an integer)

To convert to scientific notation

1. Enter the number.
2. Press =.
3. Press ENG to switch to scientific notation.

Example

Convert 1245384791: Answer: $1.245384791 \times 10^9$

Fixing decimal places or significant figures

1. Press MODE five times.
2. Select:
   • Fix – for decimal places
   • Sci – for scientific notation
   • Norm – for normal format
3. Enter the number of digits (0–9).
4. Enter the number.
5. Press =.

Example

Convert 0.0024587 into scientific notation with 3 significant figures: Answer: $2.46 \times 10^{-3}$

i. Base 10 logarithms (common log)

1. Press log.
2. Enter the number.
3. Press =.

Example

Evaluate $\log_{10}(45000)$: Answer: 4.653212514

ii. Anti-logarithm (base 10)

1. Press SHIFT then log.
2. Enter the number.
3. Press =.

Example

Find antilog(0.1284): Answer: 1.344002263

iii. Natural logarithms (base $e$)

1. Press ln.
2. Enter the number.
3. Press =.

Example

Evaluate $\ln(628)$: Answer: 6.442540166

Note: $\ln(x) = \log_e(x)$, where $e \approx 2.718281828$

Changing base

Use the change of base formula:

Example

Evaluate $\log_5 6$:

Non-programmable scientific calculators are equipped with essential functions for processing statistical data. These capabilities allow for the efficient calculation of key statistical parameters, including:

1. The sum of the data set: $\sum x$
2. The mean of the data: $\bar{x}$
3. The sum of squares: $\sum x^2$
4. The standard deviation: $\delta$

The steps below outline how to access and compute these values using your calculator's built-in features.

To compute statistical values from a set of numerical data, follow these general steps:

1. Press the SHIFT key to access secondary functions.
2. Press the key labeled 1 or 2 to access the statistics menu.
3. Select the required statistical parameter (such as $\sum x$, $\bar{x}$, or $\delta$) as prompted on the calculator screen.
4. Press the = key to display the result of the selected computation.

These steps allow the user to compute sum, mean, and standard deviation for both ungrouped and grouped data sets.

For computations involving ungrouped statistical data—where individual data points are not categorized into frequency groups—use the following procedure:

1. Press the MODE key twice to enter the mode selection menu.
2. Select the "SD" (Standard Deviation) mode by pressing the key labeled 1, as directed on the screen.
3. The calculator will now enter statistical mode. You may begin entering your data.
4. Type the first numerical value, then press the M+ key to store it in memory.
5. Repeat step 4 for each subsequent value in the data set. Each entry is stored and cumulatively processed.
6. Once all values are entered, proceed with steps outlined in the "Accessing Statistical Calculations" section above to view the desired results such as $\sum x$, $\bar{x}$, $\sum x^2$, and $\delta$.

Notes

• Always ensure that the calculator is reset or cleared of previous data before starting a new data set input.
• To delete a single entry, use the DEL or ← key before confirming it with M+.
• Refer to your calculator's manual for the exact button layout, as it may vary slightly across models.

This method ensures accurate computation of essential statistical measures for basic data analysis tasks in examinations or academic exercises using a non-programmable scientific calculator.

Example 1.14 Given data (ungrouped)

The following data set is provided:

Data: 3, 4, 6, 2, 8, 7, 5, 9, 1

Required calculations

1. Mean ($\bar{x}$)
2. Standard deviation ($\delta$)
3. Sum of the data ($\sum x$)

Solution

Using the required steps on a non-programmable scientific calculator, the results are:

1. Mean: $\bar{x} = 5$
2. Standard deviation: $\delta = 2.581988897$
3. Sum of the data: $\sum x = 45$

To calculate statistical parameters for grouped data using a non-programmable scientific calculator, follow these steps:

1. Press the MODE key twice.
2. Select the SD mode by pressing 1.
3. Enter the class mark value $x$.
4. Press the SHIFT key followed by the comma , key to enter the separator ";".
5. Enter the corresponding frequency $f$ of the class mark.
6. Press the M+ key to store the data point.
7. Repeat steps 3 to 6 for all class marks and their frequencies.
8. Press SHIFT then press either 1 or 2 to select the desired statistical value.
9. Press = to display the result.

Example

Given the following frequency distribution table, calculate the mean and standard deviation.

Solution

1. Mean: $\bar{x} = 36.92857143$
2. Standard deviation: $\delta = 17.67045268$

Note: Once statistical data are stored in the calculator, they cannot be cleared by switching the calculator off or pressing ON or AC keys. The data can only be erased by either changing the mode or following the steps below:

Steps to clear statistical data

1. Press the SHIFT key.
2. Press the MODE key.
3. Press the 3 key (for clearing statistical data).
4. Press the = key twice to confirm.

