# Sample calculation

Index
Real number calculation using advanced calculator.   Select "Real DEG" from Mode.
Example Calculation contents
Digit
mode
Arithmetical operation 10+20-30*4/6 [Execute] → 10 6
Calculation with parenthesis (2+3)*4 [Execute] → 20 6
Power calculation 3*4^2 [Execute] → 48 6
Calculation with Exponent 99*1.23E+3[Execute]→121770 6
Use of function sqrt(4) [Execute] → 2 6
Use of variable sin(x) [Execute]
(variable x) 456 [Execute] → 0.994522
6
Real
DEG
Change digit number cos(12345) [Execute] → -0.258819
[Digit 22] [Execute] → -0.2588190451025207623489
6

22
Change of calculation mode cos(123456) [Execute] → 0.9135454576426008955021
DEG

Repetition calculation

x= 1,0.1,10
1: initial value
0.1: incremental value
10: repeat count
sin(x) [Execute]
(variable x) 1, 0.1,10 [Execute]
1      0.8414709848078965066525
1.1   0.8912073600614353399518
1.2   0.9320390859672263496701
・・・
1.9   0.9463000876874144884897
22
Real

Example Calculation contents
Digit
mode
Arithmetical operation (3+4i)*(5+6i) [Execute] → -9 +38i 6
Complex
Elementary function sin(1+i) [Execute]→ 1.29846 +0.634964i
ln(-5) [Execute]→ 1.60944 +3.14159i
6
Complex
Special function gamma(2+3i) [Execute]→ -0.082395 +0.091774i 6
Complex

Example
formula and variable
Mode Chart drawing and setting
Square root

sqrt(x)
x=0,0.1,51
6
Real
DEG Sheet: width =190, height =170
X-label step: 10 per plot
Legend: down
Sine and cosine

sin(x);cos(x)
x= 0,5,101
6
Real
DEG Sheet: width =200, height =170
Y axis: Max=1, Min=-1
X-label step: 18 per plot
Legend: down
Normal distribution

normalpd(x)
x=-5,0.1,101
6
Real
DEG Sheet: width =190, height =170
Y axis: Max=0.5, Min=0
X-label step: 10 per plot
Legend: down
Bessel function
J0(x), Y0(x)

besselj(0, x);
bessely(0, x)
x=0,0.1,101
6
Real
DEG Sheet: width =190, height =180
Y axis: Max=1, Min=-1
Legend: down

High Accuracy calculation. keisan aims to provide higher calculation accuracy using variable number of digits.
Example Mode Keisan (6 digit)
Excel2003 (16 digit)
Decimal calculation
0.4-0.3-0.1
6
Real
0.4-0.3-0.1 =0
(0.4-0.3-0.1) =2.77556E-17
Error is caused by binary-to-decimal conversion.
π calculation
sin(9π)
6
Real
sin(9*pi) =0
SIN(9*PI()) =1.10263E-15
Error is caused by numerical calculation of irrational number.
Irrational number calculation
9√2-√162
6
Real
9*sqrt(2)-sqrt(162) =0
(9*SQRT(2)-SQRT(162)) =1.77636E-15
Error is caused by numerical calculation of irrational number.
Complex number calculation
(1+i)4
6
Complex
(1+i)^4 =-4
IMPOWER("1+i", 4) =-4+ 4.89842541528951E-16i
Square root calculation
√(1+x)-1
x=1.23456E-100
6
Real
sqrt(1+1.23456E-100) -1 =6.1728E-101
SQRT(1+1.23456E-100) - 1 =0
In case of x <<1, Excel shows no precision.
Exponential calculation
eln(1+x)-1
x=1.23456E-14
6
Real
e^ln(1+1.23456E-14) -1 =1.23456E-14
EXP(LN(1+1.23456E-14))- 1 =1.24345E-14
In case of x≒1E-14, Excel shows low precision.
Exponential calculation
eln(1+x)-1
x=1.23456E-15
6
Real
e^ln(1+1.23456E-15) -1 =1.23456E-15
EXP(LN(1+1.23456E-15))- 1 =0
In case of x <1E-15, Excel shows no precision.
Kinetic energy of human body
E=mc2(1-√(1-v2/c2))
m=80kg
v=100/9.77 m/s
c=299792458 m/s
6
Real 