# TI-Lists/Printable version

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# TI-86

{TOCright}} This page is to identify and detail the differences between the TI-83 and the TI-86 graphing calculators. The sections below will detail the differences in keyboard layout, differences in keystrokes, and differences in functionality.

## MenusEdit

### Edit

The LIST menu is the 2nd function on the subtraction key.

### Edit

The Stat menu is the 2nd function on the addition key.

### Edit

The List Edit menu can be accessed by two different routes. From the List menu it is at F4. From the Stat menu it is at F2. The lists originally displayed are the ones that are used for the statistical calculations: xStat, yStat, and fStat.

## Key locationsEdit

### Curly BracketsEdit

The curly brackets are in the LIST menu.

### CommaEdit

The comma key is left of the 4 key.

## FunctionalityEdit

### Saving values in a variableEdit

Pressing the STO> key on the TI-86 puts the calculator into Caps Lock mode. Since most variable names are made up of letters, this is considered handy by most TI-86 users.

### List variable namesEdit

Any variable name can be used for a list. Most people avoid single letter names, for they are mostly used for numeric varables. Press the STO> key and type in the list name.

## LinksEdit

# Creating

Lists can be created, stored and accessed in several ways. On the Home Screen, simple list operations can be performed. The editor in the STAT menu is used for more comprehensive needs.

A list must have at least one element in it. Trying to work with a list with no elements, will cause an error message.

### On the home screenEdit

To enter a list of real numbers on the home screen, start with a curly bracket: **{**. Follow this with the numbers in the list separated by commas. End the list with a closing curly bracket: **}**.

- The open-curly is the 2nd function on the open-parenthesis key, just above the 8 key. The comma key is just above the 7 key. The close-curly is on the close-parenthesis key, just above the 9 key. (TI-86)

Pressing ENTER after creating a list in this way, displays it without commas.

- {3,5,8,1}
- {3 5 8 1}

If the list is to long to display in one line, the first part is shown followed by an ellipsis (...). At this point, more of the list can be viewed by pressing the Right and Left arrow keys to move it in either direction. If something is entered on the next line, the list will no longer move.

To store a list from the home screen, use the STO key, either on the same line as creating the list or the next line. Choose a list variable L1 through L6. (2nd function on the 1 through 6 keys.) (TI-86)

- {3,5,8,1,4} STO> L1
- {3 5 8 1 4}

- {-6,0} STO> L2
- {-6 0}

To access a list after it has been stored, use the same list variable key. (Lists can be stored using other single letter variable names on the home screen, but access them is more complicated and will be covered later.)

- L1
- {3 5 8 1 4}

### In the STAT EditorEdit

**STAT:1** is used to enter a list in this editor. (Press the STAT key. It is just below the DEL key. Press ENTER or 1, to access the editor.) The first three L-variables are displayed with their values, if any. (TI-86) Use the Left and Right arrows to access more lists. Enter numbers into the list, pressing ENTER or the Down arrow, after each one.

After creating a list, you can move to the right to access another one.

The STAT Editor can also be used to modify lists.

- To delete a number from a list, move down to highlight it, and press
**DEL**. - To change a number in a list, highlight the old value, and enter the new value. When you hit ENTER or Down, it is replaced.
- To insert a new place in a list, highlight the first number that will follow the new one, and press
**2nd:INS(DEL)**. A zero value will be inserted and highlighted, which can then immediately be replaced by typing in the new number.

It is often handy to view long lists in this editor. Seven values are displayed in each of three lists, and as you scroll down all the values can be seen.

### With arithmetic or functionsEdit

In the next sections we cover lists in arithmetic operations, lists in arithmetic functions and list functions. Each time a list is used in an arithmetic calculation a new list is created. Each time a list is used in an arithmetic function a new list is created. Some of the list functions create an answer that is a list. These new lists can be stored in a list variable or used in another arithmetic or list operation.