A non-programmable scientific calculator can also be used to perform various probability calculations.

To calculate the number of combinations of $n$ items taken $r$ at a time, follow these steps:

1. Type the value of $n$.
2. Press the SHIFT key.
3. Press the ÷ key to access the combination function $nCr$.
4. Type the value of $r$.
5. Press the = key to obtain the result.

To calculate the number of permutations of $n$ items taken $r$ at a time:

1. Type the value of $n$.
2. Press the SHIFT key.
3. Press the × key to access the permutation function $nPr$.
4. Type the value of $r$.
5. Press the = key to get the result.

To find the factorial of a number $x$, denoted $x!$, use these steps:

1. Enter the value of $x$.
2. Press the SHIFT key.
3. Press the key labeled 1/x (or the key that accesses factorial function !).

Example

Calculate the following using a non-programmable scientific calculator:

1. ${}^6C_3$
2. ${}^7C_2$
3. ${}^{15}P_5$
4. ${}^{8}P_2$
5. $8! + 2 \times (2!)$

Solutions

1. ${}^6C_3 = 20$
2. ${}^7C_2 = 21$
3. ${}^{15}P_5 = 360360$
4. ${}^{8}P_2 = 56$
5. $8! + 2 \times (2!) = 40324$

You can calculate probabilities related to the normal distribution using your calculator by following these steps:

1. Press the MODE key twice.
2. Select SD mode by pressing 1.
3. Press the SHIFT key followed by 3 to access the DISTR menu.
4. Select the appropriate probability function:
   • P (press 1): $P(z \leq a)$ — cumulative probability to the left of $a$.
   • Q (press 2): $P(a \leq z \leq b)$ — probability between two z-values.
   • R (press 3): $P(z \geq a)$ — cumulative probability to the right of $a$.
5. Enter the z-score(s) as requested.
6. Press = to get the result.

Examples

• For $P(z \leq -1.83)$, the probability is approximately $0.03362$.
• For $P(z \geq 1.2)$, the probability is approximately $0.11507$.
• For $P(0 \leq z \leq 1.5)$, use the Q function, giving approximately $0.43319$.

Note: These functions allow you to find probabilities from the standard normal distribution quickly and accurately using your calculator.

A non-programmable scientific calculator can solve systems of simultaneous linear equations with either two or three unknowns.

Steps for solving simultaneous equations

1. Press the MODE key three times until the option EQN (equation) appears on the display.
2. Select EQN by pressing its corresponding number key.
3. Choose the number of unknowns: press 2 for two unknowns or 3 for three unknowns.
4. The calculator will prompt you to enter coefficients for each unknown starting with $a_1$.
5. Input each coefficient and press = after each entry.
6. After entering all coefficients and constants, press = to get the values of the unknowns.

General forms of simultaneous equations

For two unknowns $x$ and $y$, the system is:

For three unknowns $x$, $y$, and $z$, the system is:

Non-programmable scientific calculators can also solve quadratic and cubic equations efficiently.

Steps for solving quadratic and cubic equations

1. Press the MODE button three times until EQN is displayed.
2. Select EQN, then use the right arrow (▶) on the REPLAY key to select the degree of the equation:
   • Press 2 for quadratic equations
   • Press 3 for cubic equations
3. Arrange the equation in standard form:
   • Quadratic: $ax^2 + bx + c = 0$
   • Cubic: $a x^3 + b x^2 + c x + d = 0$
4. Input the coefficients starting with $a$ and press = after each one.
5. After entering the last coefficient, the calculator will display the first root $x_1$.
6. Press = repeatedly to view the other roots.

Activity

Try solving the following equations on your non-programmable scientific calculator:

1. $2x^2 + 3x - 4 = 0$
2. $x^3 + 6x^2 + 6x + 0 = 0$

Write down the steps you followed and share your results for further discussion.

To perform matrix operations, you must first store the matrix in your calculator:

1. Press the MODE key three times until MAT (Matrix Mode) appears. Select it.
2. Press SHIFT followed by 4 (MAT).
3. Choose matrix name: 1 for MatA, 2 for MatB, or 3 for MatC.
4. Set the matrix dimensions:
   • Enter the number of rows (m) and press =.
   • Enter the number of columns (n) and press =.

After naming and sizing the matrix:

1. Enter the value for each element of the matrix one-by-one, pressing = after each.
2. Entries are labeled as $A_{ij}$, where $i$ is the row number and $j$ is the column number.

Example: 2×2 Matrix A

Example: 3×3 Matrix A

To edit existing matrix values:

1. Press SHIFT then 4 (MAT).
2. Select Edit.
3. Choose the matrix name (e.g., MatA, MatB, MatC).
4. Enter new values for the matrix elements, pressing = after each.

