### Author Topic: The calculator  (Read 71158 times)

#### Mikl__

• Member
• Posts: 742
##### Re: The calculator
« Reply #30 on: June 19, 2013, 10:55:57 AM »
Ola, RuiLoureiro!
ORN = (Not A) OR (Not B) = Not (A AND B) ?

#### dedndave

• Member
• Posts: 8821
• Still using Abacus 2.0
##### Re: The calculator
« Reply #31 on: June 19, 2013, 11:46:16 AM »
Code: [Select]
`NAND = NOT (A AND B) NOR = NOT (A OR B) ORN = A OR (NOT B)`

#### RuiLoureiro

• Member
• Posts: 819
##### Re: The calculator
« Reply #32 on: June 26, 2013, 01:53:48 AM »
Olá Mikl,
It's not right.
I use A ORN B = A or (not B) = A or -B as Dave wrote

Hi
In the next Calcula55, The powerfull Calculator v2.30,

we may solve any quadratic or any system of 2,3,4 equations

where any coeficient may be a real expression as large as

a train (11 300 characters)!

Well, we may solve any system of 5,6, ..,19,20 equations

using matrices;

We may define or redefine any one of 20 constants: real, logic

or matrix.

Each matrix element is a real constant or any real expression

as large as a train!

When we want to compute a real/logic expression, we may define or

redefine a constant:

(log(2)+56.38)/12.5        (=4.5344823996531185)
or
this = (log(2)+56.38)/12.5 (=4.5344823996531185)

(the expression may contain real constants)

Using 'list r' we may edit the real constants and we may redefine them.

Using 'list l' we may edit the logic constants and we may redefine them.

Any 20x20 matrix fits in th edit box, so we may copy it from or to

the edit box.

What do you think about this version ?

Thanks
Rui
-------------------------------------------------------------------------
TYPE:   this = (log(2)+56.38)/12.5      ENTER/COMPUTE

TYPE:   a=12.3;b=-1.5;c=15;d=3.24;      ENTER/COMPUTE

Quadratic equation:

TYPE:   ax^2+bx+c=d                     ENTER/COMPUTE

Root X0=  0.0609756097560975+ i  0.9758993472640925
Root X1=  0.0609756097560975- i  0.9758993472640925

--------------------------------------------------------------------------
Quadratic equation:

TYPE:   this x^2 + bx + c=d             ENTER/COMPUTE

Root X0=  0.1653992526373845+ i  1.6019061672211593
Root X1=  0.1653992526373845- i  1.6019061672211593
--------------------------------------------------------------------------
System of 2 equations:

TYPE:   this x + by= c ; x - d y= 12;

X=  2.3196363400200491
Y= -2.9877665617222071
Determinant: -13.1917229748761039

#### RuiLoureiro

• Member
• Posts: 819
##### Re: The calculator
« Reply #33 on: July 04, 2013, 02:17:31 AM »
****-- Here it is The Calculator v3.00.1 --****

This is the new calculator with new functions
and new better and faster procedures.

A. What it does ?
---------------------

1. Conversions

2. Real constants definitions

3. Logic constants definitions

4. Function definition

5. Derivative definition

6. Matrices definitions

7. Logic operations

8. Matrices operations

9. Systems of 2 linear equations x,y

10. Systems of 3 linear equations x,y,z

11. Systems of 4 linear equations x,y,z,t

12. Quadratic equation ax^2+bx+c=d

13. Solves any real expression

B. Special functions
------------------------

B.1. list     shows the symbols defined

list r   edit the r eal constans defined

list l   edit the l ogic constants defined

list f   edit the f unction defined

list d   edit the d erivative defined

B.2. scan     shows the values of a defined function f(x)
from x=x0 to x=x1 where x0,x1 are integers.
It shows also the zeros

B.3. root     try to find the value x=x0 where the function f(x)=0

C. General rule about constants and matices
--------------------------------------------------------

1. Any constant or matrix is redefinable

2. The tables have space for 20 names

C. General rule about real constants
----------------------------------------------

1. The name cannot be any symbol already defined
like 'e' 'pi', 'log', 'sin', etc.

2. The name cannot be x, y, z, t

3. The name cannot be a matrix name

4. The matrix name cannot be any symbol already defined
and cannot be any real constant name

5. Any real constant is redefinable

6. We may define a real constant like this

name = real expression

D. General rule about logic constants
----------------------------------------------

1. The name cannot be any symbol already defined

2. The name cannot be 'd', 'h', 'b'

3. We may use real constant names or matrix names
because logic expressions doesnt use real values
or matrix names and real expressions doesnt use
logic constants

E. General rule about expressions
-------------------------------------------

We may use any constant already defined

F. Defining a matrix
-------------------------

F.1
matA =[1,2,3; 3,4,5; 2,5,9]         (without semicolon)

F.2
a=2; b=3;
matB =[3*2, a, b; 1.2, 4.5,-5.23];  (with semicolon)

F.3
matC =[2, a, b; 1, 5,-5; 0,1,-1];
matC =matC^-1;

G. Defining a real constant
----------------------------------

G.1
a=2;b=-5;c=3;

G.2
a=2;b=3;
f=round(e^-(-3+sin(3*a+12)),6)+34*5-sin(pi/b)

H. Defining a logic constant
----------------------------------

H.1
l1=2d; l2=53h; l3=11100011b;

H.2
l1=2d; l2=53h; l3=11100011b;

l4= l1 + l2 shl 1d + 45h and l3

I. System of equations
-----------------------------

1.      a=2;b=-3;c=4;d=-1;f=5;g=-2;h=-6;i=0;j=4;k=1;l=-3;

2.      aX-bY=c; dY+fX=-a;

aX-bY+cZ=d; Z-dY+fX=-a; X+Y+Z=h;

kT+aX-bY+cZ=d; Z-dY+fX=-a; X+Y+Z-T=h; Y+lZ+fT=l;

J. Quadratic equation
---------------------------

a=2;b=-3;c=4;d=-1;
aX^2 -bX +c=d

K. root function
--------------------

1. f(x)=2*x^5+3*x^4-x^3+5*x^2+x+120

2. root(x=-2,x=2)

root(x=-2,x=2; n=1000)

root(x=-2,x=2; d=0.01)

df(x)=10*x^4+12*x^3-3*x^2+10*x+1
root(x=-2,x=2; x=-2)

L. scan function
--------------------

1. f(x)=x^5-5*x^3+4*x

2. scan(x=-10,x=10)

Now type 'list' and we can see all constants defined till now.

