Math examples
On this page, you will find TI-Nspire™ CAS usage examples from different areas of mathematics.Basic calculations
Basic calculations with TI-Nspire CAS
TI-Nspire CAS handles not only complex operations but also basic arithmetic, roots, powers, and logarithms—making it easy to check and explore everyday math.
Working with fractions
Work with fractions, convert between decimals and exact forms. Try propFrac and exact.
Simplifying expressions
TI-Nspire CAS simplifies algebraic expressions, handles absolute values, logarithms and roots—making it a powerful tool for both practice and verification.
Algebra
Solving equations
With the solve command, TI-Nspire CAS solves equations symbolically or numerically. Note the use of the | operator.
Logical operators
Logical operators (and, or, not, xor, nor, nand, ⇒, ⇔) are used to work with conditions and logical statements. Parentheses are important because they determine the order in which conditions are evaluated. The same notation can be used, for example, to simplify solution sets of inequalities.
Solving systems of equations
The guided tool for solving systems of equations is found under 3: Algebra > 7: Solve System of Equations.
You can add more equations later using SHIFT ENTER on a computer keyboard, or with the 
key on the calculator. The system template 
can also be typed manually: system(eq1, eq2).
Binomial expansion with slider
Explore the coefficients and terms of the binomial expansion by changing the exponent (n) with a slider. Set the slider step to 1 (integer) to see how the coefficients from Pascal's triangle are formed for each power.
Solving a quadratic equation step by step
With CAS, calculations can be done step by step, helping to develop your own thinking. You can use ans to refer to the previous equation or copy it. Make sure to enclose the entire equation in parentheses.
Sum, product and mean of quadratic roots
Statistics commands like sum, mean and product can be used with CAS too. Note how the mean of the roots matches the x-coordinate of the parabola's vertex.
Special Tables for Mathematical Use
Use the division table to perform polynomial division like on paper. New rows are added as you move forward with the arrow keys.
In the truth table, you’ll find quick shortcuts for common logic symbols. Rows and columns are automatically added when moving with the arrow keys. You can highlight values with color.
Geometry
Circle and sphere with CAS
You can calculate the area of a circle or the volume of a sphere using definite integrals. The solve command is used to isolate y from the equation of a circle.
Basic trigonometric functions
Trigonometric functions are entered by typing sin(, cos(, tan(, arcsin(, arccos( or arctan(. The functions are also available via the TRIG key on the calculator keypad. Calculations use the angle unit defined in the settings unless a different unit is specified in the input.
Angle settings
By default, TI-Nspire uses the angle unit defined in the document settings. If the input includes a unit, it is converted to the unit specified in the settings. On Windows, the ° symbol can be entered using CTRL + *, and Mac keyboards include a dedicated key. Units can also be entered using @d or @r.
See more
Computer view: The angle unit is shown at the bottom of the screen. Double-click Settings to open the document settings. Selecting Make Default applies the same setting to new documents. Selecting OK applies the settings only to the current document.
Handheld view: The angle unit can be changed directly by tapping the unit shown at the top of the screen (e.g. RAD).
Tip: The angle setting of a math box can also be changed per calculation. This is useful when the result is needed in different units. Unit conversion can also be performed using the commands @>DD and @>Rad.
Vectors
Entering and storing vectors
Vectors can be entered using the math templates or from the computer keyboard. A row vector is typed as [1,2 using commas, and a column vector as [1;2 using semicolons. The closing ] is added automatically. Store vectors with := like other expressions.
Vector calculations
Vector operations and commands work as shown in the example.
Vector perpendicularity and parallelism
Vector problems can be solved by combining the solve command with vector commands. The image shows examples of perpendicular, parallel, and same-direction vectors.
Three points on the same line
Whether three points lie on the same line can be examined using vectors. If vectors AB and AC are parallel, the points lie on the same line. Parallelism can be checked using the solve command or alternatively using the cross product.
Vectors and Vector Operations
You can quickly add vectors using the V shortcut key. Place a vector by dragging its endpoints with the mouse or by entering an expression like v=[3, 4] or v=3i+4j. Then define other vectors using expressions such as w=-v or w=2v. A unit vector is created by entering v^0.
Angle Between Vectors
The angle between vectors object calculates and illustrates the angle between two vectors. The vectors forming the angle are drawn with dashed lines, making the angle visible even if the vectors are located in different places in the diagram. Refer to vectors by their names.
Derivative
Differentiation
Derivatives can be calculated symbolically. Use the | operator to evaluate at a specific point. A guided option is available if needed.
Graphical exploration of the derivative using a tangent
A tangent can be added to the graph from the Geometry menu in the Graphs application by clicking on the desired point of the graph. The tangent can then be moved along the curve or positioned precisely using coordinates. This method is well suited for illustrating the relationship between the value of the derivative and the graph of the function.
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Finding the location of a function’s minimum and maximum
With the commands fMin and fMax, you can determine the location of a function’s minimum and maximum values. Extrema can be studied more comprehensively using the derivative. With the solve command, the zeros of the derivative function can be determined easily.