1. Press SHIFT then 4 (MAT).
2. Use the right arrow to select Det and press 1.
3. Again, press SHIFT + 4, then select Mat and press 3 to access the stored matrix (e.g., MatA).
4. Press = to get the determinant.

Follow the same steps as for the determinant but choose Trn (Transpose) instead of Det.

To find the inverse of a square matrix:

1. Press SHIFT then 4 (MAT).
2. Select Mat and choose the matrix (e.g., MatA).
3. Once MatA appears, press x⁻¹ to compute the inverse.
4. Press = to see the first value of the inverse matrix. Use the arrow key to scroll through the rest.

The ∫dx key on the calculator is used to evaluate definite integrals of the form:

Steps

1. Press the ∫dx key.
2. Press the ALPHA key before each variable to insert the function $f(x)$.
3. Insert a comma ,, then type the lower limit a.
4. Insert another comma, then type the upper limit b.
5. Close the bracket ) if required, then press = to get the result.

Example

Evaluate $\int_1^3 (2x^2 + 1) \, dx$ Input: ∫(ALPHA x × ALPHA x × 2 + 1, 1, 3) → =

The d/dx function (accessed using SHIFT + ∫dx) is used to evaluate the derivative of a function at a specific point.

Steps

1. Press SHIFT, then press ∫dx to access d/dx.
2. Press ALPHA before each variable to insert the function $f(x)$.
3. Insert a comma ,, then type the value of $x = a$.
4. Close the bracket ) if required, then press = to display the result.

Example

Evaluate $\frac{d}{dx}(3x^2 + 2x)$ at $x = 2$ Input: d/dx(ALPHA x × ALPHA x × 3 + ALPHA x × 2, 2) → =

A non-programmable scientific calculator can perform metric unit conversions based on pre-programmed constants and conversion types. These types are listed in the calculator's manual or inside the plastic cover of the device.

1. Enter the number you want to convert.
2. Press the SHIFT key.
3. Press the CONST key (usually labeled as CONV or CONST).
4. Press the number key that corresponds to the desired conversion (refer to calculator's conversion list).
5. Press the = key to obtain the converted result.

i. Identify unit conversion types

Check the inside cover or manual of your calculator to view supported conversions such as:

• Length (e.g., ft to inches, m to km)
• Area (e.g., m² to acres)
• Speed (e.g., km/h to m/s)
• Energy (e.g., cal to J)

ii. Convert the following

(i) 10 ft to inches: Input: 10 → SHIFT → CONST → → =

(ii) 4 m² to acres: Input: 4 → SHIFT → CONST → → =

(iii) 80 km/h to m/s: Input: 80 → SHIFT → CONST → → =

(iv) 60 cal to joules (J): Input: 60 → SHIFT → CONST → → =

iii. Steps recap

1. Input the number to convert.
2. Press SHIFT + CONST.
3. Select the appropriate conversion number from your calculator's list.
4. Press = to view the result.

Many non-programmable scientific calculators provide direct access to commonly used physical constants, such as acceleration due to gravity (g), Planck's constant (h), Avogadro's number (NA), π (pi), and e (Euler's number).

Constants are usually stored in the calculator and can be accessed by using their assigned number. These numbers are listed in the calculator manual or on the inside of the calculator's plastic cover.

Example: Accessing gravity (g)

To recall the acceleration due to gravity (g), which may be assigned 35:

1. Press the CONST key (sometimes labeled SHIFT + 7).
2. Type the number 35.
3. Press the = key to view the constant's value (e.g., 9.80665).

• π (Pi) – Use the π key.
• e (Euler's number) – Usually accessed with SHIFT + ln key.
• g (gravity) – Usually constant #35.
• Speed of light (c), Planck's constant (h), etc. – Refer to the calculator manual.

i. Identify constants

Refer to the calculator's inner cover or manual. Note down constants and their corresponding numbers (e.g., g = 35, h = 36, NA = 07).

ii. Perform operations

(a) Add Two Constants: Press CONST → 35 → = + CONST → 36 → =

(b) Subtract Two Constants: Press CONST → 36 → = - CONST → 35 → =

(c) Multiply Two Constants: Press CONST → 07 → = × CONST → 36 → =

(d) Divide Two Constants: Press CONST → 36 → = ÷ CONST → 07 → =

iii. Steps recap

1. Press CONST or SHIFT + CONST.
2. Enter the number of the constant.
3. Press = to insert the constant value.
4. Perform arithmetic operations as needed.

A non-programmable scientific calculator can store vectors and perform various operations such as vector addition, subtraction, modulus (magnitude), dot product, and cross product. Below are the step-by-step instructions on how to use these features.