M. conversions
-------------------

Type 112233d or 112233h or 111000111b and press COMPUTE

note 1: type/copy and paste the examples and press compute
note 2: to get the polynomials zeros use scan() function

Try it and say something.
Good luck !
Thanks
Rui Loureiro
« Last Edit: July 06, 2013, 02:37:00 AM by RuiLoureiro »

#### RuiLoureiro

• Member
• Posts: 819
##### Re: The calculator
« Reply #34 on: July 10, 2013, 05:43:42 AM »
****-- The Calculator v3.00.2 --****
****--     The new version    --****

--------------------------------
Three things  A.1  A.2  A.3
--------------------------------

--------------------------------------------------------------------
A.1. Define a matrix as a direct operation of 2 matrices
---------------------------------------------------------------------

matA= -1.2*[1,2,3; 4,5,6; 7,8,9]

matB= [1,2,3; 4,5,6; 7,8,9]*[3; 6; 9]

matC= [1,2,3; 4,5,6; 7,8,9]-[-9,8,7; -4,0,6; 1,3,2]

Example 1

Solving a system of 5 linear equations:
Code: [Select]
`            1 X1 -2 X2 +3 X3 -1 X4 + 0 X5 = 123            0 X1 +3 X2 -2 X3 +5 X4 + 7 X5 = 15            3 X1 -4 X2 +0 X3 -2 X4 + 3 X5 = 9           -2 X1 -2 X2 +5 X3 +0 X4 + 0 X5 = -12            1 X1 +0 X2 -1 X3 +3 X4 + 9 X5 = 35        matC=[123; 15; 9; -12; 35]        matA=[1,-2,3,-1,0; 0,3,-2,5,7; 3,-4,0,-2,3; -2,-2,5,0,0; 1,0,-1,3,9]        matA1=matA^-1;             =[ 0.5230125523012552, 0.2761506276150627, 0.0711297071129707,             -0.2510460251046025,-0.2384937238493723; 0.6903765690376569,             -0.7154811715481171,-0.8661087866108786,-0.5313807531380753,              0.8451882845188284; 0.4853556485355648,-0.1757322175732217,             -0.3179916317991631,-0.11297071129707113, 0.2426778242677824;             -0.401673640167364, 1.1799163179916318, 0.8493723849372384,              0.4728033472803347,-1.200836820083682; 0.1297071129707113,             -0.4435146443514644,-0.3263598326359832,-0.1422594142259414,              0.5648535564853556];        multiplying by [123; 15; 9; -12; 35]                            = [ 63.778242677824266; 102.347280334728032; 64.050209205020915;              -71.765690376569036; 27.841004184100418];            X1=  63.778242677824266            X2= 102.347280334728032            X3=  64.050209205020915            X4= -71.765690376569036            X5=  27.841004184100418        first equation:                    63.778242677824266 -2* 102.347280334728032 + 3* 64.050209205020915               +71.765690376569036 = 122.999999999999983`
--------------------
A.2 find function
--------------------

Now we have the scan function and the find function
to study one function.

The scan function uses integer points and is useful to
study any polynomial.

The find function try to get an interval (x0,x1) where
the function changes the sign. So it may be a discontinuity
or it may be a zero (it shows the f(x) values)

Example 2
Code: [Select]
`f(x)=(x+1)/(x^2-x-1)find(x=-10, x=10)One zero was foundx= -1.0f(x)=  0`
Example 3
Code: [Select]
`f(x)=(log(x-1)-x)/(x-5)find(x=1, x=20)The function change the sign in this interval(x0,x1)=[  4.9999 ,  5.0 ]f(x0)=  43978.50866169812f(x1)= -INFINITY`
Example 4
Code: [Select]
`f(x)=x-log(1/(x-1))find(x=1, x=5)The function change the sign in this interval(x0,x1)=[  1.0826 ,  1.0827 ]f(x0)= -0.0004199526796177f(x1)=  0.0002055095525466`
Example 5
Code: [Select]
`f(x)=(x-log(1/(x-1)))/(x^2-1)find(x=1, x=10)The function change the sign in this interval(x0,x1)=[  1.0826 ,  1.0827 ]f(x0)= -0.0004199526796177f(x1)=  0.0002055095525466`

Example 6
Code: [Select]
`f(x)=x^3-log(x^2+x)+x-1find(x=0,x=1)The function change the sign in this interval(x0,x1)=[  0.1177 ,  0.1178 ]f(x0)=  0.0002288184583431f(x1)= -0.000074703978969scan(x=-2,x=2)x= -2 , f(x)=-11.3010299956639812 ; x= -1 , f(x)=+INFINITY ; x= 0 , f(x)=+INFINITY ; x= 1 , f(x)= 0.6989700043360188 ; x= 2 , f(x)= 8.2218487496163564 ; `

---------------------------------------------------------------------------
A.3  The limits of the root/find function may be real expressions
---------------------------------------------------------------------------

f(x)=x^3-x

root(x=log(2), x=log(3))
find(x=-log(3), x=log(3))

Good luck !
Rui Loureiro
« Last Edit: July 10, 2013, 08:34:49 PM by RuiLoureiro »

#### RuiLoureiro

• Member
• Posts: 819
##### Re: The calculator
« Reply #35 on: July 24, 2013, 02:15:31 AM »
****-- The Calculator v3.01.0 --****

---------------------------
COMPLEX NUMBERS
---------------------------

Now, we may solve any complex expression in the same way we do for
real expressions.

Any complex number is a number with a real part
and an imaginary part. For instance, z1=-2+i3 or z1=i3-2.

We may define it typing z1=(-2,3); or z1=-2+i3 for example.

To build a complex expression
we may define a set of real/complex constants z1, z2, z3, z4, ...
and then we write the expression

For instance,

(1-i2)*(2-i3)  or (a-ib)*(c-id) where a=12;b=-4;c=1;d=-3;
are real constants

z1+z2*z3-z4   or  z5=z1+z2*z3-z4  or z1=z1+z2*(z3/z4)+3-i2

Type:   d=7;c=1;b=-3;a=2;    press ENTER/COMPUTE
and     (a-ib)*(c-id)                 press ENTER/COMPUTE
and     Z1=(a-ib)*(c-id)           press ENTER/COMPUTE

We get:      23.0-i11.0
| Z |=  25.495097567963924
Angle= -25.559965171823808 degrees
Complex Constant defined

If you have any suggestion, please tell me

Jul. 2013

Good luck !
Thanks
Rui Loureiro
EDIT: In the next version the calculator will do all
complex functions (e.g.: ln(2-i3), etc.  )
« Last Edit: July 25, 2013, 07:54:49 AM by RuiLoureiro »

#### RuiLoureiro

• Member
• Posts: 819
##### Re: The calculator
« Reply #36 on: July 30, 2013, 04:19:25 AM »
Hi all
****-- The Calculator v3.10.1 --****

This is the last version and this is the powerful calculator
v3.10.1.

--------------------------------------------------------------
COMPLEX NUMBERS and COMPLEX FUNCTIONS
--------------------------------------------------------------

Now, we may solve any complex expression in the same way we do for
real expressions.

