Casio Fx991ms User Guide

							E-9
Imaginary axis
Real axis
kAbsolute Value and Argument
Calculation
Supposing the imaginary number expressed by the
rectangular form z = a + bi is represented as a point in the
Gaussian plane, you can determine the absolute value (r)
and argument () of the complex number. The polar form
is r
.
•Example 1:To  determine the absolute value (
r) and
argument () of 3+4i (Angle unit: Deg)
(r = 5,  = 53.13010235°)
(
r 
 5) A A R 3 + 4 i T =
( 
 53.13010235°)A a R 3 + 4 i T =
•The complex number can also be input using the polar
form r
.
•Example 2:
2 
 45 
 1  i(Angle unit: Deg)L 2 A Q 45 =
A r
kRectangular Form ↔ Polar Form
Display
You can use the operation described below to convert a
rectangular form complex number to its polar form, and a
polar form complex number to its rectangular form. Press
A r to toggle the display between the absolute value
(r) and argument ().
•Example: 1
  i ↔ 1.414213562 
 45
(Angle unit: Deg)
1 + i A Y = A r
L
 2 A Q 45 A Z = A r 
						

							E-10
BASE
•You select rectangular form (a+bi) or polar form (r
)
for display of complex number calculation results.
F...1(Disp) r
1
(a+bi): Rectangular form
2(r
): Polar form (indicated by “r
” on the display)
kConjugate of a Complex Number
For any complex number z where z = a+bi, its conjugate
(z) is z = a–bi.
•Example: To determine the conjugate of the complex
number 1.23 + 2.34
i  (Result: 1.23 – 2.34i)
A S R 1 l 23 + 2 l 34 i T =
A
 r
Base-n Calculations
Use the F key to enter the BASE Mode when you
want to perform calculations using Base-n values.
BASE ........................................................
F F 3
•In addition to decimal values, calculations can be
performed using binary, octal and hexadecimal values.
•You can specify the default number system to be applied
to all input and displayed values, and the number system
for individual values as you input them.
•You cannot use scientific functions in binary, octal,
decimal, and hexadecimal calculations. You cannot input
values that include decimal part and an exponent.
•If you input a value that includes a decimal part, the unit
automatically cuts off the decimal part.
•Negative binary, octal, and hexadecimal values are
produced by taking the two’s complement. 
						

							E-11 •You can use the following logical operators between
values in Base-
n calculations: and (logical product), or
(logical sum), xor (exclusive or), xnor (exclusive nor),
Not (bitwise complement), and Neg (negation).
•The following are the allowable ranges for each of the
available number systems.
Binary 1000000000  
x 1111111111
0  x 0111111111
Octal 4000000000  x 7777777777
0  x 3777777777
Decimal –2147483648  x 2147483647
Hexadecimal 80000000  x FFFFFFFF
0  x 7FFFFFFF
• Example 1:To  p erform the following calculation and
produce a binary result:
10111
2  110102 
1100012
Binary mode:t b0.b
10111 + 11010 =
• Example 2:To  p erform the following calculation and
produce an octal result:
7654
8 ÷ 1210 
 5168Octal mode:t o0.o
l l l 4 (o) 7654 \
l l l 1 (d) 12 =
• Example 3:To  p erform the following calculation and
produce a hexadecimal and a decimal result:
120
16 or 11012 
 12d16 
 30110
Hexadecimal mode:t h0.H
120 l 2 (or)
l l l 3 (
b)
 1101 =
Decimal mode:K 
						

							E-12
1    2    3     4
P (  Q (  R (  → t
SD
REG
SD
•Example 4:To  convert the value 2210 to its binary, oc-
tal, and hexadecimal equivalents.
(101102 , 268 , 1616 )
Binary mode:t b0.b
l l l 1(d) 22 =10110.b
Octal mode:o26.o
Hexadecimal mode:h16.H
•Example 5:To convert the value 51310 to its binary
equivalent.
Binary mode:
t b0.b
l l l 1(d) 513 =a MthERRORb
•You may not be able to convert a value from a number
system whose calculation range is greater than the cal-
culation range of the resulting number system.
•The message “Math ERROR” indicates that the result
has too many digits (overflow).
Statistical
Calculations
Normal Distribution
Use the F key to enter the SD Mode when you want
to perform a calculation involving normal distribution.
SD ...........................................................  
F F 1
• In the SD Mode and REG Mode, the | key operates as
the S key.
• Press A D, which produces the screen shown below. 
						

