Casio CFX-9970G User manual

Algebraic Expressions
The ALGBR Mode (Algebraic Mode) provides tools for expansion of
algebraic expressions, factoring, etc. In this mode, differential and
integration calculation results are displayed as mathematical
expressions instead of decimal values.
20-1 Before Using the Algebraic Mode
20-2 Inputting and Executing Calculations
20-3 ALGBR Mode Commands
20-4 Signum Function
20-5 Natural Display Notation
20-6 ALGBR Mode Error Messages
20-7 ALGBR Mode Precautions
Chapter
20

350
20-1 Before Using the Algebraic Mode
In the Main Menu, select the ALGBR icon to enter the ALGBR Mode and display
its initial screen, which contains the following items.
•{expn} ... {expansion}
•{fctor} ... {factorization}
•{diff} ... {differential}
•{
∫
(} ... {integration}
•{SOLV} ... {Solve function}
•{tExp} ... {expression transformation using the addition theorem}
•{tColl} ... {product-to-sum transformation using the addition theorem}
•{comb} ... {combination}
•{PTS
' } ... {function for line passing through specific points}
•{CPLX} ... {complex function transformation}
•{appr} ... {convert to numeric value}
•{collc} ... {collection}
•{tanL} ... {tangent expression}
The following table shows the keys that can be used in the ALGBR Mode.
• The Mkey performs its screen shot send function only. It does not perform its
fraction-decimal conversion function.

351
20-2 Inputting and Executing Calculations
The ALGBR Mode display is divided into three areas: an input area, a solution
area, and a message area (used for display of menus and error messages).
Example X + X2 + 3X – 2X2
v+vx+dv-cvx
w
• Solutions are displayed in natural display notation.
• Solutions produced in the ALGBR Mode are also stored inAns memory and
can be recalled by pressing !K.
• You can input up to 255 bytes of data for eachALGBR Mode calculation.
• If a solution does not fit within the solution area, use f, c, d, and eto
scroll the screen.
• Inputting more data while there is data in the input area and solution area
causes the previous data to be cleared from two areas automatically.
• If you clear the display by pressing A, you can recall the previous operation
by pressing dor e(Replay Function).
• The Angle item of the set up screen can be set to either "Deg" or "Rad" for
ALGBR Mode operations.
• The Display item of the set up screen can be set to "Fix", "Sci", or "Norm" for
ALGBR Mode operations. Note, however, that this setting is applied for the
approx command only.
• A displayed solution can be stored in function memory by pressing
K6(FMEM)1(STO). Next, press a function 1(f1) to 6(f6) to select a
specific function memory.
Input area
Solution area
Messages area
P.361
P.356

352
20-3 ALGBR Mode Commands
In theALGBR Mode, results are calculated in accordance with commands and
expressions you input. This section describes each of the commands available in
the ALGBR Mode.
kk
kk
kConventions Used in this Section
The following conventions are used in the command descriptions of this section.
Item Description
<expression> This item indicates a mathematical expression input by you.
The actual expression you should input depends on the type of
operation you are performing. One example of an expression
is: X+1.
<variable> This item indicates a variable input by you. The actual variable
you should input depends on the type of operation you are
performing. One example of a variable is: A.
[ ] Anything enclosed within square brackets is optional, which
means you can skip it if you want. Note the following:
expand (<expression>[)]
The above example means that the final closed parenthesis to
the right of <expression> does not need to be input for the
command to execute properly.
kk
kk
kCommands
uu
uu
uExpansion —— (expn)
This command expands an expression.
Syntax: expand (<expression>[)]
Example To expand the expression (X + 2)2
1(expn)(v+c)xw X2+ 4X + 4
uu
uu
uFactorization —— (fctor)
This command factorizes an expression.
Syntax: factor (<expression>[)]

