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

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Unit 1: Essential Skills

Simple Arithmetics

Addition / Subtraction

Expression Equivalent
a+b`a + b` a+b`a + b`
(a)+b`(-a) + b` ba`b - a`
a+(b)`a + (-b)` ab`a - b`
(a)+(b)`(-a) + (-b)` (a+b)`-(a + b)`
ab`a - b` ab`a - b`
a(b)`a - (-b)` a+b`a + b`
(a)(b)`(-a) -(-b)` (a)+b`(-a) + b`

Multiplication / Division

Signs Outcome
a×b`a \times b` Positive
(a)×b`(-a) \times b` Negative
a×(b)`a \times (-b)` Negative
(a)×(b)`(-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,xR]`[x > 3, x \isin R]`

  • | means such that

  • E or ∈ means element of

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

  • W represents Whole Numbers W={xx0,xZ}`W = \{x | x \ge 0, x \isin \mathbb{Z}\}`

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

  • Q represents Rational Numbers Q={aba,bZ,b0}`Q = \{ \frac{a}{b} |a, b \isin \mathbb{Z}, b \neq 0 \}`

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

  • a2+b2=c2`a^2+b^2=c^2`

Operations with Rationals

  • Q={aba,bZ,b0}`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:

  • 34÷214` \dfrac{3}{4} \div \dfrac{2}{14} ` Reduce to lowest terms

  • 34÷17` \dfrac{3}{4} \div \dfrac{1}{7} ` Multiple by reciprocal

  • 34×7` \dfrac{3}{4} \times 7 `

  • =214` = \dfrac{21}{4}` Leave as improper fraction

Shortcut for multiplying fractions

  • cross divide to keep your numbers small

  • Example:

  • 34×212` \dfrac{3}{4} \times \dfrac{2}{12} `

  • 12×14` \dfrac{1}{2} \times \dfrac{1}{4} `

  • =18` = \dfrac{1}{8} `

Exponent Laws

Rule Description Example
Product am×an=an+m`a^m \times a^n = a^{n+m}` 23×22=25`2^3 \times 2^2 = 2^5`
Quotient am÷an=anm`a^m \div a^n = a^{n-m}` 34÷32=32`3^4 \div 3^2 = 3^2`
Power of a Power (am)n=amn`(a^m)^n = a^mn` (23)2=26`(2^3)^2 = 2^6`
Power of a Quotient (ab)n=anbn`(\dfrac{a}{b})^n = \dfrac{a^n}{b^n}` (23)4=2434`(\dfrac{2}{3})^4 = \dfrac{2^4}{3^4}`
Zero as Exponents a0=1`a^0 = 1` 210=1`21^0 = 1`
Negative Exponents am=1am`a^{-m} = \dfrac{1}{a^m}` 110=1110`1^{-10} = \dfrac{1}{1^{10}}`
Rational Exponents anm=(am)n`a^{\frac{n}{m}} = (\sqrt[m]{a})^n` 1654=(164)5`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 5.55×102`5.55 \times 10^2` (3 significant figures).

  • In scientific notation, values are written in the form a(10n)`a(10^n)`, where a`a` is a number within 1 and 10 and n`n` is any integer.

  • Some examples include the following: 5.4×103,3.0×102`5.4 \times 10^3, 3.0 \times 10^2`, and 4.56×104`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