### LinksEdit

- Next: Arithmetic

- Back to: TI-Lists

# Arithmetic

Arithmetic can be done with Lists in a variety of ways. A single item in a list can be used in an arithmetic expression. The operations that are normally used for single numbers can work on lists of numbers. And there are a variety of functions that do list operations on lists.

### TI arithmeticEdit

After an arithmetic expression is computed, the result is displayed, and this result is available in the Ans variable. For normal arithmetic, the result is a number. If the expression is followed by a STO> command, this number is stored in the stated variable, it is displayed, and it is also stored in Ans.

- 5 + 7
- 12

- Ans + 1
- 13

- 4 STO> X
- 4

- 2 + 8
- 10

- X + Ans
- 14

If the result of an expression is a list, the entire list is displayed and the entire list is available in the Ans variable.

- {2,5,3}
- {2 5 3}

- Ans STO> L1
- {2 5 3}

- L1
- {2 5 3}

### Single Items in a ListEdit

To access or change a single item from a list, use the list variable followed by the item number in parentheses.

- {2,7,4,9} STO> L1
- {2 7 4 9}

- L1(3)
- 4

- 8 STO> L2(3)
- 8

- L2
- {2 7 8 9}

One way to put another item on the end of the list is to store the value in the next item number.

- L2
- {2 7 8 9}

- 6 STO> L2(5)
- 6

- L2
- {2 7 8 9 6}

- L2(1) + L2(4) STO> L2(6)
- 11

- L2
- {2 7 8 9 6 11}

### Arithmetic with a List and a numberEdit

The operations that use two real numbers can also be done on a number and a list. The number operates on each of the items in the list.

Addition

- 5 + {1,2,3}
- {6 7 8}

Subtraction

- 5 - {1,2,3}
- {4 3 2}

If L1 is {7,8,9}

- L1 - 5
- {2 3 4}

Multiplication

- {1,2,3} * 5
- {5 10 15}

Division

- {6,8,10} / 4
- {1.5 2 2.5}

- 60 / {3,4,5}
- {20 15 10}

- {10,12,15} / 6 >Frac
- {5/3 2 5/2}

Exponents

- L1 ^ 2
- {49 64 81}

- 3 ^ {2,3,5,6}
- {9 27 243 729}

Roots (**MATH 5**)

- 3 {27,64,1000}
- {3 4 10}

- {3,4,6} 4096
- {16 8 4}

Set the decimal places to 2 (**MODE down right right right**)

- 12 {2,6,16}
- {1.06 1.16 1.26}

### Arithmetic with 2 ListsEdit

All the arithmetic operations can be used with two lists. When two or more lists are used in an expression, each list must have the same number of values, or an error will occur. Each of the numbers in one list operates on the corresponding number in the other list. The result is a list of the same length.

Addition

- {3,5,6} + {6,2,1}
- {9 7 7}

Subtraction

- {3,5,6} - {6,2,1}
- {-3 3 5}

Multiplication

- {3,5,6} * {6,2,1}
- {18 10 6}

Division

- {3,5,6} / {6,2,1}
- {.5 2.5 6}

Exponents

- {3,5,6} ^ {6,2,1}
- {729 25 6}

Roots

- {3,5,6} {6,2,1}
- {1.82 1.15 1.00}

Multiple operations

- {5,4}({2,5}+{3,9})
- {25 56}

### Arithmetic Functions with ListsEdit

All of the functions that work on real numbers can also work on a list of numbers. The result will be a list of the same length. In these examples the decimal place has been set to 2.

Square root

- ({2,3,4})
- {1.41 1.73 2.00}

Logarithms

- log({2,3,4})
- {.30 .48 .60}

- ln({2,3,4})
- {.69 1.10 1.39}

Exponential

- e^({2,3,4})
- {7.39 20.09 54.60}

Trigonometric (in radians)

- sin({2,3,4})
- {.91 .14 -.76}

- tan
^{-1}({2,3,4})- {1.11 1.25 1.33}

Miscellaneous

- abs({2,-2})
- {2 2}

- round({π,e^(1),9/7},2)
- {3.14 2.72 1.29}

The functions that normally need two numbers can work on two lists or a list and a number. The length of the result will match the length of the list used.

