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140 lines
6.2 KiB
Markdown
140 lines
6.2 KiB
Markdown
# Unit 1: Essential Skills
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## Simple Arithmetics
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### Addition / Subtraction
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| Expression | Equivalent|
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|:----------:|:---------:|
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| $`a + b`$ | $`a + b`$ |
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| $`(-a) + b`$ | $`b - a`$ |
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| $`a + (-b)`$ | $`a - b`$ |
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| $`(-a) + (-b)`$ | $`-(a + b)`$ |
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| $`a - b`$ | $`a - b`$|
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| $`a - (-b)`$ | $`a + b`$ |
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| $`(-a) -(-b)`$ | $`(-a) + b`$|
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### Multiplication / Division
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| Signs | Outcome |
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|:-----:|:-------:|
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| $`a \times b`$ | Positive |
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| $`(-a) \times b`$ | Negative |
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| $`a \times (-b)`$ | Negative |
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| $`(-a) \times (-b)`$ | Positive |
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### BEDMAS / PEMDAS
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- Follow ```BEDMAS``` for order of operations if there are more than one operation
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| Letter | Meaning |
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|:------:|:-------:|
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| B / P | Bracket / Parentheses |
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| E | Exponent |
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| D | Divison |
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| M | Multiplication |
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| A | Addition |
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| S | Subtraction |
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- <img src="https://ecdn.teacherspayteachers.com/thumbitem/Order-of-Operations-PEMDAS-Poster-3032619-1500876016/original-3032619-1.jpg" width="300">
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## Interval Notation
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- A notation that represents an interval as a pair of numbers.
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- The numbers in the interval represent the endpoint. E.g. $`[x > 3, x \isin R]`$
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- ```|``` means ```such that```
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- ```E``` or ∈ means ```element of```
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- ```N``` represents **Natural Numbers** $`N = \{x | x \gt 0, x \isin \mathbb{Z} \}`$
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- ```W``` represents **Whole Numbers** $`W = \{x | x \ge 0, x \isin \mathbb{Z}\}`$
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- ```Z``` represents **Integers** $`Z = \{x| -\infin \le x \le \infin, x \isin \mathbb{Z}\}`$
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- ```Q``` represents **Rational Numbers** $`Q = \{ \frac{a}{b} |a, b \isin \mathbb{Z}, b \neq 0 \}`$
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| Symbol | Meaning |
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|:------:|:-------:|
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| $`(a, b)`$ | Between but not including $`a`$ or $`b`$, you also use this for $`\infty`$|
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| $`[a, b]`$ | Inclusive |
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| $`a ∪ b`$ | Union (or) |
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| $`a ∩ b`$ | Intersection (and) |
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## Pythgorean Theorem
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- a and b are the two legs of the triangle or two sides that form a 90 degree angle of the triangle, c is the hypotenuse
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- $`a^2+b^2=c^2`$
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- <img src="http://www.justscience.in/wp-content/uploads/2017/05/Pythagorean-Theorem.jpeg" width="400">
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## Operations with Rationals
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- $`Q = \{ \frac{a}{b} |a, b \isin \mathbb{Z}, b \neq 0 \}`$
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- Any operations with rationals, there are 2 sets of rules
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1. ```Rules for operations with integers```
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2. ```Rules for operations with fractions```
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- To Add / subtract rationals, find common denominator and then add / subtract numerator
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- To Multiply rationals, first reduce the fraction to their lowest terms, then multiply numerators and denominators
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- To Divide rationals, multiply them by the reciprocal
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### Example Simplify Fully:
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- $` \dfrac{3}{4} \div \dfrac{2}{14} `$ Reduce to lowest terms