1. Press MODE three times until VCT (Vector Mode) appears, then press 3.
2. Press SHIFT → 5.
3. Press 1 to set dimension.
4. Select vector name: 1 for A, 2 for B, 3 for C.
5. Enter dimension: 2 for 2D or 3 for 3D vectors.
6. Enter components: For A1, A2, A3 (if 3D) — input each and press =.

1. Press SHIFT → 5.
2. Select Edit by pressing its number.
3. Select the vector name (A, B, or C).
4. Update each component and press =.

a. Vector addition and subtraction

1. Press SHIFT → 5, select Vct.
2. Select the first vector (e.g., A).
3. Press + or -.
4. Repeat SHIFT → 5 → Vct for the second vector.
5. Press = to get the result.

b. Modulus (magnitude) of a vector

1. Press SHIFT → ) to get Abs.
2. Press SHIFT → 5 → Vct.
3. Select the desired vector.
4. Press = to get the result.

c. Dot product

1. Press SHIFT → 5 → Vct.
2. Select the first vector.
3. Use the replay arrow to select Dot by pressing 1.
4. Press SHIFT → 5 → Vct.
5. Select the second vector.
6. Press = to get the result.

d. Cross product

1. Press SHIFT → 5 → Vct.
2. Select the first vector.
3. Press × key.
4. Repeat to select the second vector.
5. Press = to get the result.

1. If a = 2i + 3j + 4k and b = -i + j + k, use your calculator to compute:

1. a + b
2. b - a

2. Given OA = 4i - 3k and OB = -2i + 4j + k, evaluate:

1. OA ⋅ OB (dot product)
2. OA × OB (cross product)

3. Use the calculator to determine the vector 2a + 3b - c for any given vectors a, b, and c.

A complex number has the form a + bi, where $a$ is the real part and $b$ is the imaginary part. Scientific calculators support various operations with complex numbers including addition, subtraction, multiplication, division, modulus, conjugate, and argument.

1. Press MODE → select CMPLX mode (press 2).
2. Type the first complex number: (a ± bi) using ENG for i.
3. Press + or -.
4. Type the second complex number similarly.
5. Press = to get the real part of the result.
6. Press SHIFT + = to get the imaginary part.

1. Activate CMPLX mode via MODE → 2.
2. Input the first complex number in brackets.
3. Use × or ÷.
4. Input the second complex number.
5. Press = for the real part and SHIFT + = for the imaginary part.

1. Activate CMPLX mode.
2. Press SHIFT + , to get Conjg.
3. Input the complex number in brackets.
4. Press = for the real part, then SHIFT + = for the imaginary part.

1. Enable CMPLX mode.
2. Press SHIFT + ) to get Abs.
3. Input the complex number inside brackets.
4. Press = to see the modulus.

1. Activate CMPLX mode.
2. Press SHIFT + ( to get arg.
3. Insert the complex number inside brackets.
4. Press = to view the argument.

a. Given

• z₁ = 2 + 3i
• z₂ = 4 − i
• z₃ = −1 − 2i

Use your calculator to evaluate:

1. z₁ + z₂ − z₃
2. z₁ × z₂
3. z₁ / z₃
4. (6z₁ − 4z₂ + z₃) / z₁

b. Given

• z₁ = i
• z₂ = −3
• z₃ = 2 + 4i

Use your calculator to compute:

1. z₁ × z₂
2. arg(z₂ / z₁)
3. Conjg(z₁ × z₂) / z₃

A non-programmable scientific calculator displays specific error messages to indicate mistakes in input or computational limitations. When an error occurs, you can use the replay button to navigate and correct your entry. In some cases, pressing the AC (All Clear) button and restarting may be faster.

i. Math error

This error occurs when the calculation is mathematically undefined or exceeds the calculator's numerical limits. For example: 5 ÷ 0 will result in Math ERROR, because division by zero is undefined.

ii. Syntax error

A syntax error occurs when the input format is incorrect or incomplete. For example: tan 45° entered as 45 tan (without correct formatting) causes a Syntax ERROR.

iii. Dimension error

This occurs when matrix or vector dimensions are not compatible for the operation. For example, trying to multiply a 3 × 1 matrix with a 3 × 3 matrix in the wrong order will display: Dim ERROR, because matrix multiplication requires that the number of columns in the first matrix equals the number of rows in the second.

iv. Stack error

This happens when the calculator's memory (stack) is overloaded with too many nested operations or values. For example, evaluating a very complex matrix expression like: T(14BC + 6CB) using large matrices B and C, may result in a Stack ERROR if the computation exceeds internal stack limits.

To fix errors, review your inputs, use the cursor to adjust where needed, or reset the entire entry with AC.