- min({4,3},{5,1})
- {4 1}

- gcd({32,27},36)
- {4 9}

- 6 nCr {2,3,4}
- {15 20 15}

Each list created as the result of an operation is available using the previous Ans.

- 2{5,8} STO L1
- {10 16}

- 3{4,2}
- {12 6}

- Ans+L1
- {22 22}

### LinksEdit

- Previous: Creating Lists

- Next: List Operation Functions

- Back to: TI-Lists

# List Operation Functions

There are several kinds of functions that use or create lists. These functions are found in the LIST menus. The LIST menus are accessed by the 2nd function of the STAT key. It is just below the DEL key. Move over to the right one or two times to access the List menus.

There are several kinds of functions that use or create lists. These functions are found in the LIST menus. The LIST menus are accessed by the 2nd function of the STAT key. It is just below the DEL key. The functions that create new lists or change existing ones are found in the LIST OPS menu.

## Functions that change lists.Edit

The functions in this section change existing lists.

### dim(Edit

The list dimension function (**2nd LIST OPS 3**) has several uses. It will tell you the length of a list (how many elements are in the list.)

- {1,3,5,7,9,11} STO> L1
- {1 3 5 7 9 11}

- dim(L1)
- 6

- dim({1,5})
- 2

The result of a dim( expression is a single number.

By storing a number in the dim( of a list you can create a new list with a certain number of elements or change the length of an existing list.

Creating a new list, fills it with elements that are all zero.

- 3 STO> dim(L1)
- 3

- L1
- {0 0 0}

A list can be made shorter thus deleting elements on the end, or longer thus adding elements with zero value onto the end.

- {2,4,6,8,10} STO> L1
- {2 4 6 8 10}

- 3 STO> dim(L1)
- 3

- L1
- {2 4 6}

- 6 STO> dim(L1)
- 6

- L1
- {2 4 6 0 0 0}

### SortingEdit

The sorting functions **SortA(** and **SortD(** are the first functions in the List Ops menu: (**2nd LIST OPS 1**) and (**2nd LIST OPS 2**). These functions sort the elements of the specified list. The first does an ascending sort (smallest item first); the second does a descending sort (largest item first).

These functions will not work on lists given with curly brackets. The parameter in the parentheses must be a list name. This is because the calculator operates on this list in memory as it is rearranged. The previous order of items is no longer available. Sorting does not create a new list.

- {3,6,2,4} STO> L1
- {3 6 2 4}

- SortA(L1)
- Done

- L1
- {2 3 4 6}

A decreasing sort works in a similar way.

- {6,8,2,9,4} STO> L2
- {6 8 2 9 4}

- SortD(L2)
- Done

- L2
- {9 8 6 4 2}

### Fill(Edit

This function (**2nd LIST OPS 4**) changes the specified list variable by replacing each element with the same value.

This function will not work on a list given with curly brackets. The parameter in the parentheses must be a list name. The previous items are no longer available. Filling does not create a new list.

- {2,7,4,5} STO> L1
- {2 7 4 5}

- Fill(-3,L1)
- Done

- L1
- {-3 -3 -3 -3}

You can start with all zeros.

- 5 STO> dim(L3)
- 5

- L3
- {0 0 0 0 0}

- Fill(4,L3)
- Done

- L3
- {4 4 4 4 4}

## Functions that create new listsEdit

The functions in this section create a new list. The new list is available in the Answer variable. If the list will be used later it should be stored in a list variable.

### seq(Edit

The sequence function (**2nd LIST OPS 5**) creates a list based on an expression and a range of values for the variable in the expression.

The first example makes a list based on 3x + 5, where x takes on the values from 1 through 6.