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- $` \dfrac{3}{4} \div \dfrac{1}{7} `$ Multiple by reciprocal
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- $` \dfrac{3}{4} \times 7 `$
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- $` = \dfrac{21}{4}`$ Leave as improper fraction
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### Shortcut for multiplying fractions
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- cross divide to keep your numbers small
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- Example:
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- $` \dfrac{3}{4} \times \dfrac{2}{12} `$
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- $` \dfrac{1}{2} \times \dfrac{1}{4} `$
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- $` = \dfrac{1}{8} `$
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## Exponent Laws
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| Rule | Description| Example |
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|:----:|:----------:|:-------:|
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|Product|$`a^m \times a^n = a^{n+m}`$|$`2^3 \times 2^2 = 2^5`$|
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|Quotient|$`a^m \divide a^n = a^{n-m}`$|$`3^4 \divide 3^2 = 3^2`$|
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|Power of a Power|$`(a^m)^n = a^mn`$|$`(2^3)^2 = 2^6`$|
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|Power of a Quotient|<img src="http://latex2png.com/output//latex_9528cf1be9ea781a9134559f27f6b94b.png" width="25"> = <img src="http://latex2png.com/output//latex_8c4a93634d0e2f5cfce05b474ebf2f02.png" width="15">|<img src="http://latex2png.com/output//latex_74d1af968da0b70a335c8d93273635e9.png" width="25"> = <img src="http://latex2png.com/output//latex_2040cef99eca664c295bd74848f0779f.png" width="15">|
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|Zero as Exponents|a<sup>0</sup> = 1|21<sup>0</sup> = 1|
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|Negative Exponents|a<sup>-m</sup> = <img src="http://latex2png.com/output//latex_0223494d8dd45b887178dcecbbfeb462.png" width="20">|1<sup>-10</sup> = <img src="http://latex2png.com/output//latex_4765f9318cc4813f30321334b18635eb.png" width="20">|
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|Rational Exponents|a<sup>n/m</sup> = <img src="http://latex2png.com/output//latex_33a019fd887e207917a831e5b5fd20e5.png" width="50">|<img src="http://latex2png.com/output//latex_af91e3845b91443f5fcf11fcf59368d3.png" width = "35"> = <img src="http://latex2png.com/output//latex_33a019fd887e207917a831e5b5fd20e5.png" width="50">|
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**Note:**
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- Exponential Form --> Expanded Form
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- 6<sup>4</sup> = 6 × 6 × 6 × 6
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## Scientific Notation
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- They convey accuracy and precision. It can either be written as its original number or in scientific notation:
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- 555 (**Exact**) or $`5.55 \times 10^2`$ (**3 significant figures**).
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- In scientific notation, values are written in the form $`a(10^n)`$, where $`a`$ is a number within 1 and 10 and $`n`$ is any integer.
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- Some examples include the following: $`5.4 \times 10^3, 3.0 \times 10^2`$, and $`4.56 \times 10^{-4}`$.
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- When the number is smaller than 1, a negative exponent is used, when the number is bigger than 10, a positve exponent is used
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- <img src="https://embedwistia-a.akamaihd.net/deliveries/d2de1eb00bafe7ca3a2d00349db23a4117a8f3b8.jpg?image_crop_resized=960x600" width="500">
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- **Remember**: For scientific notation, round to ```3 significant``` digits
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## Rates, Ratio and Percent
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- ```Ratio```: A comparison of quantities with the same unit. These are to be reduced to lowest terms.
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- Examples: ```a:b, a:b:c, a/b, a to b ```
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- ```Rates```: A comparison of quantities expressed in different units.
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- Example: ```10km/hour```
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- ```Percent```: A fraction or ratio in which the denominator is 100
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- Examples: ```50%, 240/100```
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## Number Lines
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- a line that goes from a point to another point, a way to visualize set notations and the like
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- <img src="https://i2.wp.com/mathblog.wpengine.com/wp-content/uploads/2017/03/numberlines-thumbnail.jpeg?resize=573%2C247&ssl=1" width="500">
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- A solid filled dot is used for ```[]``` and a empty dot is used for ```()```
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## Tips
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- Watch out for the ```+/-``` signs
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- Make sure to review your knowledge of the exponent laws
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- For scientific notation, watch out for the decimal point
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- Use shortcut when multiplying fractions |