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MATHEMATICAL FUNCTIONS IN
SCIENTIFIC CALCULATOR
Nandhini.V [1], Ramya.M[2], Anitta.P.J [3], Jeya sowmiya.S [4]
[1]Assistant Professor, Department of mathematics, Sri Krishna Arts and Science
College,
[2, 3,4]UG Student, Sri Krishna Arts and Science College.
[1]nandhinivelusamy@skasc.ac.in ,
[2]ramyam19bma021@skasc.ac.in
[3]anittapj19bma003@skasc.ac.in
[4]jeyasowmiya19bma012@skasc.ac.in
I. INTRODUCTION
This article assesses how to recognize a scientific calculator, what each key possesses and how
to get the accurate value in any form of the mathematical function. The proper understanding
of applying the operations in the calculator that performs the function, in the order we instruct.
scientific calculator differs a lot from other basic calculators. The basic calculator ignores the
order of operations. The main purpose of the scientific calculator has the function that we need
to perform the calculation in chemistry, physics, engineering, or trigonometric problems. It has
a significant role when the function is being worked with scientific notation. However, the
calculator model used here for referring is Casio fx-991 ES Plus. If you are using a different
calculator, you should use the corresponding features.
II. START TO KNOW YOUR CALCULATOR
Firstly, we should know how to use the calculator knowing some of the main keys sequences
for understanding the basic function, to know what may go wrong when entering the data and
calculations, and should know how to fix the calculator functions to a range of mathematical
calculations. Secondly, make sure that you are aware of the layout of keys on the keypad and
how the details are being displayed on the screen.
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Figure- 2.1 Different parts of the Casio fx-991ES Plus.
If you are using a different model of a scientific calculator you may find similar identical keys
and its function in it. To switch on the calculator using the ON key. Most of the keys have two
or more functions, for example, the main function will be on the button and the second function,
an alternate function which will be printed above the button in yellow that can be approached
by pressing the SHIFT key, and the third function will be printed in red by pressing the
ALPHA key before selecting the appropriate key. To off the calculator, press SHIFT AC
(OFF). The second function of the AC key is accessed by the SHIFT key that will off the
calculator.
III. FUNCTIONS
The following table is a list of common mathematical functions and what they define:
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OPERATOR SYMBOLS
MATHEMATICAL
FUNCTIONS
+, - , x, ÷
Arithmetic operators for basic calculations
( )
Brackets / parenthesis
x□
X raised to the power
√ or sqrt
Square root
ex
Exponent raised to the power x
Sin, cos, tan
Trigonometric function
Sin-1, cos-1, tan-1
Inverse trigonometric functions
M+
Add value to that in the ‘M’ memory
log
Logarithm to the base 10
log□ □
Logarithm function
ln
Natural logarithm
calc
Add value to that in the ‘M’ memory
Ans
Use the result of the previous calculation
hyp
Hyperbolic function
S ↔ D
Improper fraction to mixed fraction /
Decimal to fractions
⁰ ‘ “
Time calculation
x-1
X is raised to the power minus 1
RCL
The RECALL: Display the value stored in the ‘M’
memory
STO
Store a value in ‘M’ memory
ENG
Engineering mode
X 10x
Scientific notations
IV. PROCEDURES
1. CALCULATOR MODE
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➢ Mode – COMP
FUNDAMENTAL CALCULATION USING BODMAS:
In this mode, general calculations can be done ‘MODE’ ‘1’. The calculator follows BODMAS,
so it will perform the calculation in the order we instruct. According to the BODMAS rule, if
an expression contains a bracket followed by order i.e, powers and roots, then division,
multiplication, addition, and subtraction from left to right. Solving the problem in the wrong
order will result in incorrect answers. The basic calculation which includes fractions, negative
number, and exponent. The negative numbers can be performed by using ( - ) key before the
required numbers. The ‘-‘ key is used for the operation of subtraction but in some calculator
allows both purposes.
If you enter a wrong sequential order, the calculator will display it as an error. There are two
types of error:
‘Math Error’, when the calculation you entered makes mathematical sense but the result cannot
be calculated, such as attempting to divide by zero, or when the result is too large for the
calculator to handle.
‘Syntax Error’, when the calculation you entered that may not make mathematical sense.
To correct those errors, we have editing facilities by pressing those sides cursors and placing
the editing cursor at the point of the error and do the correction to be made.
