# Calculation

To enter the Calculation application, highlight the app using the directional keys. Press the ok key to enter the application. This application allows you to perform a variety of computations involving complex numbers, matrices, trigonometry, combinatorics, derivatives, integrals, and unit conversions.

## Getting started

### Performing a calculation

Begin by entering the Calculation application. Do this by highlighting the app on the home screen using the arrow keys. Press the ok key to confirm.

1. When you enter the Calculation application, the cursor will begin in the editing bar at the bottom of the screen. This is where you will enter your expression.
2. Once you have entered the desired expression, press the exe key to receive your result.

The result of your most recent calculation will appear in the calculation history located just above the editing bar. The history lines include the calculation entered, which appears on the left, and the result, which appears on the right. You will receive both an exact result and a decimal result for your calculation when applicable. Exact results are displayed in black and the decimal results are displayed in gray.

You may navigate between elements of the calculation history lines by using the updown and leftright keys. Once an element is selected, it will be highlighted in gray. Selecting an element within a history line allows you to view all information relevant to your calculation. For example, calculations using decimal numbers will return only a decimal result. To view the exact result, select your desired history line and the exact result will appear.

To facilitate reading, the result of calculations with a decimal number is always given in decimal form: $0.1+0.3$ will return $0.4$ while $\frac{1}{10}+\frac{3}{10}$ will give $\frac{2}{5}$.

### Using the most recent result in a new calculation

The exact result of your most recent calculation may be used in the expression of a new calculation. To do this, press the ans key. The expression Ans will appear in the editing line. This expression represents the result of your most recent calculation. You may then perform calculations on this result.

You may also perform a single operation directly to your most recent result by pressing an operation key. For example, you can add 5 to your last calculation by pressing the plus key followed by the five key. The calculation editor will display Ans+5. Once executed, the value of your most recent result will replace Ans.

Because the NumWorks calculator only contains one minus key, pressing the minus key will display - to represent a negative value. Press the minus again to display Ans-.

### Using an element from the calculation history in a new calculation

Results and previous entries in the calculation history may be used in the expression of a new calculation. Use the updown and leftright keys to highlight your desired element. You may choose between the exact and approximate results. Once your element is highlighted, press the ok key to confirm. The element is then displayed in the editing bar at the bottom of the screen.

### Deleting line items in the calculation history

Line items in the calculation history may be deleted. To delete a line, use the updown keys to highlight an item from the line you wish to delete. Press del to delete the line.

To delete all lines of the calculation history, press shift followed by del.

## Performing calculations with complex numbers

### Choosing the complex format of the results

With the NumWorks calculator, you may calculate using both complex and real numbers. Your results may be displayed in three different forms:

1. Cartesian: the calculation of $\sqrt{-1}$ will give the result $i$
2. Polar: the calculation of $\sqrt{-1}$ will give the result $e^{1.570796*i}$
3. Real: the calculation of $\sqrt{-1}$ will give the result “unreal”

When the calculator is using polar form, the angle in the exponent is always given in radians, even if the calculator is set in degrees mode.

To choose the form:

1. Press home to begin on the home screen.
2. Highlight the Settings application with the directional keys.
3. Press the ok key to confirm.
4. Choose Complex format and press ok to confirm.
5. Select your desired form with the updown keys and confirm with ok

### Calculating values with complex numbers

The modulus, argument, real part, imaginary part and conjugate can be calculated while in the Calculation application. There are three ways to calculate these values:

1. While in the Calculation application, press the toolbox key to view the toolbox menu.
2. Use the updown keys to scroll down the menu.
3. Select Complex numbers.
4. Confirm with the ok key.
5. You may now select which value you would like to calculate.
• Second option: Manual entry

You may manually enter the functions used to calculate these values. Enter these syntaxes to calculate your desired value:

• Modulus : abs(z)
• Argument : arg(z)
• Real part : re(z)
• Imaginary part : im(z)
• Conjugate : conj(z)

1. Enter a complex number in the editing field.
2. Press the exe key.
3. Press the up key to highlight the most recent result.
4. Press the right key to highlight the three dots.
5. Press the ok key to open the additional results.
6. Use the directional keys to navigate through all possible additional results.