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Limits
Limits can be calculated using the 
math template. If + or - is entered in the direction field, it is interpreted as a one-sided limit.
Derivative using the limit definition
The derivative can be defined as the limit of the difference quotient. First the difference quotient is formed, and then its limit is evaluated.
Using CAS, the difference quotient can be calculated symbolically and the limit taken with . The result equals the derivative, which can also be verified directly using 
.
Exploring limits with a slider
Explore limits dynamically by changing the variable's value with a slider. To get more precise results near the critical point, set a small step size for the slider. You can modify the step size and other settings by right-clicking the slider and selecting 'Settings'.
Derivatives in Lists & Spreadsheet
In the Lists & Spreadsheet application, derivatives can be computed directly in table cells. A common use case is to define expressions in one column and compute their derivatives in another. Derivatives can also be entered directly using the math template. Higher-order derivatives can be computed directly using the 
math template.
Integral
Graphical illustration of the integral concept
The integral can be explored graphically by plotting a function and selecting the integral using a start and end point. By changing the coordinates of these points, you can illustrate how the value of the integral depends on the function and the chosen interval.
Integration
You can calculate definite and indefinite integrals. Add conditions using the | operator.
Circle and sphere with CAS
You can calculate the area of a circle or the volume of a sphere using definite integrals. The solve command is used to isolate y from the equation of a circle.
Integral as a sum
An approximate value of an integral can be calculated using a sum by dividing the interval into subintervals. The number of subintervals can be adjusted with a slider, improving the accuracy of the sum. As the number of subintervals increases without bound, the limit of the sum equals the exact value of the integral.
Area between curves
Using the Analyze menu tools, you can explore the area between curves graphically.
Area between curves using integrals
The area between curves can be calculated by integrating the absolute value of the difference between the functions. Integrals involving absolute values may lead to difficult calculations, so an exact result is not always obtained directly. The problem can then be divided into parts by determining the order of the functions.
Normal distribution
To calculate probabilities for a normal distribution, you can first define the probability density function using normPdf 5: Probability > 5: Distributions > 1: Normal Pdf..., and compute the result with an integral. The same result can be obtained more directly using the guided normCdf function 5: Probability > 5: Distributions > 2: Normal Cdf....
Visualizing Numerical Integration
A graphical visualization of numerical integration helps illustrate the concept both visually and numerically. You can type or paste a function directly from the clipboard. Increase the number of intervals by adjusting the value of n. Use the arrow keys to switch between calculation methods: left sum, midpoint sum, right sum, or trapezoidal method.
Visualizing Solids of Revolution
Write or paste one or two function expressions into the input fields. If two functions are entered, the region between them is shaded to represent the volume. This tool quickly generates visualizations while studying volume calculation of solids of revolution using integrals.
Trigonometry
Basic trigonometric functions
Trigonometric functions are entered by typing sin(, cos(, tan(, arcsin(, arccos( or arctan(. The functions are also available via the TRIG key on the calculator keypad. Calculations use the angle unit defined in the settings unless a different unit is specified in the input.
Angle settings
By default, TI-Nspire uses the angle unit defined in the document settings. If the input includes a unit, it is converted to the unit specified in the settings. On Windows, the ° symbol can be entered using CTRL + *, and Mac keyboards include a dedicated key. Units can also be entered using @d or @r.
See more
Computer view: The angle unit is shown at the bottom of the screen. Double-click Settings to open the document settings. Selecting Make Default applies the same setting to new documents. Selecting OK applies the settings only to the current document.
Handheld view: The angle unit can be changed directly by tapping the unit shown at the top of the screen (e.g. RAD).
Tip: The angle setting of a math box can also be changed per calculation. This is useful when the result is needed in different units. Unit conversion can also be performed using the commands @>DD and @>Rad.
Manipulating trigonometric expressions
In addition to the standard expand() and factor() commands, you can use tExpand() and tCollect() with trigonometric functions. These allow trigonometric expressions to be represented in alternative forms.
Solving trigonometric equations
Solving trigonometric equations can be done in the CAS environment using the solve() command. Solutions can be expressed in a general form using an integer parameter or restricted to a specific interval by adding a condition. If needed, the solutions can also be converted into a list for further processing.
Integer constant (@n1)
An integer constant can be used to test the equivalence of different representations and to work with trigonometric expressions in a general form. An integer constant is written in the form @n1, where the number is a freely chosen index. This makes it possible to distinguish between multiple integer constants when more than one is used in the same expression.
Derivatives in Lists & Spreadsheet
In the Lists & Spreadsheet application, derivatives can be computed directly in table cells. A common use case is to define expressions in one column and compute their derivatives in another. Derivatives can also be entered directly using the math template. Higher-order derivatives can be computed directly using the math template.
Roots
No content found for keyword: roots
Logarithm
Calculating logarithms
Logarithms can be entered as log(number, base) or using the logarithm template 
. If the base is omitted, the logarithm is interpreted as base 10. The natural logarithm is written with ln(). The number e (Euler's number) can be typed as @e, or selected from the 
menu or symbol palette.