We may use any function:

conj(a+ib)      = a-ib
inv(a+ib)       = 1/(a+ib)

abs(a+ib)       = sqr(a^2+b^2)+i0
sqr(a+ib)       = (a+ib)^(1/2)
rnd(a+ib)       = rnd(a)+i rnd(b)
rndi(a+ib)      = rndi(a)+i rndi(b)

e^(a+ib)        = e^a * e^ib = e^a * ( cos(b)+i sin(b))

(a+ib)^(c+id)   = e^( (c+id)*ln(a+ib) )

ln(a+ib)        = ln (|a+ib|* e^i angleR )
log(a+ib)       = (ln (|a+ib|* e^i angleR )) / ln(10)

sin(a+ib)
cos(a+ib)
tan(a+ib)
sec(a+ib)
csc(a+ib)
cot(a+ib)

arcsin(a+ib)
arccos(a+ib)
arctan(a+ib)
arcsec(a+ib)
arccsc(a+ib)
arccot(a+ib)

sinh(a+ib)
cosh(a+ib)
tanh(a+ib)
sech(a+ib)
csch(a+ib)
coth(a+ib)

arcsinh(a+ib)
arccosh(a+ib)
arctanh(a+ib)
arcsech(a+ib)
arccsch(a+ib)
arccoth(a+ib)

round(a+ib, n)  - round 'a' and 'b' to n decimal places

The operation rules are the same for real numbers

The result may be a complex number, INFINITY or indeterminate form.
We may use the division by 0 to generate the infinity.

We need to use brackets when we have powers of powers

(1-i2)^(1-i)^(i2) gives "Complex power too complex- use brackets"

We need to do ((1-i2)^(1-i))^(i2) or (1-i2)^((1-i)^(i2)).

Any complex number is a number with a real part 'a'
and an imaginary part 'b'. For instance, z1=-2+i3 or z1=i3-2.

We may define it typing z1=(-2,3); or z1=-2+i3 for example.

To build a complex expression
we may define a set of real/complex constants z1, z2, z3, z4, ...
and then we write the expression

For instance,

(1-i2)*(2-i3)  or (a-ib)*(c-id) where a=12;b=-4;c=1;d=-3;
are real constants

z1+z2*z3-z4   or  z5=z1+z2*z3-z4  or z1=z1+z2*(z3/z4)+3-i2

After defining any constant/matrix we may type the constant name
to see its value. We may use also list c or list r or list l or
list.

Example 1

Type:   d=7;c=1;b=-3;a=2;    press ENTER/COMPUTE
and     (a-ib)*(c-id)        press ENTER/COMPUTE
and     Z1=(a-ib)*(c-id)     press ENTER/COMPUTE

We get:      23.0-i11.0
| Z |=  25.495097567963924
Angle= -25.559965171823808 degrees
Angle= -0.4461055489434036 radians
Complex Constant defined

Example 2

conj(-1-i2)+inv(1-i3)

-0.9+i2.3
| Z |=  2.4698178070456938
Angle=  111.370622269343183 degrees
Angle=  1.9437840485949576 radians

Example 3

zx=arctan(-1-i3.3)+sin(1.2+i3e-2)*(i*cos(1-i))^(2-i)

-1.2156119789977935-i18.15086513723519
| Z |=  18.191525986392202
Angle= -93.831529845398134 degrees
Angle= -1.6376691379855234 radians
Complex Constant defined

Example 4

zy=(ln(zx)+1-i)* (ln(zx-1)+1+i)+e^(9.1-i2.3e-1)  ; edit: ')' was not here

8732.8791809611277-i2054.6089708474515
| Z |=  8971.3207953094131
Angle= -13.2393627831433264 degrees
Angle= -0.231070471431851 radians
Complex Constant defined

Example 5

zz=zx*zy-zy/zx
-47989.335119512577-i156498.23124802209
| Z |=  163690.78370199185
Angle= -107.0478745161774 degrees
Angle= -1.8683378675690276 radians
Complex Constant defined

Example 6

zw=(e+i pi)*(pi-ie)/((ipi)* ie)

-2.0-i0.2904713703586566
| Z |=  2.0209833292231868
Angle= -171.73638751817216 degrees
Angle= -2.9973654076729973 radians
Complex Constant defined

Example 7

zr=round( (e+i pi)*(pi-ie)/((ipi)* ie), 3)

-2.0-i0.29
| Z |=  2.0209156340629363
Angle= -171.7496127710945 degrees
Angle= -2.9975962318809012 radians
Complex Constant defined

If you know any bug or something else, please post it

If you have any suggestion, please tell me

Jul. 2013

Good luck !
Thanks
Rui Loureiro
« Last Edit: July 30, 2013, 07:08:49 AM by RuiLoureiro »

#### guga

• Member
• Posts: 1019
• Assembly is a state of art.
##### Re: The calculator
« Reply #37 on: October 27, 2013, 10:43:25 AM »
Trabalho excelente, rui.

Me ajudou muito há pouco quando estava testando uma função de atan2.
Coding in Assembly requires a mix of:
80% of brain, passion, intuition, creativity
10% of programming skills
10% of alcoholic levels in your blood.

My Code Sites:
http://rosasm.freeforums.org
http://winasm.tripod.com

#### RuiLoureiro

• Member
• Posts: 819
##### Re: The calculator
« Reply #38 on: November 09, 2013, 05:06:21 AM »
Olá guga,
obrigado !
Hi all,

(1) The previous The Calculator v3.10.1 (calcula59.exe)
dosn't solve trigonometric functions,
when we are working with angles in DEGREES:

sind(a+ib)
cosd(a+ib)
tand(a+ib)
secd(a+ib)
cscd(a+ib)
cotd(a+ib)

arcsind(a+ib)
arccosd(a+ib)
arctand(a+ib)
arcsecd(a+ib)
arccscd(a+ib)
arccotd(a+ib)

If we try, we get "Complex expression error"

(2) The results of some complex trigonometric functions
are not correct
because there is a bug in one internal procedure
(i wanted «fstp OperandZ» but i did «fstp OperandI» !!!)

(3) there is a problem when we try to define variables x1, x2, x3, etc

All these problems are solved in
the NEXT version v3.10.2 (calcula60.exe)
that i will post soon.