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➢ Mode-CMPLX
In this mode complex number calculations can be performed ‘MODE’ ‘2’. Operations can be
carried out using an imaginary part ‘i’ by pressing ‘SHIFT’ ‘ENG’ to get the imaginary part.
i. ARGUMENTS:
By pressing ‘SHIFT’ ‘2’ ‘1’(arguments), enter the required complex number and press ‘=’
(equal to) key to get the answer.
ii. VALUE OF r AND ϴ
By entering the required complex number and then press ‘SHIFT’ ‘2’ ‘3’(r< ϴ) ‘=’ (equal to)
key to get the answer.
iii. CONJUGATE:
By pressing ‘SHIFT’ ‘2’ ‘2’(Conjugate), enter the required complex number and press ‘=’
(equal to) key to get the answer.
iv. COMPLEX NUMBER FROM r AND ϴ:
By entering the required r < ϴ, then press ‘SHIFT’ ‘2’ ‘3’(a+ib) ‘=’ (equal to) key to get the
complex number.
➢ Mode- STAT
This mode, ‘MODE’ ‘3’(STAT) allows us to input data in multiple forms and will give you
the procedures required to calculate many values from the data sets.
SINGLE VARIABLE
Compute and on pressing ‘SHIFT’ ‘1’ interrupt the mean, median, standard deviation, and
normal distribution and entering our data one by one on our calculator by pressing ‘MODE’
‘3’ ‘2’, where only one X variable is used.
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Application: Single-variable statistical calculations are used in a broad range of fields,
including engineering, business, and economics. They are most often applied to analysis in
atmospheric observations and physics experiments, as well as for quality control in factories.
3:Sum:
5:Distr:
REGRESSION AND CORRELATION
y=A+Bx is the estimated regression line of Y on X.
Correlation can reveal how strongly two variables are related. This can be done by obtaining
the coefficient correlation, r. The coefficient correlation, r is a measure of the linear
relationship between pair of variables x and y. the value of r will be between -1 to 1.
By pressing ‘MODE’ ‘3’ ‘2’, enter the data one by one for the variables x and y respectively.
Interrupt the mean, median, standard deviation, r, minimum and maximum values, a
(intercepts) and b (reg-coefficient), on pressing ‘SHIFT’ ‘1’.
3:Sum
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4:Var:
5:Reg:
6: 1:minX: 2:maxX:
3:minY: 4:maxY:
➢ Mode- BASE-N
The Mode BASE-N, ‘MODE’ ‘4’, will change the number base between the many number
modes available. Changing the Mode alters the legends on buttons that have functions specific
to different number modes. Changing number mode changes the calculator display, but any
data stored in the calculator is retained. Conversion between different number systems by
entering data and then selecting the new number base. However, only the decimal system has
a decimal point, the other number bases work for integers only. Also note that a number in one
system will require a larger number of digits in a lower number base, which may cause an
overflow. This is especially likely when converting to binary. To change the conversion press
‘SHIFT’ ‘3’ for having and, or, xor, xnor, not and neg.
➢ Mode- EQN
Solving equations for x is possible in this mode, ‘MODE’ ‘5’. The quadratic equation is solved
easily using the quadratic formula for finding the value of X, which is stored in the calculator.
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The roots of the equation will be calculated and will be displayed on the screen as X1 and X2.
The calculator allows the equations for x which have complex numbers as their roots.
By pressing ‘MODE’ ‘5’, will be changed to equation mode, you will be presented with the
following options:
1:
2:
3:
4:
Enter the values of a, b, c respectively, and press ‘=’ to solve the equation for X.
1:System of equations with two unknowns:
2:System of equations with three unknowns:
3:Quadratic equation:
4:Cubic equation:
➢ Mode-MATRIX
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To do matrix addition or subtraction or to find matrices multiplications can be easily done in
the scientific calculator by matrix mode ‘MODE’ ‘6’ ‘1’ (Mat A), select your dimension (m x
n) and enter your data. Matrices can be carried out matrices up to 3 x 3. To perform the matrices
calculation, we must give special variables (Mat A, Mat B, Mat C), which will be used then a
variable in the calculations.
