Julia, the giant calculator

Simple arithmetic

Addition, subtraction, multiplication, division and powers by arithmetic operators



In [9]:

    
#Addition using +
2 + 2









    Out[9]:





4



In [10]:

    
#Subtraction using -
3 - 4









    Out[10]:





-1



In [11]:

    
#Multiplication using *
3 * 4









    Out[11]:





12



In [12]:

    
#Division using /
12 / 4









    Out[12]:





3.0



In [13]:

    
#Powers using ^
2^4









    Out[13]:





16

Simple arithmetic using in-built functions, as well as fractions



In [15]:

    
#typeof() (we will learn more about types in an upcoming lesson)
typeof(3.0)









    Out[15]:





Float64



In [16]:

    
#typeof(+)
typeof(+)









    Out[16]:





Function



In [18]:

    
#Addition using +()
+(2, 2, 4, 6, 3, 4, 7)









    Out[18]:





28



In [19]:

    
#Subtraction using -()
-(3, 4)









    Out[19]:





-1



In [20]:

    
#Multiplication using *()
*(3, 4)









    Out[20]:





12



In [21]:

    
#Division using /()
/(12, 4)









    Out[21]:





3.0



In [22]:

    
#Powers using ^()
^(2, 4)









    Out[22]:





16



In [23]:

    
#Fractions using //
 3 // 4









    Out[23]:





3//4



In [24]:

    
#The numerator num()
num(3 // 4)









    Out[24]:





3



In [25]:

    
#The denominator den()
den(3 // 4)









    Out[25]:





4



In [26]:

    
#Simplification using fractions, ie.e 6 // 9
6 // 9









    Out[26]:





2//3

The remainder is what is left after division, as in $ \frac{5}{3} = 1 \frac{2}{3} $
The $ 1 $ is the floor and the $ 2 $ is the remainder



In [27]:

    
#Calculating the remainder of division using rem()
rem(5, 3)









    Out[27]:





2



In [28]:

    
#Calculating both the floor and the remainder using divrem()
divrem(5, 3)









    Out[28]:





(1,2)

Association

The order of operations are
- .
- ::
- ^ (and its element-by-elemnt equivalent .^
- // and .//
- - and / and %
- Bitshifts
- - and -
- Syntax such as : .. |>
- Comparisons
- Control flow
- Assignments



In [29]:

    
#Forcing order through the use of parentheses
2 - 4 * 3









    Out[29]:





-10



In [30]:

    
#Now with parentheses
(2 - 4) * 3









    Out[30]:





-6

Rounding numbers



In [31]:

    
#Rounding to the nearest integer using the round() function
#Example using 3.4
round(3.4)









    Out[31]:





3.0



In [32]:

    
#To the nearest integer using the round() function
#Example using 3.5
round(3.5)









    Out[32]:





4.0



In [33]:

    
#Towards positive infinity using the ceil() function
#Example using 3.001
ceil(3.001)









    Out[33]:





4.0



In [34]:

    
#Towards positive infinity using the ceil() function
#Example using -3.999
ceil(-3.999)









    Out[34]:





-3.0



In [35]:

    
#Towards negative infinity using floor()
#Example using 3.999
floor(3.999)









    Out[35]:





3.0



In [36]:

    
#Towards negative infinity using the floor() function
#Example using -3.001
floor(-3.001)









    Out[36]:





-4.0



In [37]:

    
#Towards zero using the trunc() function
#Example using 3.999
trunc(3.999)









    Out[37]:





3.0



In [38]:

    
#Towards zero using the trunc() function
#Example using -3.999
trunc(-3.999)









    Out[38]:





-3.0

Greatest common divisor and least common multiple



In [39]:

    
#Finding the greatest common divisor using the gcd() function
#What is the greatest numbers that will divide into 4, 8, and 20?
gcd(4, 8, 20)









    Out[39]:





4



In [40]:

    
#Finding the least common multiple using the lcm() function
#What is the least common multiple of 2, 3, 5, 10 (takes successive multiple of all these values and find
#the least (lowest) common value)?
lcm(2, 3, 5, 10)









    Out[40]:





30

Absolute values

$ \left| -a \right| = a $



In [41]:

    
#Calculating the absolute value using the abs() function
abs(-10)









    Out[41]:





10

$ {\left| -a \right|}^{2} $



In [42]:

    
#Calculating the absolute valued squared using the abs2() function
abs2(-10)









    Out[42]:





100

Trigonometric functions

$ \sin \cos \ \tan \\ \csc \sec \cot \\ \sinh \cosh \tanh $ and many more



In [43]:

    
#Calculating the sine of pi using sin(pi)
sin(pi)









    Out[43]:





1.2246467991473532e-16



In [44]:

    
#Usingthe d-ending for degrees, i.e. sind(180)
sind(180)









    Out[44]:





0.0



In [45]:

    
#Calculating the hypotenuse of a right-angled triangle using the hypot() function
hypot(3, 4)









    Out[45]:





5.0

The sign of a value (as in positive or negative or none for zero)

Returns +1 for positive values
Returns -1 for negative values
Returns 0 for 0



In [47]:

    
#Using sign() for the sign of the sine of pi
sign(sind(180))









    Out[47]:





0.0

Numeric comparisons

Using Boolean logic to find out if values are equal on unequal, i.e. greater than or less than each other