--------------------------------------------------------------------
These are some values that we get, when we use the
next version v3.10.2 (calcula60.exe)
--------------------------------------------------------------------
--------------------------------------
about trigonometric functions
inverse trigonometric functions
RADIANS
--------------------------------------
Code: [Select]
`TYPE:   z1=pi/4+i3WE GET:0.7853981633974483+i3.0| Z |=  3.101104686247803Angle=  75.329256947468645 degreesAngle=  1.314743556814141 radiansComplex Constant definedTYPE:   z2=sin(z1)WE GET:7.1189120679085489+i7.0837072942502342| Z |=  10.0427993139974647Angle=  44.857978252688267 degreesAngle=  0.7829194162974231 radiansComplex Constant definedTYPE:   w1=arcsin(z2)WE GET:0.7853981633974483+i3.0                 ->>>> is = z1 (correct)| Z |=  3.101104686247803Angle=  75.329256947468645 degreesAngle=  1.314743556814141 radiansComplex Constant definedTYPE:   z2=cos(z1)WE GET:7.1189120679085491-i7.0837072942502341| Z |=  10.0427993139974647Angle= -44.857978252688266 degreesAngle= -0.7829194162974231 radiansComplex Constant definedTYPE:   w1=arccos(z2)WE GET:0.7853981633974483+i3.0                 ->>>> is = z1 (correct)| Z |=  3.101104686247803Angle=  75.329256947468645 degreesAngle=  1.314743556814141 radiansComplex Constant definedTYPE:   z2=tan(z1)WE GET:0.0049574738935603+i0.9999877116507955| Z |=  1.0Angle=  89.715956505376533 degreesAngle=  1.5658388325948463 radiansComplex Constant definedTYPE:   w1=arctan(z2)WE GET:0.7853981633974411+i3.0000000000000079  ->>>> is = z1 (correct)| Z |=  3.1011046862478089Angle=  75.32925694746881 degreesAngle=  1.3147435568141439 radiansComplex Constant definedTYPE:   z2=csc(z1)WE GET:0.0705836414544171-i0.0702345879617375| Z |=  0.099573830834817Angle= -44.857978252688267 degreesAngle= -0.7829194162974231 radiansComplex Constant definedTYPE:   w1=arccsc(z2)WE GET:0.7853981633974483+i3.0000000000000004  ->>>> is = z1 (correct)| Z |=  3.1011046862478034Angle=  75.329256947468646 degreesAngle=  1.3147435568141411 radiansComplex Constant definedTYPE:   z2=sec(z1)WE GET:0.0705836414544171+i0.0702345879617375| Z |=  0.099573830834817Angle=  44.857978252688266 degreesAngle=  0.7829194162974231 radiansComplex Constant definedTYPE:   w1=arcsec(z2)WE GET:0.7853981633974482+i3.0000000000000004  ->>>> is = z1 (correct)| Z |=  3.1011046862478034Angle=  75.329256947468647 degreesAngle=  1.3147435568141411 radiansComplex Constant definedTYPE:   z2=cot(z1)WE GET:0.0049574738935603-i0.9999877116507955| Z |=  1.0Angle= -89.715956505376533 degreesAngle= -1.5658388325948463 radiansComplex Constant definedTYPE:   w1=arccot(z2)WE GET:0.7853981633974554+i3.0000000000000079  ->>>> is = z1 (correct)| Z |=  3.1011046862478125Angle=  75.329256947468554 degreesAngle=  1.3147435568141395 radiansComplex Constant defined`
--------------------------------------
about trigonometric functions
inverse trigonometric functions
RADIANS
--------------------------------------
Code: [Select]
`--------------First quadrant--------------z1=pi/4+i3    = 0.7853981633974483+i3.0y1=sin(z1)    = 7.1189120679085489+i7.0837072942502342w1=arcsin(y1) = 0.7853981633974483+i3.0 = z1y1=cos(z1)    = 7.1189120679085491-i7.0837072942502341w1=arccos(y1) = 0.7853981633974483+i3.0 = z1y1=tan(z1)    = 0.0049574738935603+i0.9999877116507955w1=arctan(y1) = 0.78539816339744 11+i3.00000000000000 79 = z1---------------Second quadrant---------------z2=-pi/4+i3   = -0.7853981633974483+i3.0y2=sin(z2)    = -7.1189120679085489+i7.0837072942502342w2=arcsin(y2) = -0.7853981633974483+i3.0 = z2y2=cos(z2)    = 7.1189120679085491+i7.0837072942502341w2=arccos(y2) = -0.7853981633974483+i3.0 = z2y2=tan(z2)    = -0.0049574738935603+i0.9999877116507955w2=arctan(y2) = -0.78539816339744 11+i3.00000000000000 79 = z2--------------Third quadrant--------------z3=-pi/4-i3 = -z1y3=sin(z3)    = -7.1189120679085489-i7.0837072942502342w3=arcsin(y3) = -0.7853981633974483-i3.0 = z3y3=cos(z3)    = 7.1189120679085491-i7.0837072942502341w3=arccos(y3) =  0.7853981633974483+i3.0 = z1y3=tan(z3)    = -0.0049574738935603-i0.9999877116507955w3=arctan(y3) = -0.78539816339744 11-i3.00000000000000 79 = z3---------------Fourth quadrant---------------z4=pi/4-i3  = -z2y4=sin(z4)    = 7.1189120679085489-i7.0837072942502342w4=arcsin(y4) = 0.7853981633974483-i3.0 = z4y4=cos(z4)    = 7.1189120679085491+i7.0837072942502341w4=arccos(y4) = -0.7853981633974483+i3.0 = z2y4=tan(z4)    = 0.0049574738935603-i0.9999877116507955w4=arctan(y4) = 0.78539816339744 11-i3.00000000000000 79 = z4`
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about trigonometric functions
inverse trigonometric functions
DEGREES
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Code: [Select]
`TYPE:   z1=45+i3WE GET:| Z |=  45.099889135118725Angle=  3.8140748342903542 degreesAngle=  0.0665681637758238 radiansComplex Constant definedTYPE:   z2=sind(z1)WE GET:7.118912067908549+i7.0837072942502342| Z |=  10.0427993139974647Angle=  44.857978252688266 degreesAngle=  0.7829194162974231 radiansComplex Constant definedTYPE:   w1=arcsind(z2)WE GET:45.0+i3.0                           ->>>> is = z1 (correct)| Z |=  45.099889135118724Angle=  3.8140748342903542 degreesAngle=  0.0665681637758238 radiansComplex Constant definedTYPE:   z2=cosd(z1)WE GET:7.118912067908549-i7.0837072942502342| Z |=  10.0427993139974647Angle= -44.857978252688266 degreesAngle= -0.7829194162974231 radiansComplex Constant definedTYPE:   w1=arccosd(z2)WE GET:45.0+i3.0                           ->>>> is = z1 (correct)| Z |=  45.099889135118725Angle=  3.8140748342903542 degreesAngle=  0.0665681637758238 radiansComplex Constant definedTYPE:   z2=tand(z1)WE GET:0.0049574738935603+i0.9999877116507955| Z |=  1.0Angle=  89.715956505376533 degreesAngle=  1.5658388325948463 radiansComplex Constant definedTYPE:   w1=arctand(z2)WE GET:44.99999999999959+i3.0000000000000079   ->>>> is = z1 (correct)| Z |=  45.099889135118316Angle=  3.8140748342903989 degreesAngle=  0.0665681637758245 radiansComplex Constant definedTYPE:   z2=cscd(z1)WE GET:0.0705836414544171-i0.0702345879617375| Z |=  0.099573830834817Angle= -44.857978252688266 degreesAngle= -0.7829194162974231 radiansComplex Constant definedTYPE:   w1=arccscd(z2)WE GET:45.000000000000002+i3.0000000000000004  ->>>> is = z1 (correct)| Z |=  45.099889135118727Angle=  3.8140748342903545 degreesAngle=  0.0665681637758238 radiansComplex Constant definedTYPE:   z2=secd(z1)WE GET:0.0705836414544171+i0.0702345879617375| Z |=  0.099573830834817Angle=  44.857978252688266 degreesAngle=  0.7829194162974231 radiansComplex Constant definedTYPE:   w1=arcsecd(z2)WE GET:44.999999999999998+i3.0000000000000004  ->>>> is = z1 (correct)| Z |=  45.099889135118723Angle=  3.8140748342903549 degreesAngle=  0.0665681637758238 radiansComplex Constant definedTYPE:   z2=cotd(z1)WE GET:0.0049574738935603-i0.9999877116507955| Z |=  1.0Angle= -89.715956505376533 degreesAngle= -1.5658388325948463 radiansComplex Constant definedTYPE:   w1=arccotd(z2)WE GET:45.00000000000041+i3.0000000000000079   ->>>> is = z1 (correct)| Z |=  45.099889135119134Angle=  3.8140748342903296 degreesAngle=  0.0665681637758233 radiansComplex Constant defined`
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about hyperbolic functions
inverse hyperbolic functions
---------------------------------
Code: [Select]
`TYPE:   z1=2.3+i0.25WE GET: 2.3+i0.25| Z |=  2.3135470602518549 Angle=  6.2034479016918352 degreesAngle=  0.10827059086045604 radiansComplex Constant definedTYPE:   z2=sinh(z1)WE GET:4.783483618904492+i1.2462283322677304| Z |=  4.9431569455636773Angle=  14.6025145321957282 degreesAngle=  0.2548619576571349 radiansComplex Constant definedTYPE:   w1=arcsinh(z2)WE GET:2.3+i0.25                               ->>>> is = z1 (correct)| Z |=  2.3135470602518549Angle=  6.2034479016918353 degreesAngle=  0.10827059086045604 radiansComplex Constant definedTYPE:   z2=cosh(z1)WE GET:4.8806256579738629+i1.2214238973804283| Z |=  5.0311413367510157Angle=  14.0502468762936514 degreesAngle=  0.2452230687093726 radiansComplex Constant definedTYPE:   w1=arccosh(z2)WE GET:2.3+i0.25                               ->>>> is = z1 (correct)| Z |=  2.3135470602518549Angle=  6.2034479016918353 degreesAngle=  0.10827059086045604 radiansComplex Constant definedTYPE:   z2=tanh(z1)WE GET:0.9824664000943306+i0.0094701778128957| Z |=  0.9825120414438294Angle=  0.5522676559020767 degreesAngle=  0.0096388889477623 radiansComplex Constant definedTYPE:   w1=arctanh(z2)WE GET:2.3000000000000002+i0.249999999999998   ->>>> is = z1 (correct)| Z |=  2.3135470602518549Angle=  6.2034479016917859 degreesAngle=  0.10827059086045518 radiansComplex Constant definedTYPE:   z2=csch(z1)WE GET:0.1957651997848019-i0.0510021895924765| Z |=  0.2022998684873777Angle= -14.6025145321957282 degreesAngle= -0.2548619576571349 radiansComplex Constant definedTYPE:   w1=arccsch(z2)WE GET:2.3000000000000003+i0.2499999999999997  ->>>> is = z1 (correct)| Z |=  2.3135470602518552Angle=  6.2034479016918292 degreesAngle=  0.10827059086045593 radiansComplex Constant definedTYPE:   z2=sech(z1)WE GET:0.1928157309006095-i0.0482540063543098| Z |=  0.1987620567713517Angle= -14.0502468762936514 degreesAngle= -0.2452230687093726 radiansComplex Constant definedTYPE:   w1=arcsech(z2)WE GET:2.3000000000000005+i0.2499999999999998  ->>>> is = z1 (correct)| Z |=  2.3135470602518554Angle=  6.2034479016918313 degreesAngle=  0.10827059086045597 radiansComplex Constant definedTYPE:   z2=coth(z1)WE GET:1.017751950195062-i0.0098103018452776| Z |=  1.0177992307661405Angle= -0.5522676559020767 degreesAngle= -0.0096388889477623 radiansComplex Constant definedTYPE:   w1=arccoth(z2)WE GET:2.3000000000000003+i0.2499999999999993  ->>>> is = z1 (correct)| Z |=  2.3135470602518551Angle=  6.2034479016918193 degreesAngle=  0.10827059086045576 radiansComplex Constant defined`
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hyperbolic functions
inverse hyperbolic functions
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Code: [Select]
`TYPE:   z1=2.3+i1.58 = Z + iI  ( I > pi/2)WE GET: 2.3+i1.58| Z |=  2.790412155936825Angle=  34.487372830078654 degreesAngle=  0.6019182062477074 radians Complex Constant definedTYPE:   y1=sinh(z1)WE GET:-0.045437541593796+i5.0370073053376468| Z |=  5.0372122413306062Angle=  90.516836405985953 degreesAngle=  1.5798168237735256 radiansComplex Constant definedTYPE:   w1=arcsinh(y1)WE GET:-2.3+i1.5615926535897932     -> is not = z1| Z |=  2.7800308659699467Angle=  145.82532569791053 degreesAngle=  2.5451320662216368 radiansComplex Constant defined`
---------------------------------
about hyperbolic functions
inverse hyperbolic functions
---------------------------------
Code: [Select]
`--------------First quadrant--------------z1=2.3+i0.25y1=sinh(z1)    = 4.783483618904492+i1.2462283322677304 w1=arcsinh(y1) =  2.3+i0.25 = z1y1=cosh(z1)    = 4.8806256579738629+i1.2214238973804283 w1=arccosh(y1) = 2.3+i0.25 = z1y1=tanh(z1)    = 0.9824664000943306+i0.0094701778128957w1=arctanh(y1) = 2.3000000000000002+i0.249999999999998 = z1---------------Second quadrant---------------z2=-2.3+i0.25  = -z4y2=sinh(z2)    = -4.783483618904492+i1.2462283322677304w2=arcsinh(y2) = -2.3+i0.25 = z2y2=cosh(z2)    = 4.8806256579738629-i1.2214238973804283w2=arccosh(y2) = 2.3-i0.25 = z4y2=tanh(z2)    = -0.9824664000943306+i0.0094701778128957w2=arctanh(y2) = -2.3000000000000002+i0.249999999999998 = z2--------------Third quadrant--------------z3=-2.3-i0.25 = -z1y3=sinh(z3)    = -4.783483618904492-i1.2462283322677304w3=arcsinh(y3) = -2.3-i0.25 = z3y3=cosh(z3)    = 4.8806256579738629+i1.2214238973804283w3=arccosh(y3) = 2.3+i0.25 = z1y3=tanh(z3)    = -0.9824664000943306-i0.0094701778128957w3=arctanh(y3) = -2.3000000000000002-i0.249999999999998 = z3---------------Fourth quadrant---------------z4=2.3-i0.25  = -z2y4=sinh(z4)    = 4.783483618904492-i1.2462283322677304 w4=arcsinh(y4) = 2.3-i0.25 = z4y4=cosh(z4)    = 4.8806256579738629-i1.2214238973804283w4=arccosh(y4) = 2.3-i0.25 = z4y4=tanh(z4)    = 0.9824664000943306-i0.0094701778128957w4=arctanh(y4) = 2.3000000000000002-i0.249999999999998 = z4`
« Last Edit: November 10, 2013, 06:05:13 AM by RuiLoureiro »

