highschool/Grade 9/Math/MPM1DZ/Unit 1: Essential Skills.md

Unit 1: Essential Skills

Simple Arithmetics

Addition / Subtraction

Expression	Equivalent
$`a + b`$	$`a + b`$
$`(-a) + b`$	$`b - a`$
$`a + (-b)`$	$`a - b`$
$`(-a) + (-b)`$	$`-(a + b)`$
$`a - b`$	$`a - b`$
$`a - (-b)`$	$`a + b`$
$`(-a) -(-b)`$	$`(-a) + b`$

Multiplication / Division

Signs	Outcome
$`a \times b`$	Positive
$`(-a) \times b`$	Negative
$`a \times (-b)`$	Negative
$`(-a) \times (-b)`$	Positive

BEDMAS / PEMDAS

• Follow BEDMAS for order of operations if there are more than one operation

Letter	Meaning
B / P	Bracket / Parentheses
E	Exponent
D	Divison
M	Multiplication
A	Addition
S	Subtraction

• 

Interval Notation

• A notation that represents an interval as a pair of numbers.
  

• The numbers in the interval represent the endpoint. E.g. $`[x > 3, x \isin R]`$
  

• | means such that
  

• E or ∈ means element of
  

• N represents Natural Numbers $`N = \{x | x \gt 0, x \isin \mathbb{Z} \}`$
  

• W represents Whole Numbers $`W = \{x | x \ge 0, x \isin \mathbb{Z}\}`$

• Z represents Integers $`Z = \{x| -\infin \le x \le \infin, x \isin \mathbb{Z}\}`$

• Q represents Rational Numbers $`Q = \{ \frac{a}{b} |a, b \isin \mathbb{Z}, b \neq 0 \}`$

  Symbol	Meaning
  $`(a, b)`$	Between but not including $`a`$ or $`b`$, you also use this for $`\infty`$
  $`[a, b]`$	Inclusive
  \(`a ∪ b`\)	Union (or)
  $`a ∩ b`$	Intersection (and)

Pythgorean Theorem

• 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
  

• $`a^2+b^2=c^2`$

• 

Operations with Rationals

• $`Q = \{ \frac{a}{b} |a, b \isin \mathbb{Z}, b \neq 0 \}`$

• Any operations with rationals, there are 2 sets of rules

  1. Rules for operations with integers
  2. Rules for operations with fractions

• To Add / subtract rationals, find common denominator and then add / subtract numerator
  

• To Multiply rationals, first reduce the fraction to their lowest terms, then multiply numerators and denominators
  

• To Divide rationals, multiply them by the reciprocal

Example Simplify Fully:

• $` \dfrac{3}{4} \div \dfrac{2}{14} `$ Reduce to lowest terms

• $` \dfrac{3}{4} \div \dfrac{1}{7} `$ Multiple by reciprocal

• $` \dfrac{3}{4} \times 7 `$

• $` = \dfrac{21}{4}`$ Leave as improper fraction

Shortcut for multiplying fractions

• cross divide to keep your numbers small
  

• Example:
  

• $` \dfrac{3}{4} \times \dfrac{2}{12} `$

• $` \dfrac{1}{2} \times \dfrac{1}{4} `$

• $` = \dfrac{1}{8} `$

Exponent Laws

Rule	Description	Example
Product	$`a^m \times a^n = a^{n+m}`$	$`2^3 \times 2^2 = 2^5`$
Quotient	$`a^m \div a^n = a^{n-m}`$	$`3^4 \div 3^2 = 3^2`$
Power of a Power	$`(a^m)^n = a^mn`$	$`(2^3)^2 = 2^6`$
Power of a Quotient	$`(\dfrac{a}{b})^n = \dfrac{a^n}{b^n}`$	$`(\dfrac{2}{3})^4 = \dfrac{2^4}{3^4}`$
Zero as Exponents	$`a^0 = 1`$	$`21^0 = 1`$
Negative Exponents	$`a^{-m} = \dfrac{1}{a^m}`$	$`1^{-10} = \dfrac{1}{1^{10}}`$
Rational Exponents	$`a^{\frac{n}{m}} = (\sqrt[m]{a})^n`$	$`16^{\frac{5}{4}} = (\sqrt[4]{16})^5`$

Note:
- Exponential Form –> Expanded Form
- 6⁴ = 6 × 6 × 6 × 6

Scientific Notation

• They convey accuracy and precision. It can either be written as its original number or in scientific notation:

• 555 (Exact) or $`5.55 \times 10^2`$ (3 significant figures).
  

• 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.

• Some examples include the following: $`5.4 \times 10^3, 3.0 \times 10^2`$, and $`4.56 \times 10^{-4}`$.
  

• When the number is smaller than 1, a negative exponent is used, when the number is bigger than 10, a positve exponent is used

• 

• Remember: For scientific notation, round to 3 significant digits

Rates, Ratio and Percent

• Ratio: A comparison of quantities with the same unit. These are to be reduced to lowest terms.
  

• Examples: a:b, a:b:c, a/b, a to b

• Rates: A comparison of quantities expressed in different units.
  

• Example: 10km/hour

• Percent: A fraction or ratio in which the denominator is 100
  

• Examples: 50%, 240/100

Number Lines

• a line that goes from a point to another point, a way to visualize set notations and the like
  
• 
  
• A solid filled dot is used for [] and a empty dot is used for ()

Tips

• Watch out for the +/- signs
  
• Make sure to review your knowledge of the exponent laws
  
• For scientific notation, watch out for the decimal point
• Use shortcut when multiplying fractions