Simplifying logarithmic expressions
Many logarithmic expressions can be simplified simply by pressing Enter. If the expressions include variables, it is often necessary to define a domain to allow simplification. You can also try using expand and factor. These tools can be helpful, but may sometimes lead to unwanted forms, such as factorizations of base numbers.
Domain of the logarithmic function
The command domain(expression, variable) shows the values of the variable for which the expression is defined.
Graphical connection between logarithmic and exponential functions
You can plot the graph of an exponential function and explore points. Add a point using the shortcut key P or from the geometry menu. Clicking a point on the graph allows it to move along the curve. You can examine coordinates more closely by double-clicking and entering for example log(3,2).
Change of base for logarithms
Use the 
logbase() or ln command to convert the base of a logarithm. These can be found under 3: Algebra > A. Convert Expression. The arrow can be typed as @>.
Probability
Factorial
Factorials are calculated by simply appending the ! symbol. You can also calculate the factorial of a list or use it in spreadsheets. In simple cases, expressions with factorials can be simplified using CAS.
Permutations
Permutations can be calculated using factorials or directly with the nPr command. You can find the command in the menu under 5: Probability > 2: Permutations. CAS commands can also be used in these calculations.
Combinations
Combinations can be calculated using factorials or directly with the nCr command. You can find the command in the menu under 5: Probability > 3: Combinations. CAS commands can also be used in these calculations.
Normal distribution
To calculate probabilities for a normal distribution, you can first define the probability density function using normPdf 5: Probability > 5: Distributions > 1: Normal Pdf..., and compute the result with an integral. The same result can be obtained more directly using the guided normCdf function 5: Probability > 5: Distributions > 2: Normal Cdf....
Equations related to the normal distribution
The guided normCdf command, found in the menu 5: Probability > 5: Distributions > 2: Normal Cdf..., allows you to calculate probabilities. Combined with solve, you can find missing values such as upper bound, mean, or standard deviation. Some equations require an initial guess to find a solution. Alternatively, you can use the density function and an integral.
Binomial distribution
The binomial probability function can be defined using the binomPdf command 5: Probability > 5: Distributions > A: Binomial Pdf.... To calculate the probability for a range of values, you can use a sum or the guided binomCdf command 5: Probability > 5: Distributions > B: Binomial Cdf..., which evaluates the sum for you.
Statistics
Statistical calculations in the Notes app
Data can be defined by storing it in a variable or by entering it into a table. After that, you can calculate summary statistics using the menu 6: Statistics > 3: List Math. Alternatively, you can calculate all summary statistics at once by selecting 6: Statistics > 1: Stat Calculations > 1: One-Variable Statistics.
Statistical calculations in a table
When the data has been entered into a table and the column has been named, one-variable statistics can be performed by selecting 6: Statistics > 1: Stat Calculations > 1: One-Variable Statistics.
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One-variable statistics with a frequency list
If the data is given as values with frequencies in a separate column, one-variable statistics can be computed by selecting 6: Statistics > 1: Stat Calculations > 1: One-Variable Statistics and setting the Frequency List to the frequency column.
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Calculating a frequency table
If a frequency list is not given, it can be created using the frequency command.
Number theory
Greatest Common Divisor and Least Common Multiple
The commands `gcd` and `lcm` are used to compute the greatest common divisor and the least common multiple of integers.
Factorization and primality
The factor command can be used to write an integer in factored form. The isPrime command can be used to check whether a number is prime.
Remainders and modular arithmetic
The remainder can be computed using the mod() command. The remain() command also computes a remainder; the difference appears for negative numbers. If the numtheory library is available, the pwrmod command can be used to compute remainders of large numbers that exceed the numeric limits.
Remainders in a table
A table can be used to examine number sequences and their remainders. The seq() command generates a sequence, and mod() can be applied to an entire column by entering the command in the formula bar.
Diophantine equation
There is no dedicated command for solving a Diophantine equation, but the integer n1 can be used to construct and verify a solution. The integer constant can be entered from the keyboard as @n1, where 1 is an index used to distinguish multiple integer constants.
Base conversions
In TI-Nspire, integers can be converted between different number systems, such as decimal, binary, and hexadecimal. Binary numbers use the prefix 0b and hexadecimal numbers use 0h. Without a prefix, a number is interpreted as decimal.
Conversions can be performed using Base commands or by typing, for example, 10@>Base2 from the keyboard.
If the document’s base setting is changed, results are shown by default in the selected base. Individual results can still be converted separately if needed.
Logic
Logical operators
Logical operators (and, or, not, xor, nor, nand, ⇒, ⇔) are used to work with conditions and logical statements. Parentheses are important because they determine the order in which conditions are evaluated. The same notation can be used, for example, to simplify solution sets of inequalities.
Logical simplifications and tautologies
Logical expressions can be simplified and tested for tautologies. Implication can be written as => and equivalence as <=>.
Special Tables for Mathematical Use
Use the division table to perform polynomial division like on paper. New rows are added as you move forward with the arrow keys.
In the truth table, you’ll find quick shortcuts for common logic symbols. Rows and columns are automatically added when moving with the arrow keys. You can highlight values with color.