## Performing calculations with matrices

### Entering a matrix

When entering a matrix in the Calculation application, a template will appear in the editing bar. Grey boxes indicate a space that can be filled with a number. Yellow boxes indicate a space that must be filled to complete the matrix. There are two ways to enter a matrix:

1. While in the Calculation application, press the toolbox key to view the toolbox menu.
2. Use the updown keys to scroll down the menu.
3. Select Matrices and vectors.
4. Confirm with the ok key.
5. Select New matrix.
6. Confirm with the ok key. The matrix template will appear in the editing bar.
• Second option: Manual entry
1. Press the shift key.
2. Press the exp key. You can also press ln. The matrix template will appear in the editing bar.

### Storing a matrix

You may store a matrix and access it in the variables menu. This allows you to use the matrix in later calculations. To store a matrix:

2. Use the right key to navigate to the right of your matrix.
3. Press shift
4. Press pow. An arrow should appear in the editing bar.
5. Enter your desired letter or string. Press alpha followed by the letter(s).
6. Confirm by pressing exe

To use a capital letter, press shift then alpha. Your desired letter will now be entered as a capital letter.

### Performing calculations with matrices

You can perform calculations using more than one matrix:

• Addition of two matrices: M1+M2
• Subtraction of two matrices: M1-M2
• Multiplication of two matrices (matrix product) : M1*M2

You can also perform calculations with a number and a matrix:

• Multiplication of a matrix by a scalar: 4*M1
• Power of a matrix: M1^5
• Inverse of a matrix: M1^(-1)

### Operations and important values of matrices

The inverse, determinant, transpose, trace and size can be found for a matrix. There are three ways to determine these values:

1. While in the Calculation application, press the toolbox key to view the toolbox menu.
2. Use the updown keys to scroll down the menu.
3. Select Matrices and vectors.
4. Confirm with the ok key.
5. Select which option you would like to find. Additional options are found in the Matrices submenu.
• Second option: Manual entry

You may manually enter the functions used to calculate these values. Enter these syntaxes to calculate your desired value:

• Inverse : inverse(M)
• Determinant : det(M)
• Transpose : transpose(M)
• Trace : trace(M)
• Size : dim(M)
• Row echelon form: ref(M)
• Reduced row echelon form: rref(M)

1. Enter a matrix in the editing field.
2. Press the exe key.
3. Press the up key to highlight the most recent result.
4. Press the right key to highlight the three dots.
5. Press the ok key to open the additional results.
6. Use the directional keys to navigate through all possible additional results.

## Performing calculations with lists

### Creating a list

There are three ways to create a list of values:

1. While in the Calculation application, press the toolbox key to view the toolbox menu.
2. Use the updown keys to scroll down the menu.
3. Select Lists.
4. Select the New list option.
5. Enter the values of your list using the number keys.
6. Use the comma key to enter a comma between each value.

You can also generate a list using a function:

1. Within the Lists section of the Toolbox, select the {f(k)} option.
3. Enter your upper bound. For example k≤5 will create a list of 5 values using your function.
• Second option: Manual entry

1. While in the Calculation application, press the shift key followed by the log key to start your list.
2. Enter the values of your list using the number keys and a comma between each value.
3. End your list by pressing the shift key followed by the i key.
• Third option: Data tables

Lists are automatically created for the columns of the data tables in the Statistics (N1, V1, etc) and Regression (X1, Y1, etc) applications.

### Storing a list

You may store a list in the variables menu. This allows you to use the list in later calculations or within other applications. To store a list:

2. Use the right key to navigate to the right of your matrix.
3. Press shift
4. Press pow. An arrow should appear in the editing bar.
5. Enter your desired letter or string. Press alpha followed by the letter(s).
6. Confirm by pressing exe

To use a capital letter, press shift then alpha. Your desired letter will now be entered as a capital letter.

### Call an entry

You can call an element using parentheses. In all lists, the index of the first element is 1. To call the fourth element of the list L, you can therefore type L(4).

### Performing calculations

You can perform calculations using one or more lists:

Assume L is a stored list.

• Add a value to each entry of a list: L+2
• Subtract a value to each entry of a list: L-2
• Multiply each entry of a list by a value: L*2
• Divide each entry of a list by a value: L/2

Assume L and R are stored lists with the same number of elements.