- seq(2X+3,X,1,6)
- {5 7 9 11 13 15}

Notice that the first parameter is the expression. The second parameter is the variable that assumes the range of values. The third and fourth parameters are the beginning and ending values for the range. The numbers in the range can be incremented by an amount other than 1, by using an optional fifth parameter.

There are several ways to create the same list as in the first example. Since the expression is linear, the range can be used to produce it.

- seq(N,N,5,15,2)
- {5 7 9 11 13 15}

The expression can contain any mathematical calculation, and the parameters for the range can be negative or decimals.

- seq(round(sin(θ),2),θ,-0.7,π/8,0.3)
- {-.64 -.39 -.1 .2}

Since this function creates a new list, if you want do something else with it, you must either use **Ans** or **STO>** it to a list variable.

We can do the first example again in several steps.

- seq(J,J,1,6)
- {1 2 3 4 5 6}

- Ans * 2
- {2 4 6 8 10 12}

- Ans + 3
- {5 7 9 11 13 15}

Here's another example that stores the non-rounded list:

- seq(1/R,R,1,6) STO> L1
- (1 .5 .33333333...

- round(L1,2)
- {1 .5 .33 .25 .2 .17}

### cumSum(Edit

The cumulative sum function (**2nd LIST OPS 6**) adds up the items in the list and creates a new list with the partial sums. The first item in the new list is the same as the first item in the original list. The second item in the new list is the sum of the first two items in the original list. The third item in the new list is the sum of the first three items in the original list. And so on. The last item in the new list is the sum of the entire original list.

- cumSum({1,2,3,4,5,6})
- {1 3 6 10 15 21}

- cumSum(Ans) STO> L1
- {1 4 10 20 35 56}

Once again, if the new list is to be used rather than just observed, it must be stored, or used immediately from the Ans variable.

### ΔList(Edit

The list difference function (**2nd LIST OPS 7**) subtracts adjacent list items and forms a list of the differences. This new list will be one item shorter than the original.

- L1
- {1 4 10 20 35 56}

- ΔList(L1)
- {3 6 10 15 21}

- ΔList(Ans)
- {3 4 5 6}

- ΔList(Ans)
- {1 1 1}

- ΔList(Ans)
- {0 0}

Using this function on a list with only one element will result in an error.

### augment(Edit

The list connecting function (**2nd LIST OPS 9**) has two list parameters. The result is a list with the two lists connected together.

- {1,2 3} STO> L1
- {1 2 3}

- {6,7,8,9} STO> L2
- {6 7 8 9}

- augment(L1,L2)
- {1 2 3 6 7 8 9)

- augment(L2,L1)
- {6 7 8 9 1 2 3}

- augment({10,14,8},L1)
- {10 14 8 1 2 3}

- augment(Ans,{5,1,7}) STO> L3
- {10 14 8 1 2 3 5 1 7}

The new list must be stored if it is to be used later.

- SortA(L3)
- Done

- L3
- {1 1 2 3 5 7 8 10 14}

## LinksEdit

- Previous: Arithmetic

- Back to: TI-Lists

# Mathematical List Functions

There are several kinds of functions that use or create lists. These functions are found in the LIST menus. The LIST menus are accessed by the 2nd function of the STAT key. It is just below the DEL key. There are a variety of mathematical operations that can be done on lists. These are found in the LIST MATH menu. These functions result in a single number answer.

### min( and max(Edit

The minimum function (**2nd LIST MATH 1**) results in the smallest number in the list. The maximum function (**2nd LIST MATH 2**) gives the largest number in the list.

- {4,7,2,5} STO> L1
- {4 7 2 5}

- min(L1)
- 2

- max(L1)
- 7

- min({-3,-8,-1,-4})
- -8

- max({-3,-8,-1,-4})
- -1

### mean(Edit

The averaging function (**2nd LIST MATH 3**) calculates the arithmetic mean of the numbers in the list by adding them up and dividing by the number of items added.

