︠b083473a-e807-45be-bbb9-686425fa54d2i︠
%md

_Sage_ (or _SageMath_) stands for "System for Algebra and Geometry Experimentation"
* Free and open-source algebra system.
* Built on Python.
* Available on CoCalc---but you can also install it on your own machine.
* Lots of built-in functions available for mathematics.

A basic documentation can be found [here](http://jephianlin.github.io/SageBasics.pdf).
︡48783d82-27d9-44da-9dd7-9c825d744f96︡{"done":true,"md":"\n_Sage_ (or _SageMath_) stands for \"System for Algebra and Geometry Experimentation\"\n* Free and open-source algebra system.\n* Built on Python.\n* Available on CoCalc---but you can also install it on your own machine.\n* Lots of built-in functions available for mathematics.\n\nA basic documentation can be found [here](http://jephianlin.github.io/SageBasics.pdf)."}
︠a071d6ab-f698-40e4-9325-3e486c6810b0s︠
### Sage examples

### press shift+enter to evaluate the cell

1+1
︡31ee386a-15bc-4183-a28a-bfdfb6be3f89︡
︠d7fe3c63-1a39-47dd-ba26-52da2f4a3ac7s︠
### use tab to auto-complete your typing
### (move your cursor to the end of "facto" and press tab)

factor(12)
︡63b80c2e-685c-42c3-9114-994bcc38b42e︡
︠e0b6a975-2b48-43e3-a1a4-8dbb81371719s︠
### use equal sign to assign a value

a=3;
f(x)=x^2+1;
M=matrix(
[
[1,2],
[3,4]
]);

print M
︡ebd1ba6e-e403-4eb9-9cc5-c3253298d93c︡
︠34b5bd24-770e-41a8-8e59-4f753c606d88s︠
### use print to output the value

print M; ### Python 2 only
print(M); ### Python 2 or Python 3
︡0273ff68-dea0-4278-8b65-0fb7cbc75f19︡
︠3989fd9f-9e3d-4237-80fb-957d1228d7e7︠
### now M is a matrix
### type M. and press tab to see available functions

M.determinant()
︡e69366fe-ecc6-4397-99a6-d7e0297dfb05︡
︠1e3668b5-e804-4549-9333-8dbdca3cbdcfs︠
### "inverse" is a function under the object "matrix"
### evaluate M.inverse? to see the document

M.inverse?
︡5edbffec-ee0f-4966-b664-085ffe0ddc0e︡
︠f8f3ce24-2821-43f9-911a-3ca90825eb37︠
### evaluate M.inverse?? to read the source code

M.inverse??
︡6ce42667-49f1-4fc9-a017-6c8b6339ec7f︡
︠bf3fe100-27fc-49cc-bf26-accaf414c730︠
### Exercise:
###     Find the Taylor expansion of g at x=0.
###     [Hint: use f. tab to see available functions
###            or Google "taylor expansion, sage"]

f(x)=x^2+1
g(x)=exp(-x^2); ### This does not hold in Python
g.show();

︡00dafb09-100c-425a-8021-4baf7d870526︡
︠2cc997dc-ae4a-41d3-8481-6e7129705694s︠
### Exercise:
###     Find the determinant and the eigenvalues of X.

X=matrix(3,range(9));
show(X);
︡fda2d7e3-b393-4105-ab1e-23c6a24f896a︡
︠67c7f251-914d-4898-bf1c-3b67d70327bd︠
### Exercise:
###     Think of a math concept and search online
###     to see how to define it in Sage.  Then
###     see the available functions under it.
###     (E.g., g=graphs.CompleteGraph(5);)
︡79f8448e-25da-4fe0-9715-cbc678b0253c︡
︠adfbbfe3-384d-43a2-94d3-325c6d304471s︠
### list is a sequence of values

f=[1,1,2,3,5,8];
︡1bb01bbf-99df-4479-b943-a9c595e11488︡
︠1549c476-3346-4939-a4c6-56a348ac43d2s︠
### Use a[i] to call the value of a on index i

f[4]
︡80f0f7d1-91fa-4551-bd3a-01c9d89c0a1b︡
︠d4baec18-2103-4e6e-af3a-66a553d5b90bs︠
### range(a,b) gives a list from a to b-1

range(5,10)
︡7aaef645-fda4-411f-a4fa-82e52a262dde︡
︠b444d6fe-b67e-484d-a7d9-2098c0fee56cs︠
### called "for loop"

### for list:
###     do something;  # indentation is important!  Usually, 4 spaces or a tab.

for i in range(5,10):
    print i^2;  ### indentation is important!!!!
︡813db54f-ed64-44b3-979f-5eee73e17781︡
︠cf65726b-ff0d-487c-a93e-2a9e66e55bea︠
### called "if statement"

### if test:
###     do something;

k=10;
### k % 2 means the remainder of k divided by 2
if k % 2 == 0:
    print "k is even"
︡118bd8f0-f4d0-4fa0-9119-add6c77c08d4︡
︠a9b7499d-7fc7-4c81-80a7-c477131d0ac0s︠
### combining for loop and if statment can do many things

for i in range(1,11):
    if i % 2 == 1:
        print i;
︡5eefdc0a-1a8e-4122-8bc7-109a18c282dc︡
︠25c58b8c-e988-4cce-9202-b1b780cc31fas︠
### a counter help you to count

counter = 0;
for i in range(1,11):
    if i % 2 == 1:
        counter = counter + 1;
print counter;
︡49c08b40-2b32-4ffd-baa6-9a1f0ec4fd4b︡
︠c4cd238d-0cb1-4ab9-a3e4-403ac0d56726s︠
### def function_name(input_parameters):
###     do something using input_parameters;
###     return something;

def how_many_odd_numbers(n):
    counter = 0;
    for i in range(1,n+1):
        if i % 2 == 1:
            counter = counter + 1;
    return counter;
︡5cfc689c-25b0-421f-96c4-d45439ff4747︡
︠16312829-16b3-491d-b329-e77f62665e54s︠
how_many_odd_numbers(15)
︡0a6c6312-d25c-4c7f-88c7-18ca472c4c01︡
︠773cbfbb-b5b4-485b-8d69-d4b566744ed5︠
### greatest common divisor

gcd(6,4)
︡5393bac4-6867-4c39-bb38-cb502103af07︡
︠45535785-5fef-40cf-8d3c-8e132e113cb5s︠
### prime decomposition

factor(60)
︡6fb12408-6266-4bd5-b9b0-29e6e771325b︡
︠800282c0-f41c-4be0-b092-41d370bbb389s︠
### extract the factors and the powers

list(factor(60))
︡bb00d41e-efe8-4645-8ae6-345e515baffa︡
︠9f20d4a5-6626-47fd-9308-d8da5f58503di︠
%md

## Project 1
Write a function to compute the Euler's totient function $\phi(n)$.

_Do not use the function_ `euler_phi` _in Sage._
︡1491331a-ff0e-4231-ae95-9c18acf6be1f︡{"done":true,"md":"\n## Project 1\nWrite a function to compute the Euler's totient function $\\phi(n)$.\n\n_Do not use the function_ `euler_phi` _in Sage._"}
︠e8a40d27-ed26-4b3c-81aa-4bc1b950f580︠
def phi(n):
    ### Your answers here
    counter=0;
    for k in range(1,n+1):
        if gcd(k,n)==1:
            counter+=1;
    return counter;

### test;
for n in range(1,11):
    print "phi({})={}, {}=correct answer".format(n,phi(n),euler_phi(n))
︡17f55808-1e21-49f3-9691-e614adb9a2de︡
︠9521f063-083a-4bb6-b448-aef313add05ci︠
%md
## Project 2
Write a function to compute the Moebius function $\mu(n)$.

_Do not use the function_ `moebius` _in Sage._
︡bc1a6203-7757-4240-9efd-494b052c3ed8︡{"done":true,"md":"## Project 2\nWrite a function to compute the Moebius function $\\mu(n)$.\n\n_Do not use the function_ `moebius` _in Sage._"}
︠fc083c04-5715-45cc-aa5a-d4fc7ea91a40︠
def mu(n):
    ### Your answers here
    f = list(factor(n));
    for p in f:
        if p[1] >= 2:
            return 0;
    ### now the powers are either 0 or 1
    r = sum([p[1] for p in f]);
    return (-1)^r;

### test;
for n in range(1,11):
    print "mu({})={}, {}=correct answer".format(n,mu(n),moebius(n))
︡9b5c85d3-f6c0-4ce8-affe-2cb07272364a︡
︠6a6551c2-dba9-4378-809b-1fc31e17c0aa︠