Math 9
I bet that like literally every previous year ever, Math will be about 90% repetitive.
The Number System
- Complex numbers include pretty much everything. They include real and imaginary numbers.
- Imaginary numbers involve the variable i (the square root of -1). You’ll learn more about this later in life.
- Real numbers have rational and irrational rational numbers. Any real number can be represented on the number line.
- Irrational numbers are infinitely precise/repeating numbers with no identifiable pattern (can’t be represented as a “normal” fraction).
- Rational numbers can be written as the quotient of two integers. They consist of integers.
- Integers are any non-decimal number, whether it’s positive or negative.
- Whole numbers are positive integers, including zero.
- Natural numbers are whole numbers, without zero.
Powers and Exponents
- The “standard form” of a power is pretty much just the answer.
- To calculate the perfect square root of a fraction, find equivalent fractions until both the numerator and denominator are perfect squares. (sqrt(18/32) = sqrt(9/16) = 3/4)
- To calculate the perfect square root of a decimal, put the number into a fraction where the denominator is a perfect square base ten number. (sqrt(2.25) = sqrt(225/100) = 15/10 = 3/2 = 1.5)
- The square roots of whole numbers can be estimated by benchmarking them, or finding the previous and next perfect squares. (sqrt(45) -> between sqrt(36) and sqrt(49), but closer to the next -> about 6.7)
- To estimate large numbers with lots of zeros at the end, pair each two zeros into one, and then use the square root of the first digit(s) to prefix the result. (sqrt(800,000,000) -> 2.8, 0, 0, 0, 0 -> about 28,000)
- To estimate small decimal numbers with lots of zeros at the start, pair each two zeros past the point into one, and then use the square root of the last digit(s) to suffix the result. (sqrt(0.000008) -> 0, 0, 2.8 -> about 0.0028)
- There are two types of negative powers, both of which comply with the standard order of operations:
- In an expression like -3^4, the negative isn’t included with the base, so the answer is -81 (-(3 * 3 * 3 * 3)).
- In an expression like (-3)^4, the negative is included with the base, so the answer is 81 (-3 * -3 * -3 * -3).
- A power with the exponent zero is equal to one and changes according to sign. This is caused by a pattern that can be observed by comparing the division of the result of each expression by the base while decrementing the exponent each time. (5^3 = 125 -> 125 / 5 = 25 -> 5^2 = 25 -> 25 / 5 = 5 -> 5 / 5 = 1 -> 5^0 = 1)
- Numerous exponent laws exist to make expressions easier to evaluate, and they predominately rely on the fact that the multiplication of numbers in any order doesn’t change the answer.
- When multiplying powers with the same base, the answer can be simplified by adding the exponents. (5^3 * 5^2 = 5^5)
- When dividing powers with the same base, the answer can be simplified by subtracting the exponents. (5^6 / 5^2 = 5^4)
- A power of a power can be simplified by multiplying the exponents. ((5^2)^4 = 5^8)
- If two powers with different bases but the same exponent are being multiplied, the two bases can be multiplied and brought up to the power of the common exponent. (3^4 * 2^4 = 6^4)
- If two powers with different bases but the same exponent are being divided, the two bases can be divided and brought up to the power of the common exponent. (18^3 / 9^3 = 2^3)
Equations and Inequalities
- To solve an equation with fractions, find a common denominator, multiply each term by the common denominator, simplify (divide), and then solve for the variable. (x/3 - 3x/2 = 1/6 - x -> 2x/6 - 9x/6 = 1/6 - 6x/6 -> 2x - 9x = 1 - 6x -> -7x = 1 - 6x -> 0 = 1 + x -> x = -1)
- An inequality is like an equality, except it isn’t an equality (inequality moment (I have no idea what I’m writing right now (ooh nested parentheses))). They use less than (<), greater than (>), less than or equal (<=) to, or greater than or equal to (>=) symbols.
- To solve one, replace the inequality sign with an equal sign, try to move the variable to one of the sides, solve for the variable, replace the equal sign with an inequality sign, and if a number is multiplied or divided, flip the inequality symbol in the opposite direction. (7x + 4 < 5x + 8 -> 2x + 4 < 8 -> 2x < 4 -> x < 2)
- To solve a word problem that seems to involve multiple variables that are in some way related to each other, try to simplify into one variable in one equation. (three consecutive numbers that add up to 60 -> x + y + z = 60 -> x + (x + 1) + (x + 2) = 60 -> 3x + 3 = 60 -> 3x = 57 -> x = 19 -> first = 19, second = 20, third = 21)
Linear Relations
- One way to see linear relations is as y = mx + b, where m is the slope (steepness) and b is the intercept (where the line crosses the graph).