							E-13
COMPDifferential
Calculations
The procedure described below obtains the derivative of
a function.
Use the 
F key to enter the COMP Mode when you
want to perform a calculation involving differentials.
COMP............................................................
F 1
•Three inputs are required for the differential expression:
the function of variable x, the point (a) at which the dif-
ferential coefficient is calculated, and the change in
x (∆x).A J expression P a P ∆x T
•Example: To determine the derivative at point x = 2 for
the function y = 3x2– 5x + 2, when the increase or de-
crease in x is ∆x = 2 × 10–4 (Result: 7)
A J 
3 p x K , 5 p x + 2 P 2 P2 e D 4 T =
•Input a value from 1 to 4 to select the probability
distribution calculation you want to perform.
P(t)R(t) Q(t)
•Example: To  determine the normalized variate (→t) forx = 53 and normal probability distribution P(t) for the
following data: 55, 54, 51, 55, 53, 53, 54, 52
(
→t = 0.284747398, P(t) = 0.38974 )
55 
S 54 S 51 S 55 S
53 S S 54 S 52 S
53 A D 4(→t) =
A
 D 1( P( ) D 0.28 F = 
						

							E-14
COMP
•You can omit input of ∆x, if you want. The calculator
automatically substitutes an appropriate value for ∆x if
you do not input one.
•Discontinuous points and extreme changes in the value
of x can cause inaccurate results and errors.
•Select Rad (Radian) for the angle unit setting when
performing trigonometric function differential calculations.
Integration
Calculations
The procedure described below obtains the definite integral
of a function.
Use the 
F key to enter the COMP Mode when you
want to perform integration calculations.
COMP............................................................
F 1
•The following four inputs are required for integration
calculations: a function with the variable x; a and b, which
define the integration range of the definite integral; and
n, which is the number of partitions (equivalent to N =
2n) for integration using Simpson’s rule.
d expression P a P b P n F
Note!
•You can specify an integer in the range of 1 to 9 as the
number of partitions, or you can skip input of the number
of partitions entirely, if you want.
•Internal integration calculations may take considerable
time to complete.
•Display contents are cleared while an integration
calculation is being performed internally.
•Select Rad (Radian) for the angle unit setting when
performing trigonometric function integration calculations. • Example:
∫     (2x2 + 3x + 8) dx = 150.6666667(Number of partitions n = 6)
d 2 p x K + 3 p x +
8 P 1 P 5 P 6 T =
 5
1 
						

							E-15
kCreating a Matrix
To create a matrix, press A
 j
 1(Dim), specify a matrix
name (A, B, or C), and then specify the dimensions
(number of rows and number of columns) of the matrix.
Next, follow the prompts that appear to input values that
make up the elements of the matrix.
You can use the cursor keys to move about the matrix in
order to view or edit its elements.
To exit the matrix screen, press 
t.
MATMatrix Calculations
The procedures in this section describe how to create
matrices with up to three rows and three columns, and
how to add, subtract, multiply, transpose and invert
matrices, and how to obtain the scalar product,
determinant, and absolute value of a matrix.
Use the 
F key to enter the MAT Mode when you want
to perform matrix calculations.
MAT.....................................................
F F F 2
Note that you must create one or more matrices before
you can perform matrix calculations.
•You can have up to three matrices, named A, B, and C,
in memory at one time.
•The results of matrix calculations are stored automatically
into MatAns memory. You can use the matrix in MatAns
memory in subsequent matrix calculations.
•Matrix calculations can use up to two levels of the matrix
stack. Squaring a matrix, cubing a matrix, or inverting a
matrix uses one stack level. See “Stacks” in the separate
“User’s Guide” for more information.
MatA23
2 rows and 3 columns 
						