• Addition of the corresponding elements of two lists: L+R
• Subtraction of the corresponding elements of two lists: L-R
• Multiplication of the corresponding elements of two lists: L*R
• Division of the corresponding elements of two lists: L/R

### Computing statistics

The following statistical values can be computed for a given list:

• Mean : mean(L)
• Standard deviation : stddev(L)
• Sample standard deviation: samplestddev(L)
• Median : med(L)
• Variance : var(L)

To compute a statistical value of a list:

1. Press the toolbox key to view the toolbox menu.
2. Select Lists.
3. Select Statistics.
4. Select the option for the statistical value you need.

### Performing operations

The following operations can be performed on a given list:

• List length : dim(L)
• Minimum : min(L)
• Maximum: max(L)
• Ascending sorting : sort(L)
• Sum of elements : sum(L)
• Product of elements : prod(L)

To perform an operation on a list:

1. Press the toolbox key to view the toolbox menu.
2. Select Lists.
3. Select Operations.
4. Select the operation you need.

## Performing calculations with units and constants

### Entering a calculation with units and constants

You may use units and constants in your calculations. Once you perform your calculation, the result will be returned along with the most suitable unit. There are two ways to enter units and constants:

1. While in the Calculation application, press the toolbox key to view the toolbox menu.
2. Use the updown keys to scroll down the menu.
3. Select Units and constants.
4. Confirm with the ok key.
5. Choose from multiple constants and types of measurement: length and angle, time and frequency, volume and area, mass, electricity, force and pressure, energy and power, temperature, other units and constants.
6. Once you have chosen your desired constant or unit of measurement, confirm with the ok key.
7. Your constant or unit of measurement will appear in the editing bar. Use the leftright keys to enter the desired numbers before your units, if needed.
• Second option: Manual entry
If you know your desired unit or constant’s abbreviation, you may choose to enter your calculation manually. To do this:
1. Press alpha
2. Enter your unit or constant abbreviation using the alpha keys.

To lock your keyboard in alphabetical mode, press alpha twice. This will allow you to enter letters without pressing alpha each time. Press alpha to unlock alpha mode.

### Converting units

To convert units, use the arrow on the keyboard accessible via shift and then pow. For example, to convert 185 minutes to hours, type: 185min→h.

You may also choose a→b in the toolbox menu.

## Displaying additional results on a calculation

When you complete a calculation, the NumWorks calculator searches for relevant additional results.

It is possible to navigate in the calculation history to show additional information on the results of particular calculations. If three dots appears on the right side of a calculation when an item is highlighted in the calculation history, additional results are available. Highlight the three dots and press the ok key to display the additional results.

### Additional results for integer solutions

When the result is an integer, the calculator gives the following additional results:

• scientific notation
• decimal form
• binary form
• prime factorization

### Additional results for solutions with fractions

When the result is a fraction, the calculator gives the following additional results:

• mixed fraction
• Euclidean division of the numerator by the denominator

### Additional results for basic functions

When the result is the output of a basic function, the calculator gives the following additional results:

• a graphical representation of the curve of the basic function
• the point on the curve
• the curve equation

### Additional results for calculations using trigonometric functions

When the result or the input involves a trigonometric funcion like sin(x) or arccos(x), the calculator gives the following additional results:

• a representation of the unit circle on a polar grid
• the angle in both radians and degress
• the value of cosine, sine and tangent evaluated at the angle

### Additional results for solutions with complex numbers

When the result is a complex number, the calculator gives the following additional results:

• a representation in the complex plane
• absolute value (modulus)
• argument
• real part
• imaginary part

When the result is a matrix, and if possible, the calculator gives the following additional results:

• the determinant
• the inverse
• the row echelon form
• the reduced row echelon form
• the trace

When the result is a vector, the calculator gives the following additional results:

• a graphical representation
• the magnitude
• the corresponding unit vector
• the angle with the positive x-axis

### Additional results for calculations with units

When the result contains units, the calculator gives the following additional results:

• results written as an imperial decomposition (for example, a result of 38in will return 1yd+2in in the additional results window).
• results in base SI units (for example, a result of 38in will return 0.9662m in the additional results window).
• a comparison with up to two reference values, one larger and one smaller than your result.