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