#### RuiLoureiro

• Member
• Posts: 819
##### Re: The calculator
« Reply #39 on: November 10, 2013, 05:34:30 AM »
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about cos(w)=z and the inverse function w=arccos(z)
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Code: [Select]
`By definition                cos(w)= (e^iw+e^-iw)/2 = z   => w=arccos(z)    From                     e^iw+e^-iw                 z=--------------   doing t=e^iw                            2    we get                         1                2z= t + ---  <=>   2z t = t^2 + 1  ( t<>0 )                         t                <=> t^2 -2z t + 1 = 0                <=>   t= z + SQRT(z^2-1)        - positive solution I                   or                       t= z - SQRT(z^2-1)         - negative solution I    From            SQRT(z^2-1)= SQRT[(-1).(1-z^2)]= i SQRT(1-z^2)    we have                      t= z + i SQRT(1-z^2)      - positive solution II                   or                       t= z - i SQRT(1-z^2)      - negative solution II    From             t=e^iw  we get w=-i ln(t)    Solutions:                w=arccos(z)= -i ln(z +  SQRT(z^2-1))       - positive solution I            w=arccos(z)= -i ln(z -  SQRT(z^2-1))       - negative solution I            w=arccos(z)= -i ln(z +i SQRT(1-z^2))       - positive solution II            w=arccos(z)= -i ln(z -i SQRT(1-z^2))       - negative solution II    Because cos(w)=z and cos(-w)=z            arccos(z)= w or -w`
Here are some results
Code: [Select]
`-----------------------------------The calculator negative solution Iarccos(z) = -i * ln(z - sqrt(z^2-1)-----------------------------------z1=pi/4+i3 = 0.7853981633974483+i3 = -z3y1=cos(z1)=  7.1189120679085491-i7.0837072942502341w1=arccos(y1)= pi/4+i3 = z1z2=-pi/4+i3 = -z4y2=cos(z2)= 7.1189120679085491+i7.0837072942502341w2=arccos(y2)= -pi/4+i3 = z2z3=-pi/4-i3 = -z1y3=cos(z3)= 7.1189120679085491-i7.0837072942502341w3=arccos(y3)= pi/4+i3 = z1z4=pi/4-i3 = -z2y4=cos(z4)= 7.1189120679085491+i7.0837072942502341w4=arccos(y4)= -pi/4+i3 = z2------------------------------------The calculator positive solution Iarccos(z) = -i * ln(z + sqrt(z^2-1))------------------------------------z1=pi/4+i3 = 0.7853981633974483+i3 = -z3y1=cos(z1)=  7.1189120679085491-i7.0837072942502341w1=arccos(y1)= -pi/4-i3 = z3z2=-pi/4+i3 = -z4y2=cos(z2)= 7.1189120679085491+i7.0837072942502341w2=arccos(y2)= pi/4-i3 = z4z3=-pi/4-i3 = -z1y3=cos(z3)= 7.1189120679085491-i7.0837072942502341w3=arccos(y3)= -pi/4-i3 = z3z4=pi/4-i3 = -z2y4=cos(z4)= 7.1189120679085491+i7.0837072942502341w4=arccos(y4)= pi/4-i3 = z4-------------------------------------The calculator positive solution IIarccos(z) = -i * ln(z +i sqrt(1-z^2))-------------------------------------z1=pi/4+i3 = 0.7853981633974483+i3 = -z3y1=cos(z1)=  7.1189120679085491-i7.0837072942502341w1=arccos(y1)= pi/4+i3 = z1z2=-pi/4+i3 = -z4y2=cos(z2)= 7.1189120679085491+i7.0837072942502341w2=arccos(y2)= pi/4-i3 = z4z3=-pi/4-i3 = -z1y3=cos(z3)= 7.1189120679085491-i7.0837072942502341w3=arccos(y3)= pi/4+i3 = z1z4=pi/4-i3 = -z2y4=cos(z4)= 7.1189120679085491+i7.0837072942502341w4=arccos(y4)= pi/4-i3 = z4-------------------------------------The calculator negative solution IIarccos(z) = -i * ln(z - i sqrt(1-z^2)-------------------------------------z1=pi/4+i3 = 0.7853981633974483+i3 = -z3y1=cos(z1)=  7.1189120679085491-i7.0837072942502341w1=arccos(y1)=-pi/4-i3 = z3z2=-pi/4+i3 = -z4y2=cos(z2)= 7.1189120679085491+i7.0837072942502341w2=arccos(y2)= -pi/4+i3 = z2z3=-pi/4-i3 = -z1y3=cos(z3)= 7.1189120679085491-i7.0837072942502341w3=arccos(y3)=-pi/4-i3 = z3z4=pi/4-i3 = -z2y4=cos(z4)= 7.1189120679085491+i7.0837072942502341w4=arccos(y4)= -pi/4+i3 = z2arccos(z) = +w or -w    because cos(w)=z and cos(-w)=z`
From these results,
the calculator uses the positive solution II.
Code: [Select]
`            w=arccos(z)= -i ln(z +i SQRT(1-z^2))    - positive solution II`
Here some more results
Code: [Select]
`--------------------------------------The calculator negative solutionarccos(z) = -i * ln(z - sqrt(z^2-1))--------------------------------------x1=pi/4x2=3 pi/4z1=x1+ix1    = 0.7853981633974483+i0.7853981633974483y1=cos(z1)   = 0.9366400694314301-i0.6142431274865956w1=arccos(y1)= 0.7853981633974482+i0.7853981633974482  = z1z2=-x1+ix1   = -0.7853981633974483+i0.7853981633974483y2=cos(z2)   = 0.9366400694314301+i0.6142431274865956w2=arccos(y2)= -0.7853981633974482+i0.7853981633974482 = z2z3=-x1-ix1   = -0.7853981633974483-i0.7853981633974483 y3=cos(z3)   = 0.9366400694314301-i0.6142431274865956w3=arccos(y3)=  0.7853981633974482+i0.7853981633974482 = z1z4=x1-ix1    = 0.7853981633974483-i0.7853981633974483y4=cos(z4)   = 0.9366400694314301+i0.6142431274865956w4=arccos(y4)=-0.7853981633974482+i0.7853981633974482  = z2 ------------------------------------------------------------z1=x2+ix1    = 2.3561944901923449+i0.7853981633974483y1=cos(z1)   = -0.93664006943143-i0.6142431274865956w1=arccos(y1)= -2.3561944901923449-i0.7853981633974482 = z3z2=-x2+ix1   = -2.3561944901923449+i0.7853981633974483y2=cos(z2)   = -0.93664006943143+i0.6142431274865956w2=arccos(y2)= 2.3561944901923449-i0.7853981633974482  = z4z3=-x2-ix1   = -2.3561944901923449-i0.7853981633974483y3=cos(z3)   = -0.93664006943143-i0.6142431274865956w3=arccos(y3)= -2.3561944901923449-i0.7853981633974482 = z3z4=x2-ix1    = 2.3561944901923449-i0.7853981633974483 y4=cos(z4)   = -0.93664006943143+i0.6142431274865956w4=arccos(y4)= 2.3561944901923449-i0.7853981633974482  = z4--------------------------------------The calculator positive solution IIarccos(z) = -i * ln(z +i sqrt(1-z^2))--------------------------------------x1=  pi/4 = 0.7853981633974483x2=3 pi/4 = 2.3561944901923449x3=5 pi/4 = 3.9269908169872415x4=7 pi/4 = 5.4977871437821381z1=x1+ix1    = 0.7853981633974483+i0.7853981633974483y1=cos(z1)   = 0.9366400694314301-i0.6142431274865956w1=arccos(y1)= 0.7853981633974482+i0.7853981633974482 = z1z2=-x1+ix1   = -0.7853981633974483+i0.7853981633974483y2=cos(z2)   = 0.9366400694314301+i0.6142431274865956w2=arccos(y2)= 0.7853981633974482-i0.7853981633974482 = z4z3=-x1-ix1   = -0.7853981633974483-i0.7853981633974483 y3=cos(z3)   = 0.9366400694314301-i0.6142431274865956w3=arccos(y3)= 0.7853981633974482+i0.7853981633974482 = z1z4=x1-ix1    = 0.7853981633974483-i0.7853981633974483 y4=cos(z4)   = 0.9366400694314301+i0.6142431274865956w4=arccos(y4)= 0.7853981633974482-i0.7853981633974482 = z4------------------------------------------------------------z1=x2+ix1    = 2.3561944901923449+i0.7853981633974483y1=cos(z1)   = -0.93664006943143-i0.6142431274865956w1=arccos(y1)=  2.3561944901923449+i0.7853981633974482  =z1z2=-x2+ix1   = -2.3561944901923449+i0.7853981633974483y2=cos(z2)   = -0.93664006943143+i0.6142431274865956w2=arccos(y2)= 2.3561944901923449-i0.7853981633974482   =z4z3=-x2-ix1   = -2.3561944901923449-i0.7853981633974483 y3=cos(z3)   = -0.93664006943143-i0.6142431274865956w3=arccos(y3)= 2.3561944901923449+i0.7853981633974482  = z1z4=x2-ix1    = 2.3561944901923449-i0.7853981633974483 y4=cos(z4)   = -0.93664006943143+i0.6142431274865956w4=arccos(y4)= 2.3561944901923449-i0.7853981633974482  = z4 ------------------------------------------------------------z1=x1+ix2    = 0.7853981633974483+i2.3561944901923449y1=cos(z1)   = 3.7637541395008347-i3.6967343997925613w1=arccos(y1)= 0.7853981633974483+i2.3561944901923449   = z1 z2=-x1+ix2   = -0.7853981633974483+i2.3561944901923449y2=cos(z2)   = 3.7637541395008347+i3.6967343997925613w2=arccos(y2)= 0.7853981633974483-i2.3561944901923449   = z4z3=-x1-ix2   = -0.7853981633974483-i2.3561944901923449y3=cos(z3)   = 3.7637541395008347-i3.6967343997925613w3=arccos(y3)= 0.7853981633974483+i2.3561944901923449   = z1z4=x1-ix2    =  0.7853981633974483-i2.3561944901923449y4=cos(z4)   =  3.7637541395008347+i3.6967343997925613w4=arccos(y4)= 0.7853981633974483-i2.3561944901923449   = z4------------------------------------------------------------z1=x2+ix2    = 2.3561944901923449+i2.3561944901923449y1=cos(z1)   = -3.7637541395008346-i3.6967343997925614w1=arccos(y1)= 2.3561944901923449+i2.3561944901923449   = z1z2=-x2+ix2   = -2.3561944901923449+i2.3561944901923449y2=cos(z2)   = -3.7637541395008346+i3.6967343997925614w2=arccos(y2)=  2.3561944901923449-i2.3561944901923449  = z4z3=-x2-ix2   = -2.3561944901923449-i2.3561944901923449y3=cos(z3)   = -3.7637541395008346-i3.6967343997925614w3=arccos(y3)= 2.3561944901923449+i2.3561944901923449   = z1z4=x2-ix2    = 2.3561944901923449-i2.3561944901923449 y4=cos(z4)   = -3.7637541395008346+i3.6967343997925614w4=arccos(y4)= 2.3561944901923449-i2.3561944901923449   = z4-------------------------------------------------------------x1=  pi/4 = 0.7853981633974483x2=3 pi/4 = 2.3561944901923449x3=5 pi/4 = 3.9269908169872415x4=7 pi/4 = 5.4977871437821381z1=x3+ix1    =  3.9269908169872415+i0.7853981633974483y1=cos(z1)   = -0.9366400694314301+i0.6142431274865956w1=arccos(y1)=  2.356194490192345-i0.7853981633974482   = x2-ix1 z2=-x3+ix1   = -3.9269908169872415+i0.7853981633974483y2=cos(z2)   = -0.9366400694314301-i0.6142431274865956w2=arccos(y2)=  2.356194490192345+i0.7853981633974482   = x2+ix1z3=-x3-ix1   = -3.9269908169872415-i0.7853981633974483y3=cos(z3)   = -0.9366400694314301+i0.6142431274865956w3=arccos(y3)=  2.356194490192345-i0.7853981633974482   = x2-ix1z4=x3-ix1    =  3.9269908169872415-i0.7853981633974483y4=cos(z4)   = -0.9366400694314301-i0.6142431274865956w4=arccos(y4)=  2.356194490192345+i0.7853981633974482   = x2+ix1------------------------------------------------------------------z1=x3+ix2    =  3.9269908169872415+i2.3561944901923449y1=cos(z1)   = -3.7637541395008349+i3.6967343997925611w1=arccos(y1)=  2.356194490192345-i2.3561944901923449   = x2-ix2 z2=-x3+ix2   = -3.9269908169872415+i2.3561944901923449y2=cos(z2)   = -3.7637541395008349-i3.6967343997925611w2=arccos(y2)=  2.356194490192345+i2.3561944901923449   = x2+ix2z3=-x3-ix2   = -3.9269908169872415-i2.3561944901923449y3=cos(z3)   = -3.7637541395008349+i3.6967343997925611w3=arccos(y3)=  2.356194490192345-i2.3561944901923449   = x2-ix2z4=x3-ix2    =  3.9269908169872415-i2.3561944901923449y4=cos(z4)   = -3.7637541395008349-i3.6967343997925611w4=arccos(y4)=  2.356194490192345+i2.3561944901923449   = x2+ix2; -------------------------------------------------------------------x1=  pi/4 = 0.7853981633974483x2=3 pi/4 = 2.3561944901923449x3=5 pi/4 = 3.9269908169872415x4=7 pi/4 = 5.4977871437821381z1=x4+ix1    = 5.4977871437821381+i0.7853981633974483 = -z3y1=cos(z1)   = 0.93664006943143+i0.6142431274865956w1=arccos(y1)= 0.7853981633974483-i0.7853981633974482 = x1-ix1 z2=-x4+ix1   = -5.4977871437821381+i0.7853981633974483 = -z4y2=cos(z2)   =  0.93664006943143-i0.6142431274865956w2=arccos(y2)=  0.7853981633974483+i0.7853981633974482 = x1+ix1z3=-x4-ix1   = -5.4977871437821381-i0.7853981633974483 = -z1y3=cos(z3)   =  0.93664006943143+i0.6142431274865956w3=arccos(y3)=  0.7853981633974483-i0.7853981633974482  = x1-ix1z4=x4-ix1    = 5.4977871437821381-i0.7853981633974483  = -z2y4=cos(z4)   = 0.93664006943143-i0.6142431274865956w4=arccos(y4)= 0.7853981633974483+i0.7853981633974482  = x1+ix1----------------------------------------------------------------z1=x4+ix2    = 5.4977871437821381+i2.3561944901923449y1=cos(z1)   = 3.7637541395008344+i3.6967343997925616w1=arccos(y1)= 0.7853981633974483-i2.3561944901923449   =x1-ix2 z2=-x4+ix2   = -5.4977871437821381+i2.3561944901923449y2=cos(z2)   = 3.7637541395008344-i3.6967343997925616w2=arccos(y2)= 0.7853981633974483+i2.3561944901923449   =x1+ix2z3=-x4-ix2   = -5.4977871437821381-i2.3561944901923449y3=cos(z3)   = 3.7637541395008344+i3.6967343997925616w3=arccos(y3)= 0.7853981633974483-i2.3561944901923449   =x1-ix2z4=x4-ix2    = 5.4977871437821381-i2.3561944901923449y4=cos(z4)   = 3.7637541395008344-i3.6967343997925616w4=arccos(y4)= 0.7853981633974483+i2.3561944901923449   =x1+ix2------------------------------z1=x4+ix3    = 5.4977871437821381+i3.9269908169872415y1=cos(z1)   = 17.951221702159904+i17.937289667062213w1=arccos(y1)= 0.7853981633974484-i3.9269908169872415   =x1-ix3 z2=-x4+ix3   = -5.4977871437821381+i3.9269908169872415y2=cos(z2)   = 17.951221702159904-i17.937289667062213w2=arccos(y2)= 0.7853981633974484+i3.9269908169872415   =x1+ix3z3=-x4-ix3   = -5.4977871437821381-i3.9269908169872415y3=cos(z3)   =  17.951221702159904+i17.937289667062213w3=arccos(y3)= 0.7853981633974484-i3.9269908169872415   =x1-ix3z4=x4-ix3    = 5.4977871437821381-i3.9269908169872415y4=cos(z4)   = 17.951221702159904-i17.937289667062213w4=arccos(y4)= 0.7853981633974484+i3.9269908169872415   =x1+ix3----------------------------------------------------------------x1=  pi/4 = 0.7853981633974483x2=3 pi/4 = 2.3561944901923449x3=5 pi/4 = 3.9269908169872415x4=7 pi/4 = 5.4977871437821381z1=x4+ix4    = 5.4977871437821381+i5.4977871437821381y1=cos(z1)   = 86.321884181857327+i86.318987996303526w1=arccos(y1)=  0.7853981633974483-i5.4977871437821381  =x1-ix4 z2=-x4+ix4   = -5.4977871437821381+i5.4977871437821381y2=cos(z2)   = 86.321884181857327-i86.318987996303526w2=arccos(y2)= 0.7853981633974483+i5.4977871437821383   =x1+ix4z3=-x4-ix4   = -5.4977871437821381-i5.4977871437821381y3=cos(z3)   = 86.321884181857327+i86.318987996303526w3=arccos(y3)= 0.7853981633974483-i5.4977871437821381   =x1-ix4z4=x4-ix4    = 5.4977871437821381-i5.4977871437821381y4=cos(z4)   = 86.321884181857327-i86.318987996303526w4=arccos(y4)= 0.7853981633974483+i5.4977871437821383   =x1+ix4`

#### Gunther

• Member
• Posts: 3585
• Forgive your enemies, but never forget their names
##### Re: The calculator
« Reply #40 on: November 10, 2013, 05:38:29 AM »
Hi Rui,

only one question: What algorithm do you use by solving a linear equation system?

Gunther
Get your facts first, and then you can distort them.

#### RuiLoureiro

• Member
• Posts: 819
##### Re: The calculator
« Reply #41 on: November 10, 2013, 05:46:50 AM »
Hi Rui,
only one question: What algorithm do you use by solving a linear equation system?
Gunther

Hi Gunther

Where is the linear equation system ?

Do you want to say, for example:
a X+ bY=c and dX+eY=f  ? etc.
I use matrices.

#### dedndave

• Member
• Posts: 8821
• Still using Abacus 2.0
##### Re: The calculator
« Reply #42 on: November 10, 2013, 06:20:54 AM »
Hi Rui,

the standard form for a linear equation is called the "slope-intercept" form:

Y=mX+b

m is the slope (=rise/run=(Y2-Y1)/(X2-X1))
b is the y-intercept (the value of Y when X=0)

many calculators let you enter 2 (X,Y) points (some allow m and b)
then, you can enter an X or Y value, and it will spit out the opposite

nearly any straight line can be described using the slope-intercept form
however, it gets a little tricky as the slope approaches infinity
those are lines that graph stright up and down, like the line: X=0

#### RuiLoureiro

• Member
• Posts: 819
##### Re: The calculator
« Reply #43 on: November 10, 2013, 07:49:32 AM »
Hi Rui,

the standard form for a linear equation is called the "slope-intercept" form:

Y=mX+b

m is the slope (=rise/run=(Y2-Y1)/(X2-X1))
b is the y-intercept (the value of Y when X=0)

many calculators let you enter 2 (X,Y) points (some allow m and b)
then, you can enter an X or Y value, and it will spit out the opposite

nearly any straight line can be described using the slope-intercept form
however, it gets a little tricky as the slope approaches infinity
those are lines that graph stright up and down, like the line: X=0
Hi Dave,

i know all about that
Gunther asked for linear equation system

#### Gunther

• Member
• Posts: 3585
• Forgive your enemies, but never forget their names
##### Re: The calculator
« Reply #44 on: November 10, 2013, 07:57:45 AM »
Rui,

Do you want to say, for example:
a X+ bY=c and dX+eY=f  ? etc.
I use matrices.

that's clear. The direction of my question was: Do you use the Gaussian elimination?

Gunther
Get your facts first, and then you can distort them.