353
Example To factorize the expression X2 - 4X + 4
2(fctor)vx-ev
+ew(X – 2)2
• You can also factorize a value into its prime factors.
Example To factorize 64 into its prime factors
2(fctor)gew26
uu
uu
uAddition Theorems —— (tExp)
This command uses trigonometric addition theorems to transform an expression.
Syntax: tExpand (<expression>[)]
Example To transform sin(A+B) using addition theorems
6(g)1(tExp)
s(aA+aBwcos(B) • sin(A) + sin(B) • cos(A)
uu
uu
uProduct-to-Sum Transformation —— (tColl)
This command uses addition theorems to perform product-to-sum transformation.
Syntax: tCollect (<expression>[)]
Example To perform product-to-sum transformation on sin(A)cos(B)
using addition theorems
6(g)2(tColl)
saAcaBw
ALGBR Mode Commands 20 - 3
sin(A + B) sin(A – B)
+
2 2

354
uu
uu
uIntegration —— ( ()
This command can be used to determine the primitive function or calculate the
definite integral for an expression.
Syntax 1: ∫(<expression>, <variable> [, <integration constant>] [)]
Syntax 2: ∫(<expression>[, <variable>, <integration constant>] [)]
Syntax 3: ∫(<expression>, <variable> , <start>, <end> [)]
<integration constant>...........Integration constant
<start> ............ Start point of the integration interval
<end> ............. End point of the integration interval
• A default variable of X is used when specification of a variable is skipped in
Syntax 2.
• Syntax 3 calculates the definite interval in accordance with the specified
integration interval.
• Multiple integral calculations can also be performed.
Example To integrate the expression X2for variable X
4(∫( )vx,vw
• A default value of 0 is automatically assumed for the integration constant.
Inputting a symbol name such as C for the integration constant produces a
result in a form that is the same as the indefinite integral.
uu
uu
uDifferential —— (diff)
This command can be used to determine the derivative or calculate the value of
the derivative for an expression.
Syntax 1: diff (<expression>, <variable>, <nth>[, <differential coefficient>] [)]
Syntax 2: diff (<expression>, <variable>[, <nth>, <differential coefficient>] [)]
Syntax 3: diff (<expression>[, <variable>, <nth>, <differential coefficient>] [)]
<nth> .............. Specifies differential of nth order. nmust be a positive integer.
<differential coefficient>
....................... Any value specified as the differential coefficient is substituted
in the function for calculation of the result.
20- 3 ALGBR Mode Commands
X3
3

355
• Syntax 1 determines the derivative in accordance with a specified expression,
variable and order. Specifying a differential coefficient calculates a result in
accordance with the input value.
• A default order of 1 is used when specification of the order is skipped in Syntax
2.
• A default variable of X is used when specification of a variable is skipped in
Syntax 3.
Example To differentiate the expression X6for variable X
3(diff)vMgw6X5
uu
uu
uTangent Expression —— (tanL)
This command calculates the tangent expression of another expression.
Syntax: tanLine (<expression>, <variable>, <contact point>[)]
<contact point>
....................... The contact point is specified using the <variable> name.
Example To calculate the tangent expression when X = 2 for the
expression X3
6(g)6(g)3(tanL)
vMd,v,cw12X – 16
uu
uu
uSolve —— (solve)
This command calculate solutions for an expression. Solutions are displayed as
mathematical expressions.
Syntax: solve (<expression> [ =<expression>] [, <variable>] [)]
The second expression can be preceded by any of the following operators: =
(equals), < (less than), > (greater than), H(less than or equal to), or I(greater
than or equal to).
• A default variable of X is used when specification of a variable is skipped.
ALGBR Mode Commands 20 - 3