- {2,3,5,6,14} STO> L1
- {2 3 5 6 14}

- mean(L1)
- 6

- mean({40,23,64,14,51})
- 38.4

### median(Edit

The median function (**2nd LIST MATH 4**) finds an average in a different way from the mean. If the list of numbers were listed in order from smallest to largest, the median is the middle number.

- {2,3,5,6,14} STO> L1
- {2 3 5 6 14}

- median(L1)
- 5

- median({40,23,64,14,51})
- 40

If there are an even number of numbers, there is no single middle number. The median in this case if the mean of the two middle numbers. (Add them and divide by 2.)

- median({1.3,1.8,1.1,1.6})
- 1.45

The result, 1.45, is half way between 1.3 and 1.6.

### sum( and prod(Edit

The sum function (**2nd LIST MATH 5**) adds up all the items in a list. The product function (**2nd LIST MATH 6**) multiplies all the items in a list together.

- sum({3,5,6})
- 14

- prod({3,5,6})
- 90

- sum({-4,-2,5,10})
- 9

- prod({-4,-2,5,10})
- 400

- prod({4,3,2,1,0,-1})
- 0

### stdDev( and variance(Edit

The standard deviation function (**2nd LIST MATH 7**) and the variance function (**2nd LIST MATH 8**) give the results of their statistical calulations.

- stdDev({4,5,6,8,11})
- 2.77

- variance({4,5,6,8,11})
- 7.7

## LinksEdit

- Previous: List Operation Functions

- Next: Graphing with Lists

- Back to: TI-Lists

# Graphing

Lists can enhance the graphing process.

### LinksEdit

- Previous: Mathematical List Functions

- Next: Programming with Lists

- Back to: TI-Lists

# Programming

Programs can be written using Lists to build variety of structures.

### LinksEdit

- Previous: Graphing with Lists

- Back to: TI-Lists

# Applications

This section contains several examples of mathematics problems that can be solved using lists. These applications come for a variety of levels of math. Some of these problems can be solved in other ways on the calculator.

### Reducing fractionsEdit

Let's reduce to lowest terms by seeing which prime numbers divide into both the numerator and denominator. In a list we can divide tham both at the same time.

Enter the numerator and denominator of the fraction as a list

- {29302,75803}
- {29302 75803}

Divide by 2.

- Ans / 2
- {14651 37901.5}

As we might have guessed the top comes out even, but the bottom does not. So, multiply by 2 to get the original fraction back.

- Ans * 2
- {29302 75803}

Let's try dividing by 3.

- Ans / 3
- {9767.333 25267.667}

Neither one works. Multiply the 3 back in.

- Ans * 3
- {29302 75803}

Looking at the ones digit tells us not to bother with 5. Seven is the next prime.

- Ans / 7
- {4186 10829}

Both number came out even so the original fraction reduces by 7. Now we must see if there is another factor of 7 in both numbers. We can hit enter again to do the same operation again: dividing the previous answer by 7.

- Enter
- {598 1547}

Seven is a factor of both numbers again; what about a third time?

- Enter
- {85.429 221}

The denominator is dividible by 7 a third time, but not the numerator. We must put the 7 back in.

- Ans * 7
- {598 1547}

Next try 11.

- Ans / 11
- {54.364 140.636}

No good.

- Ans * 11
- {598 1547}

13 is next.

- Ans / 13
- {46 119}

They both reduce by 13.

Now since 13 time 13 is more than 100 and thus more than 46, if 13 divided into 46 evenly the answer would be smaller than 13. But we tried all the primes smaller than 13. This fraction is in lowest terms.

### Checking for prime factorsEdit

We can check any three digit number to see if is a prime by dividing it by all the primes less that the square root of 1000. Create a list in L1 for these primes.

- {2,3,5,7,11,13,17,19,23,29,31} Sto> L1

It might be easier to create this list in the Editor.

Is 997 a prime number? Divide 997 by all those primes.

- 997 / L1 Sto→ L2
- {498.5 332.333...

