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 * b	Positive
(-a) * b	Negative
a * (-b)	Negative
(-a) * (-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 ∈ 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 ∞
  [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:

• \(` \frac{3}{4} \div \frac{2}{14} `\) Reduce to lowest terms

• \(` \frac{3}{4} \div \frac{1}{7} `\) Multiple by reciprocal

• \(` \frac{3}{4} \times 7 `\)

• \(` = \frac{21}{4}`\) Leave as improper fraction

Shortcut for multiplying fractions

• cross divide to keep your numbers small
  

• Example:
  

• \(` \frac{3}{4} \times \frac{2}{12} `\)

• \(` \frac{1}{2} \times \frac{1}{4} `\)

• \(` = \frac{1}{8} `\)

Exponent Laws

Rule	Description	Example
Product	am × an = an+m	23 × 22 = 25
Quotient	am ÷ an = an-m	34 ÷ 32 = 32
Power of a Power	(am)n = amn	(23)2 = 26
Power of a Quotient	=	=
Zero as Exponents	a0 = 1	210 = 1
Negative Exponents	a-m =	1-10 =
Rational Exponents	an/m =	=

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