NOTE: ‘Mat ANS’ stands for matrix answer memory.
i. DETRERMINANENT
Go to the matrix option by pressing ‘SHIFT’ ‘4’ ‘7’ (DET)and choose the matrix you assign
‘SHIFT’ ‘4’ ‘3’ (Mat A) and then press ‘=’ to get the answer.
ii. TRANSPOSE
Go to the matrix option by pressing ‘SHIFT’ ‘4’ ‘8’ (Trn)and choose the matrix you assign
‘SHIFT’ ‘4’ ‘3’ (Mat A) and then press ‘=’ to get the answer.
iii. INVERSE
Choose the matrix you assign ‘SHIFT’ ‘4’ ‘3’ (Mat A), press X-1 then press ‘=’ to get the
inverse matrix.
iv. ABSOLUTE VALUE OF A MATRIX
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We can obtain the absolute value of a matrix by pressing ‘SHIFT’ ‘HYP’ (Abs), choose the
matrix you assign ‘SHIFT’ ‘4’ ‘3’ (Mat A) and then press ‘=’ to get the answer
v. MATRIX MULTIPLICATIONS
Go to the matrix option by pressing ‘SHIFT’ ‘4’ and choose the matrix you assign ‘SHIFT’
‘4’ ‘3’ (Mat A), then press ‘X’(multiplication option), select Mat B‘SHIFT’ ‘4’ ‘4’ and then
press ‘=’ to get the answer.
➢ Mode- TABLE
This mode ‘MODE’ ‘7’ provides you with a table of values. This mode helps us to find the
value of a function (y) at each given value of the variable (x). type the equation of the function
into the f(x)= (input field) and press the ‘=’ button. Table Mode shares the same set of
functions with Graph Mode within a single workspace.
Now, by Entering the value of X at which the table should begin and end, the difference
between each value and the table can be generated using its function.
➢ Mode- VECTOR
This mode ‘MODE’ ‘8’ carries out the calculation on 2D and 3D vectors. Selecting vector
mode, we must give data to choose a vector memory slot (Vct A, Vct B, Vct C), which will be
used then as a variable in the calculation, we’ll be prompt to choose the dimension of the
vector(2 or 3).
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To enter data for another vector press ‘SHIFT’ ‘5’(VECTOR), which will be now called a
vector option, and choose ‘2’ (DATA). Proceed by choosing a different memory slot(Vct B,
Vct C)
✓ CROSS PRODUCT
Go into vector option by pressing ‘SHIFT’ ‘5’ and choose the vector you assign your data
‘3’(Vct A) to show up on the main screen and press ‘X’(multiplication option), select another
vector Vct B, ‘SHIFT’ ‘5’ ‘4’ and finally ‘=’ to get the answer.
✓ DOT PRODUCT
Select vector A (Vct A) ‘SHIFT’ ‘5’ ‘3’ and re-enter into the vector option ‘SHIFT’ ‘5’ and
select ‘7’(DOT), go to vector option again select vector B (Vct B), press ‘=’ to get the answer.
✓ MAGNITUDE OF A VECTOR
Choose your vector by entering to vector option and then press ‘SHIFT’ ‘HYP’(Abs) and
finally press ‘=’ to get the magnitude of a vector A, |A|.
➢ SETUP – (‘SHIFT’ ‘MODE’)
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SET UP
MATHEMATICAL
FUNCTION
PROCEDURE
EXAMPLE
MthIO
The answers will have
a natural display in the
form of fractions
‘SHIFT’ ‘MODE’ ‘1’
LineIO
It will be displayed in
a linear form as
decimals
‘SHIFT’ ‘MODE’ ‘2’
Deg
Specify degrees as the
angle unit
‘SHIFT’ ‘MODE’ ‘3’
Rad
Specify radians as the
angle unit
‘SHIFT’ ‘MODE’ ‘4’
Gra
Specify grads as the
angle unit
‘SHIFT’ ‘MODE’ ‘5’
Fix
Control the number of
decimal places in the
results that we obtain
‘SHIFT’ ‘MODE’ ‘6’
Sci
Control the number of
significant digits in the
results that we obtain
‘SHIFT’ ‘MODE’ ‘7’
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Norm
Non-exponential form
in the specified range
‘SHIFT’ ‘MODE’ ‘8’
ab/c
set the calculator to use
mixed numbers by
default,
‘SHIFT’ ‘MODE’ ‘▼’
‘1’
(SETUP)
(ab/c)
d/c
set the calculator to use
improper or top-heavy
fractions by default
‘SHIFT’ ‘MODE’ ‘▼’
‘2’
(SETUP) (d/c).
V. MATHEMATICAL FUNCTIONAL KEYS
➢ PERCENTAGE:
A percentage is a number, or a ratio expressed in a fraction of 100. Input the value using the
number pad and press ‘SHIFT’ ‘(‘, the percentage sign should appear on the screen and press
‘=’ to get the required answer.
➢ FACTORIAL:
The factorial of a positive integer of n is denoted by n! it’s a product of all positive integer less
than or equal to n. To find the factorial of a scientific calculator, enter any value and press
‘SHIFT’ ‘X!’ and press ‘=’ to get the required answer.
Application: Used in statistics and mathematics. In statistics, this function is used in
calculations involving combinations and permutations
➢ SUMMATION:
The summation is the addition of a sequence of any kind of numbers called summands, the
result is their sum.
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In the scientific calculator, find the summation of natural numbers 1 to 10. Press the ‘SHIFT’
button, then ‘log□ □’ the summation sign,
will be displayed on the screen. Input the
function in terms of X and enter the value for x and value for n.
➢ INTEGRATION and DIFFERENTIATION:
It can be calculated with respect to x. Integration can be performed by pressing the button
‘
’ a button from your calculator. It will display the integration symbol with upper and
lower limits and the integrand and dx at the end. To perform integration, enter your integrand
equations in terms of x and enter your limits and press ‘=’ to display your answer.
To perform differentiation, press
. Enter the equation to be differentiated in terms of x and
enter the value for x in which you want to find the value for the gradient, press ‘=’ to get the
required answer.
➢ TRIGONOMETRIC FUNCTION VALUE:
To find the values of the trigonometric ratios of angles you need to ensure that various units in
which an angle can be measured, in degrees or radians. To calculate the trigonometric values
using the sine, cosine, or tangent of an angle, press the ‘SIN’ ‘COS’ or ‘TAN’ key and then
type the measurement of an angle. after entering the angle press ‘=’ to get the trigonometric value.
Application: Trigonometric functions are useful in mathematics and various engineering
calculations. They are often used in astronomical observations, civil engineering, and in
calculations involving electrical circuits, as well as in calculations for physics such as parabolic
motion and wave motion.
➢ INVERSE TRIGONOMETRIC VALUE:
Finding angles from trigonometric ratios, we can be obtained by taking inverse trigonometric
functions. To calculate inverse trigonometric values can be found using the second
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functions , and , by pressing ‘SHIFT’ ‘SIN’, ‘COS’, or ‘TAN’ to get the
required angle.
➢ EQUATION SOLVE:
The equation for X is possible with the functionality of ‘SHIFT-solve’ when a quadratic
equation which is not in their standard forms or any equations. The calculator finds the value
for x by using the Newton- Raphson method. It only allows the equation to have a single
variable. Type the equation and press ‘SHIFT’ ‘CAlC’ (SOLVE), to solve for x.
➢ PERMUTATION and COMBINATION:
1. nPr: This function finds the number of different possible orderings in selecting r objects from
a set of n objects. Input the number, n, and press ‘SHIFT’ ‘X’ (MULTIPLICATION SIGN), the
letter P will be displayed which indicates for permutation and input the value for r and then
finally press ‘=’ to get the answer.
2. nCr: This function finds the number of ways of selecting r objects from a set of n objects. Input
the number, n, and press ‘SHIFT’ ‘⁒’ (DIVISION SIGN), the letter C will be displayed which
indicates for combination and input the value for r and then finally press ‘=’ to get the answer.
Applications: Used to find the expansion of polynomial, also used in statistics (probability
calculations) and simulation hypotheses in fields such as medicine, pharmaceutics, and physics.
Also, can be used to determine the chances of winning in lotteries.
➢ LOGARITHM, NATURAL LOGARITHM, and e POWER X:
1. LOGARITHM:
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‘log’ button in your calculator will calculate the logarithm, the exponent of the power to which
10 must be raised to equal the given value. By pressing the ‘log’ button and enter any value,
press ‘=’ to get the log value of that number.
‘log□ □’ button in your calculator will calculate the logarithm in which the base can be entered
that must be raised to the given value.