In [48]:

    
#Comparison using ==
3 == 3









    Out[48]:





true



In [49]:

    
#Is greater than using >
3 > 3









    Out[49]:





false



In [50]:

    
#Is greater than or equal to using >=
3 >= 3









    Out[50]:





true



In [51]:

    
#Is less than using <
3 < 3









    Out[51]:





false



In [52]:

    
#Is less than or equal to using <=
3 <= 3









    Out[52]:





true



In [53]:

    
#Is not equal to using !=
3 != 3









    Out[53]:





false

The special cases of NaN, Inf and -Inf



In [54]:

    
#What is the type of NaN using typeof(NaN)
typeof(NaN)









    Out[54]:





Float64



In [55]:

    
#Is NaN equal to NaN?
NaN == NaN









    Out[55]:





false



In [56]:

    
#Is NaN greater than NaN
NaN > NaN









    Out[56]:





false



In [57]:

    
#What is the type of Inf?
typeof(Inf)









    Out[57]:





Float64



In [59]:

    
#Is Inf + 1 greater than Inf?
(Inf + 1) < Inf









    Out[59]:





false

Logical tests using functions



In [60]:

    
#Comparing values using the isequal(x, y) function
#Just FYI, isequal(NaN, NaN) is true
isequal(3, 3)









    Out[60]:





true



In [61]:

    
#Checking to see of a value is finite using the isfinite() function
isfinite(3)









    Out[61]:





true



In [62]:

    
#Checking to see if a number is infinite using the isinf() function
isinf(3)









    Out[62]:





false



In [63]:

    
#Checking to see if a value is NaN using the isnan() function
isnan(NaN)









    Out[63]:





true



In [64]:

    
#The peculiarity of -0.0 == 0.0 vs isequal(-0.0, 0.0)
-0.0 == 0









    Out[64]:





true



In [65]:

    
#
isequal(-0.0, 0)









    Out[65]:





false



In [66]:

    
#We could also use ==() instead of isequal(), etc...
==(3, 3)









    Out[66]:





true

Special powers (roots) and logarithms



In [67]:

    
#Calculating the square roots using the sqrt() function
sqrt(9)









    Out[67]:





3.0



In [68]:

    
#Calculating the cube roots using the cbrt() function
cbrt(27)









    Out[68]:





3.0



In [69]:

    
#Calculating the natural exponential using the exp() function
exp(1)









    Out[69]:





2.718281828459045



In [70]:

    
#For more accuracy when x is near zero in the case of exp(x-1) use the expm1(x) function
expm1(0.0001)









    Out[70]:





0.00010000500016667084



In [71]:

    
#Calculating the natural logarithm using the log() function
log(100)









    Out[71]:





4.605170185988092



In [72]:

    
#Specifying a base for the logarithm using the log(base, value) function (there are also the log2() and log10() functions)
log(10, 100)









    Out[72]:





2.0



In [73]:

    
#For more accuracy when x is near zero in the case of log(1+x) use the log1p() function
log1p(0.0001)









    Out[73]:





9.999500033330834e-5

Special functions

The error function at x
- erf(x)
The inverse function to the error function
- erfinv(x)
The gamma function at x
- gamma(x)
The beta function at x
- beta(x)
Many others...

The complex numbers



In [1]:

    
#The notation for a complex number using im
2 + 2im









    Out[1]:





2 + 2im



In [2]:

    
#Constructing a complex number using the complex() function
complex(2, 2)









    Out[2]:





2 + 2im



In [3]:

    
#The real part of a complex number using the real() function
real(2 + 2im)









    Out[3]:





2



In [5]:

    
#The imaginary part of a complex number using the imag() function
imag(2 + 4im)









    Out[5]:





4



In [7]:

    
#The complex conjugate of a complex number using the conj() function
conj(2 - 2im)









    Out[7]:





2 + 2im



In [9]:

    
#The absolute value of a complex number using the abs() function
abs(3 + 4im)









    Out[9]:





5.0



In [10]:

    
#The square of the absolute value using the abs2() function
abs2(3 + 4im)









    Out[10]:





25



In [11]:

    
#The argument (in radians) of a complex number using the angle() function
angle(3 + 4im)









    Out[11]:





0.9272952180016122



In [12]:

    
#Calculating the square root of -1 using sqrt(1im^2), that is, calculating the imaginary number
sqrt(1im^2)









    Out[12]:





0.0 + 1.0im

Simple plotting using Gadfly



In [1]:

    
using Gadfly



In [2]:

    
f(x) = sin(x)









    Out[2]:





f (generic function with 1 method)



In [3]:

    
g(x) = cos(x)









    Out[3]:





g (generic function with 1 method)



In [4]:

    
plot([f, g], -2pi, 2pi)









    Out[4]:



In [88]:

    
plot([sin, cos], -2pi, 2pi)









    Out[88]:



In [83]:

    
h(x) = (x - 1)^3 + 1









    Out[83]:





h (generic function with 1 method)



In [84]:

    
plot(h, -5, 5)









    Out[84]:



In [85]:

    
plot(x -> 1 / x, -4, 4)









    Out[85]:



In [86]:

    
plot(x -> 1 / x, 0, 5)









    Out[86]:



In [87]:

    
plot(x -> erf(x), -3, 3)









    Out[87]:



In [ ]: