R Basics

class: center, middle, inverse, title-slide

.title[
# R Basics
]

---

# R as a calculator

- Addition: +

- Subtraction: -

- Multiplication: *

- Division: /

- Exponentiation: ^

Put the following to the console in Rstudio

```r
(3+4)^2 - 3^4 - 1/2 - 3*4
```

```
## [1] -44.5
```

---
# Data structures in R

|    | Homogeneous | Heterogeneous |
|----|-------------|---------------|
| 1d | Vector      | List          |
| 2d | Matrix      | Data Frame    |
| nd | Array       |               |

---
# Vector

- Numeric vectors

```r
x <- c(2,3,4,5,6)
x
```

```
## [1] 2 3 4 5 6
```

- Use `<-` to assign *something* to a variable/an object

---
# Vector

- Character vectors

```r
x <-  c("Monday", "Tuesday", "Wednesday", "Thursday", "Friday")
x
```

```
## [1] "Monday"    "Tuesday"   "Wednesday" "Thursday"  "Friday"
```

- Logical vectors

```r
x <- c(TRUE, FALSE, TRUE)
sum(x)
```

```
## [1] 2
```

---
# Operations on Numeric Vectors

- Operations on Vectors are **element-to-element**

- You can add, subtract, multiply, and divide vectors

```r
x <- c(2, 3, 4, 5, 6)
y <- c(1, 0, 0, 0, 99)
x+y
```

```
## [1]   3   3   4   5 105
```

```r
x-y
```

```
## [1]   1   3   4   5 -93
```

```r
x*y
```

```
## [1]   2   0   0   0 594
```

---
# Operations on Numeric Vectors

- You can apply a function to vector

```r
x <- c(2, 3, 4, 5, 6)
y <- c(1, 0, 0, 0, 99)
x^2
```

```
## [1]  4  9 16 25 36
```

```r
log(x)
```

```
## [1] 0.6931472 1.0986123 1.3862944 1.6094379 1.7917595
```

```r
sin(x)
```

```
## [1]  0.9092974  0.1411200 -0.7568025 -0.9589243 -0.2794155
```

---
# Some functions with vectors

```r
x <- c(2, 3, 40, 5, 6)
mean(x)
```

```
## [1] 11.2
```

```r
sum(x)
```

```
## [1] 56
```

```r
max(x)
```

```
## [1] 40
```

```r
length(x)
```

```
## [1] 5
```

---
# Some functions with vectors

```r
x <- c(2, 3, 40, 5, 6)
# Standard Deviation
sd(x)
```

```
## [1] 16.17714
```

```r
# Index of the maximum element
which.max(x)
```

```
## [1] 3
```

```r
# Index of the element equaling 40
which(x==40)
```

```
## [1] 3
```

---
# Quickly generate vectors

```r
c(1:10)
```

```
##  [1]  1  2  3  4  5  6  7  8  9 10
```

```r
rep(1, 10)
```

```
##  [1] 1 1 1 1 1 1 1 1 1 1
```

```r
rep(c(1,2,3), 10)
```

```
##  [1] 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
```

```r
# Arithmetic sequences
seq(1, 2, .2)
```

```
## [1] 1.0 1.2 1.4 1.6 1.8 2.0
```

---
class: inverse, middle, center

# Live Coding

solve problems, (talk about variable environment, variables, shortcuts...)

present the results in an rmarkdown and

publish it to Github Page

---
class:inverse
# Problem

- **Calculate**: 
$$
S_{1} = 1 + 2 +...+  100
$$
--

- **Solution**

```r
x <- c(1:100)

sum(x)
```

```
## [1] 5050
```

---
class:inverse
# Problem

- **Calculate**
$$
S_{2} = 1^2 + 2^2 +...+  100^2
$$
--

- **Solution**

```r
x <- c(1:100)
sum(x^2)
```

```
## [1] 338350
```

---
class:inverse
# Problem

- **Calculate**
$$
S = \frac{1^3 + 2^3 +...+  100^3}{100}
$$
--

- Solution*

```r
x <- c(1:100)
sum(x^3)/100
```

```
## [1] 255025
```

- **Another way?**

```r
# Or

mean(x^3)
```

```
## [1] 255025
```

---
class:inverse
# Problem

- **Calculate**
$$
S = 1\cdot 2 + 3 \cdot 4 + 5 \cdot 6 + 7 \cdot 8 +...+ 99 \cdot 100
$$
--

- **Solution**

```r
x <- seq(1, 99, 2)
y <- seq(2, 100, 2)

sum(x*y)
```

```
## [1] 169150
```

---
class:inverse
# Problem

- **Calculate**
$$
S = \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + ...+  \frac{100}{101}
$$
--

- **Solution**

```r
x <- c(1:100)
y <- c(2:101)

sum(x/y)
```

```
## [1] 95.80272
```

---
class:inverse
# Problem

- Is this summation *convergent* or *divergent*?

$$
S = 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+...
$$

---
class:inverse
# Solution

- Rewrite `$S$`:

$$
S = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...
$$
--

- The sum of the first 1000 terms

```r
n <-  1000
x <-  c(1:n)
sum(1/x^2)
```

```
## [1] 1.643935
```
--

- The sum of the first 10000 terms

```r
n <-  10000
x <-  c(1:n)
sum(1/x^2)
```

```
## [1] 1.644834
```

---
class: center, middle, inverse

### The series converges!

---
class: center, middle, inverse

### YOUR TURN

This is also your assignment 2

https://bryantstats.github.io/math421/assignments/assignment2.html

---
class:inverse
# Problem

***Guideline***: Use the rmarkdown [here](https://bryantstats.github.io/math421/assignments/assignment2.Rmd) as a template to write your codes. Open the Rmarkdown file in Rstudio. After each question, insert  a code chunk (you can use the hotkey `Ctrl + Alt + I` to add a code chunk) and code the solution for the questions. `Knit` the rmarkdown file (hotkey `Ctrl + Alt + K`) to export an html.  Then Publish the html to your Githiub Page.

***Submissions***: Submit the link on Github of the assignment to Canvas under Assignment 2.

**Go the the next page for the problems**

---
class:inverse
# Problem

- **Calculate**

$$
S = 1 + 4 + 7 + 11+... + 100
$$
$$
S = 1^{100} + 2^{100} + 3^{100} +...+100^{100}
$$
$$
S = \frac{1}{1\cdot 2}+\frac{1}{2 \cdot 3}+...+\frac{1}{100\cdot 101}
$$

$$
S = \frac{1}{1\cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+...+\frac{1}{100\cdot 101 \cdot 102}
$$

$$
S = 1^1 + 3^3 + 5^5 + 7^7 +....+101^{101}
$$