2. NATURAL LOGARITHM
‘ln’ button computes the value of the natural logarithm, the exponent of the power to which e
must be raised to equal the given value.
3. e POWER X
By pressing the ‘SHIFT’ ‘ln’ button or by pressing ‘ALPHA’ ‘’ ‘ so that will be
displayed on the screen. Here it calculates powers based on the constant e (2.718281828).
4. 10 TO THE POWER of X
‘’ button calculates the value of 10 raised to the power x
➢ TIME CALCULATION:
Inputs values in sexagesimal notation (degrees, minutes, seconds). You must input zero even
the degree or minute value is zero.
Application: Used in calculations of angles and angular velocity in physics, and latitude and
longitude in geography.
➢ FRACTION and DECIMALS:
Fractions are entered using the button in the left-hand column of the function key area.
Mixed numbers can be entered similarly using the mixed number template obtained using the key
sequence ‘SHIFT’ or by entering our fraction and press ‘SHIFT’ ‘S↔D’ to get a
mixed fraction. Conversion of a fraction to decimal can be done by entering a fraction and press
‘S↔D’ to get the respective decimal answer.
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By pressing ‘SHIFT’ ‘MODE’(SETUP) ‘▼’ ‘5’ Specify the decimal point for calculation
results.
Application: There is a wide variety of applications for this function because fractions are such
a basic part of mathematics. This function is useful for calculations involving electrical circuit
resistance.
➢ HYPERBOLIC FUNCTION:
The hyperbolic function is defined by using natural exponents in trigonometric functions.
Arc hyperbolic functions are defined by using natural logarithms in trigonometric functions
Applications: Hyperbolic and arc hyperbolic functions are very useful in electrical engineering
and physics.
➢ ABSOLUTE VALUE:
To the absolute value of any number or a function can be done, by pressing ‘SHIFT’ ‘HYP’ (Abs)
and entering the function or any number to get the required answer.
➢ IMPORTANT SCIENTIFIC CONSTANT
A scientific constant that can be used inside of calculation by pressing Shift then 7 in any
mode except Base-N, it is possible to recall one of 40 constants stored on the calculator by
inputting a number from 01 to 40. Some important scientific constants are
Press ‘SHIFT’ ‘7’ (CONSTANT 01 – 40)
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01 - Mass of proton
02 - Mass of neutron
03 - Mass of electron
06 - Planck's constant
17 - Atomic mass unit
24 - Avogadro's constant
27 - Molar gas constant
28 - Speed of light
35 - Gravitational field strength
38 - Difference between Kelvin and Celsius
39 - Gravitational Constant
40 - Atmospheric pressure
➢ CONVERSION OF UNITS
Conversion of units from one unit to another can be easily done in the scientific calculator by
inputting the value that is to be converted and press ‘SHIFT’ ‘7’ (CONSTANT 01 – 40)
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➢ POWERS AND SURDS
1. Taking the Power:
enter any number and press x□, input to which power has to be calculated. To get any
number to be squared press x2 and similarly for any number to be powered by a cube
press ‘SHIFT’ ‘x2’.
2. SURDS:
roots of any number can also be found directly. Square roots can be calculated using the button
and similarly to get any root of any number press ‘SHIFT’ ‘x□’ to display (
), which
is the second function of the key ‘x□’.
3. Π- Pi:
‘π’ is used for scientific calculation. You could enter an approximate value for by hand,
but this is time-consuming and won't be accurate and leads to error. The calculator has an
approximation value for built into it, which is obtained using the key sequence ‘SHIFT’
‘X’ (π).
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➢ CALCULATOR MEMORY:
To store the result of an expression just calculated (i.e. an answer displayed in the output area of
the calculator screen) in the ‘M’ calculator memory, use the key ‘M+’. After selecting the store
function, we need to tell the calculator which memory the value is to be stored in. These memories
are labelled in red on some of the calculator keys. Once ‘RCL’ (or ‘SHIFT’ ‘RCL’(STO)) has
been pressed, the display indicator RCL (or STO) is shown on the display to indicate that the
calculator is waiting to know which memory to recall (store) the value.
VI. CONCLUSION
In the above article, we understood the importance and realise how powerful this unassuming
scientific calculator actually is. It most sophisticated and comprehensive scientific calculator with an
equation solver and are widely used in both education and professional settings.
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