356
Example To solve AX+B = 0 for X
5(SOLV)1(solve)aA
v+aB,vw
• Other solve functions are available that produce numeric calculation results .
uu
uu
uConvert to Numeric Value —— (appr)
This command converts an expression to a numeric value.
Syntax: approx <expression>
Example To convert the expression 2 to a numeric value
6(g)6(g)1(appr)
!9cw1.414213562
• Any command to the left of approx causes an error.
Example 1+ approx ( 2 ) (Causes an error.)
• Inputting anotherALGBR Mode command or the signum( function into the
approx command causes an error.
Example approx approx 2 (Causes an error.)
kk
kk
kDifference Between "approx" and Standard Calculations
approx differs from standard calculations (calculations that do not use natural
display notation) in the number of display digits and handling of variables. With
standard calculations, calculation results are displayed without using exponential
notation.
Example jMcaw12157665459056928801
When part of the expression includes a variable, the variable is processed as a
variable regardless of whether or not it has been assigned a value.
Example f*aA+dw5A + 3
20- 3 ALGBR Mode Commands
{–B }
X =
A
P.107

357
With approx, calculation results are displayed using exponential notation. As with
the RUN Mode, the mantissa can have up to 10 digits and the exponent up to two
digits. The number of digits that can be input for approx depends on the setting of
the set up screen's Display item.
Example 6(gg
gg
g)6(gg
gg
g)1(appr)jMcaw(Display: Norm1)
1.215766546E + 19
When part of the expression includes a variable, the calculation is performed by
substituting the value for the variable. The following shows the calculation whenA
= 0.
Example 6(gg
gg
g)6(gg
gg
g)1(appr)f*aA+dw
3
uu
uu
uCollection —— (collc)
This command arranges the terms of an expression, focusing on a particular
variable.
Syntax: collect (<expression>[, <variable>] [)]
• A default variable of X is used when specification of a variable is skipped.
Example To arrange the terms of the expression X2+ AX + BX, focusing
on the variable X
6(g)6(g)2(collc)vx+
aAv+aBvw X2+ (A + B)X
uu
uu
uCombine —— (comb)
This command produces a fraction made up of a fully expanded numerator over a
fully expanded denominator.
Syntax: combine (<expression>[)]
Example To combine the expressions (X+1) / (X+2) + X × (X+3)
6(g)3(comb)(v+b)
/(v+c)+v*
(v+d)w
ALGBR Mode Commands 20 - 3
X3+ 5X2+ 7X + 1
X + 2

358
20- 3 ALGBR Mode Commands
N353N
– N2++15
6 6
uu
uu
uSequence —— (sequ)
This command creates the function that describes the relationship between the
variable and the value of the expression, if the value of the expression is entered
when the variable is assigned the first specified <value>, the second specified
<value>, and so on.
• The function is a linear algebra expression.
Syntax 1: sequence ({<value>, <value>, ...} [,<variable>] [)]
• A default variable of X is used when specification of a variable is skipped.
Example To obtain the expression when 1 through 4 is {23, 30, 37, 45}
6(g)4(PTS
' )1(sequ)
!{cd,da,dh,
ef!},aNw
• If List 1 = {23, 30, 37, 45}, the same result can be obtained by inputting the
following: sequence(List 1, N).
Syntax 2: sequence ({<value>, <value>, ...},{<value>, <value>, ...} [,<variable>] [)]
The values input with this syntax are handled as lists, with the first value of the
first list paired with the first value of the second list, the second value with the
second value, and so on. This syntax creates a function using this relationship.
Example To obtain an expression for variable values {2, 4, 6, 8} and
expression values {23, 30, 37, 44}
6(g)4(PTS ' )1(sequ)
!{c,e,g,i
!},!{cd,da
,dh,ee!},
aNw
• If List 1 = {2, 4, 6, 8} and List 2 = {23, 30, 37, 44}, the same result can be
obtained by inputting the following: sequence(List 1, List 2, N).
uu
uu
uSum of Sequence —— (smSq)
This command obtains a function that expresses the sum up to the nth term of a
sequence of numbers.
• The function is a linear algebra expression.
7N +16
2

359
ALGBR Mode Commands 20 - 3
Syntax: sumSeq ({<value>, <value>, ...} [,<variable>] [)]
• A default variable of X is used when specification of a variable is skipped.
Example To obtain an expression that expresses the sum up to the nth
term when terms 1 through 4 are the following sequence of
values: {23, 30, 37, 45}
6(g)4(PTS
' )2(smSq)
!{cd,da,dh
,ef!},aNw
• If List 1 = {23, 30, 37, 45}, the same result can be obtained by inputting the
following: sumSeq(List 1, N).
uu
uu
uComplex Exponential-to-Trigonometric Transformation ——
(expTo)
This command transforms an exponential function whose exponent includes an
imaginary number to a trigonometric function.
Syntax: expToTrig (<expression>[)]
Example To transform the following function to a trigonometric function:
eiX
6(g)5(CPLX)1(expTo)
!e(3(i)v)wcos(X) + i • sin(X)
uu
uu
uComplex Trigonometric-to-Exponential Transformation —— (trgTo)
This command transforms a trigonometric function whose argument is an
imaginary number to an exponential function.
Syntax: trigToExp (<expression>[)]
Example To transform the following function to an exponential function:
cos iX
6(g)5(CPLX)2(trgTo)
c(3(i)v)w
N4N395N277N
–++
24 424 4
1
eX+
X
e
2

360
20-4 Signum Function
The signum function described in this section is available in the ALGBR Mode.
Syntax: signum (<expression>[)]
• A solution can be obtained only when <expression> is a numeric value.
Definition: 1 (real number, A > 0)
Undefined (A = 0)
–1 (real number, A < 0)
|A|
signum(A)
(A = imaginary number)
A

Example To solve signum (3.1)
K5(sign)d.bw1
Example To solve signum (–4)
K5(sign)-ew– 1

361
B
A
20-5 Natural Display Notation
Most calculators use their own symbols, such as ABS for absolute values and ^ for
powers, in place of standard mathematical notation. Expressions in theALGBR
Mode are displayed using "natural display notation," which uses standard
mathematical notation as shown below.
Absolute Values |A|
Powers X4
5
Fractions __
3
Square Roots 2 2
Roots 3X
Integration
∫
sin(cos(X))dx
dn
Differentials ___ (X3)
dxn

362
20-6 ALGBR Mode Error Messages
A number of error messages are unique to theALGBR Mode. The following lists
the error messages and explains the meaning of each one.
• Error messages unique to theALGBR Mode appear in the message area of the
display.
uUndefined
No solution exists for the operation being performed.
Example 1/0
uOverflow ERROR
The result of the operation being performed exceeds the range of the calculator.
Example 99999^99999
uDomain ERROR
Input value is outside the domain of the operation being performed.
Example (–4)!
uNon-Real ERROR
Only real numbers have been input and the result is a complex number while the
set up screen's Answer Type item is specified as "Real".
Example (–1)^(1/2)
uNo Solution
No solution can be obtained using the Solve Function.
Example solve(X^2 = –1, X), when Answer Type = "Real"
uMa ERROR
Attempt to use approx with an expression that generates an error unique to the
ALGBR Mode.
Example approx(1/0)
uOther Errors
Stk, Syn, Mem, Arg, and Dim errors have the same meanings as they do in the
RUN Mode. See "Overflow and Errors" for details.
P.7
P.19

363
20-7 ALGBR Mode Precautions
• When an input expression cannot be processed any further, the expression
displayed as the result of an operation will be identical to the input expression.
• It may take a considerable amount of time for a result to appear. This does not
indicate malfunction.
• Note that there may be a variety of different formats that can be used to
express a result. Because of this, even if the format of a result may displayed
by the calculator does not match the format that you need for your purposes, it
does not necessarily mean that the result is wrong.
Regardless of whether intervals are continuous or discontinuous, this calculator
performs definite integral calculations by first obtaining an indefinit integral. Based
on this result, it then obtains a definite integral.
f(x)
F(x): primitive function of f(x)
∫
f(x)dx = F(b) – F(a)
b
a