							E-16
kEditing the Elements of a Matrix
Press A
 j
 2(Edit) and then specify the name (A, B, or
C) of the matrix you want to edit to display a screen for
editing the elements of the matrix.
kMatrix Addition, Subtraction, and
Multiplication
Use the procedures described below to add, subtract,
and multiply matrices.
•Example: To multiply Matrix A =                by
Matrix B =
(Matrix A 32)A
 j
 1(Dim)
 1(A)
 3 =
 2 =
(Element input)1 =
 2 =
 4 = 0 =
 D 2 =
 5 = t
(Matrix B 23)A
 j
 1(Dim)
 2(B)
 2 =
 3 =
(Element input)D 1 = 0 = 3 = 2 =
 D 4 = 1 = t
(MatAMatB)A
 j
 3(Mat)
 1(A)
 -
A
 j
 3
(Mat)
 2(B)
 =
•An error occurs if you try to add, subtract matrices whose
dimensions are different from each other, or multiply a
matrix whose number of columns is different from that of
the matrix by which you are multiplying it.
kCalculating the Scalar Product of a
Matrix
Use the procedure shown below to obtain the scalar
product (fixed multiple) of a matrix.
•Example: Multiply Matrix C =                  by 3. 2–1
–5 3
[      ][       ](         )6–3
–15 9
[     ]
12
4 0
–2  5
[          ](              )
3–8 5
–4 0 12
12 –20 –1 –1 0 3
2–4 1
[         ] 
						

							E-17
(Matrix C 2
2)A
 j 1
 (Dim)
 3(C)
 2 = 2 =
(Element input)2 =
 D 1 =
 D 5 = 3 = t
(3
MatC)3 -
 A
 j
 3(Mat)
 3(C)
 =
kObtaining the Determinant of a Matrix
You can use the procedure below to determine the
determinant of a square matrix.
•Example: To obtain the determinant of
Matrix A =                         (Result:
 73)
(Matrix A 3
3)A
 j
 1(Dim)
 1(A)
 3 = 3 =
(Element input)2 =
 D 1 = 6 = 5 = 0 = 1 =3 = 2 = 4 = t
(DetMatA)A
 j
 r
 1(Det)
A
 j
 3(Mat)
 1(A)
 =
•The above procedure results in an error if a non-square
matrix is specified.
kTransposing a Matrix
Use the procedure described below when you want to
transpose a matrix.
•Example: To transpose Matrix B =
(Matrix B 2
3)A
 j
 1(Dim)
 2(B)
 2 = 3 =
(Element input)5 = 7 = 4 = 8 = 9 = 3 = t
(TrnMatB)A
 j
 r
 2(Trn)
A
 j
 3(Mat)
 2(B)
 =
2–1 6
501
324
[          ]
574
893[         ]58
7 9
4 3
[     ](        ) 
						

							E-18
VCT
0.4 1 0.8
1.5 0.5 1.5
0.8 0 0.6
[             ](                 )
kInverting a Matrix
You can use the procedure below to invert a square matrix.
•Example: To invert Matrix C =
(Matrix C 3
3)A
 j
 1(Dim)
 3(C)
 3 = 3 =
(Element input)D 3 = 6 =
 D 11  = 3 =
 D 4 =6 = 4 =
 D 8 = 13 = t
(MatC–1)A
 j
 3(Mat)
 3(C) a
 =
•  The above procedure results in an error if a non-square
matrix or a matrix for which there is no inverse
(determinant = 0) is specified.
kDetermining the Absolute Value of a
Matrix
You can use the procedure described below to determine
the absolute value of a matrix.
•Example: To determine the absolute value of the matrix
produced by the inversion in the previous example.
(AbsMatAns)A
 A
 A
 j
 3(Mat)
 4(Ans) =
Vector Calculations
The procedures in this section describe how to create a
vector with a dimension up to three, and how to add, sub-
tract, and multiply vectors, and how to obtain the scalar
product, inner product, outer product, and absolute value
of a vector. You can have up to three vectors in memory at
one time.–3 6 –11
3–4 6
4–8 13
[          ]
–0.4 1 –0.8
–1.5 0.5 –1.5
–0.8 0 –0.6
[              ](                   )