Since this contains lots of numbers with ten decimal digits, look at L2 in the list editor, and scroll down it. We see that none of the divisions come out as even integers, so 997 is prime.

Try 991

- 991 / L1 Sto→ L2

Can you see that 991 is also a prime?

Try 989.

- 989 / L1 Sto→ L2

Can you see in the editor that the factors of 989 are 23 × 43?

### Graphing families of curvesEdit

A list of numbers can be used to display several related graphs.

For example put 0, 1, 2, and 3 into a list. We will use this list of numbers in several ways.

- {0,1,2,3,} STO> L1
- {0 1 2 3}

- Start with a set of linear graphs.

Press Y= and enter

- Y1=L1*X+1
- ZOOM 6(ZStandard)

Notice that the lines have different slopes. Each graph is steeper that the previous one. But the y-intercept is 1 in each case.

- Next look at a family of quadratic curves.

Change to

- Y1=1+L1*X+0.25X^2
- GRAPH

This time each of these parabolas have the same shape, due to the second degree term, and the same y-intercept, due to the constant (1). The list draws each graph in a different place on the axes.

- How will changing the coefficient on an exponential function affect the graph?

- Y1= L1*e^(X/4)
- GRAPH

Why do you only see three graphs? The list has affected the y-intercept. What does multiplying by zero do to the rest of the formula?

- And now for some trigonometry.

Hit the MODE key and move down and over to Degree; hit ENTER.

- Y=L1*3*sin(36X)
- GRAPH

Once again there are only three visible graphs. (I guess zero wasn't the best number to put in our list.) The sine waves are similar with different amplitudes.

### Solving systems of linear equationsEdit

One good way of solving a system of equations is called the elimination method. The plan is to multiply each equation by a different number. The multipliers are chosen so that one of the variables in each equation has the same coefficient, except for opposite sign. This allows us to add the equations together and eliminate that variable with opposite cofficients.

Example

- 8x + 7y = 35
- 6x + 5y = 27

Put the three numbers in each equation into a list.

- {8,7,35} STO> L1
- {8 7 35}

- {6,5,27} STO> L2
- {6 5 27}

35 is the smallest number that the y-coefficients are both multiples of; and 24 is the smallest common multiple of 8 and 6. Let's start with 24. We need a positive 24x in one equation and a negative 24x in the other.

We can multiply the first equation by 3:

- 3 * L1 STO> L3
- {24 21 105}

and the second equation by -4:

- -4 * L2 STO> L4
- {-24 -20 -108}

Add the two new lists.

- L3 + L4
- {0 1 -3}

This list represents 0x + 1y = -3 or y = -3.

Next let's fix the y coeficients to 35 and -35

- 5 * L1 STO> L3
- {40 35 175}

- -7 * L2 STO> L4
- {-42 -35 -189}

And add

- L3 + L4
- {-2 0 -14}

Which represents -2x = -14, so divide the list by -2

- Ans / -2
- {1 0 7}

Solution: x = 7, y = -3

### Calculating both solutions to quadratic equationsEdit

This process is begun by having a list containing a positive one and a negative 1.

- {1,-1} STO> L6
- {1 -1}

When this list is multiplied by a number, the result is a list with plus and minus the number.

- 1.8 * L6
- {1.8 -1.8}

Next, since the answers to a quadratic equation are often irrational, they need to be rounded off. So set the calculator to 3 decimal digits.

Press the MODE Key and move down and over to the 3 and press ENTER.

Example - Solve x2 + 5x + 3 = 0

Using the quadratic formula gives us:

Start with the calculation inside the square root:

- 5 ^ 2 - 4 * 1 *3
- 13.000

Next do the square root and get both positive and negative answers:

- L6 * √ (Ans)
- {3.606 -3.606}

Add -5

- Ans - 5
- {-1.394 -8.606}

And divide by 2

- Ans / 2
- {-.697 -4.303}

That list contains both of the answers.