- Another way would be called standard form, which is Ax + By = C. (3x - 3y = 6 -> y = x - 2)
Polynomials
- Terms are separated by addition/subtraction, and a polynomial is a collection of these terms. A monomial has one term, a binomial has two, and a trinomial has three. (3x^2 + 2x - 4 is a trinomial)
- The degree of a term is the total exponent count of the powers in a term, including implicit exponents, while the degree of a polynomial is the highest degree of any of its terms. (the degree of 8x^2y^1 + 10x^1 - 2x^0 is 3)
- To represent a squared variable with number tiles, use a filled or unfilled square (positive and negative). (4x^2 + 7x - 1 is 4 filled square tiles, 7 filled long tiles, and 1 unfilled small tile)
- To simplify a polynomial, combine like terms (this should be straightforward by now). This is also how you add two polynomials. ((2x^2 + 3x - 8) + (-3x^2 - x + 2) = -1x^2 + 2x - 6)
- To subtract two polynomials, simply flip the signedness of each of the second polynomial’s terms before adding. ((2x^2 + 3x - 8) - (-3x^2 - x + 2) = 5x^2 + 4x - 10)
- To multiply a polynomial by some constant or monomial, just multiply every term by it. (-4x(2x^2 - 4x + 5) = -8x^3 + 16x^2 - 20x)
- To divide a polynomial by some constant or monomial, just divide every term by it. ((4x^5 - 12x^3 + 16x^2) / -4x^2 = -x^3 + 3x - 4)
Similarity and Symmetry
- Line symmetry can be either vertical, horizontal, or oblique (diagonal). It represents the number of repetitions in a shape or design.
- The order of rotation of something is for how many times it can fit in itself in a full 360 degree rotation.
- The angle of rotational symmetry is the smallest incremental change in rotation (in degrees) for a figure to exactly overlap itself.
- For example, a square has four lines of symmetry, an order of rotation of four, and an angle of rotational symmetry of 45 degrees.
- The scale factor of an enlargement or reduction is represented as the ratio of the scale diagram to the original diagram. (a 3x3 square scaled to a 9x9 square is using an enlarging scale factor of 3:1)
- When one polygon is an enlargement or reduction of another polygon, it’s said that they’re both similar. They must have the same corresponding angles, and their corresponding sides must be proportional. (a triangle of 6x6x8.48 is similar to a triangle of 3x3x4.24)
Circle Geometry
- You can do a lot of math in circles and lines by using the radius of a circle, the Pythagoras theorem, and the fact that all angles in a triangle add up to 180. Figure it out yourself because I’m too lazy to explain all the things you can do.
- When referring to an angle of a triangle, it’s written like <ABC, where B is the angle being addressed.
- A tangent is a line intersecting the circle at only one point (just touching the edge outside).
- The point of tangency is the point where a tangent intersects the circle.
- In a circle, a chord is a line segment that joins two points on a circle.
- A perpendicular bisector comes from the center of a circle to divide a chord into two equal parts.
- A section of the circumference of a circle is an arc. The shorter arc is the minor arc, while the longer arc is the major arc.
- The central angle is the angle formed by joining the endpoints of an arc to the center of its circle. It’s facing away from the inscribed angle.
- The inscribed angle is the angle formed by joining the endpoints of an arc to some other point on its circle.
- The inscribed angle is equal to the central angle divided by two.
- One says that the central and inscribed angles of a circle are subtended by their minor arc.
Statistics and Probability
- There are three different types of probability:
- Theoretical probability uses the number of favorable outcomes over the number of possible outcomes. (1/6 chance to roll a 5 on a normal dice)
- Experimental probability calculates the probability of an event from experimental results. (there was a 1% light bulb was faulty)
- Subjective probability is decided by a “gut feeling”. (going to a casino because you’re feeling lucky)
- There are some possible problems with collecting data using a survey:
- A biased question influences responses to favor something related to a topic. (Do you think the price of a movie ticket is too high?)
- The use of a question’s language can lead people to give a particular answer. (If someone drops a wallet, your return it. Agree or disagree?)
- A question can be badly timed, leading to varying answers. (What do you think of winter tires? (conducted in summer))
- Surveys can be too long, causing some people not do the survey. (a 100 question survey)
- A survey can be very expensive to conduct. (a survey for every person in a nation (that isn’t the Vatican City or something))
- Questions may delve into private matters that people don’t want to reveal, leading to people not doing the survey. (Which of your parents do you like more?)
- A question might not be aware of other cultures that may not work too well with the question. (What’s your favorite way of cooking ham?)
- Yet another list of words! There are also some methods of sampling a population:
- With simple random sampling, each member of a population has an equal chance of being selected. (thirty random students being chosen from a school)
- Stratified random sampling has some members from each group of a population being randomly selected. (a random student is chosen from each class in a school)
- With cluster sampling, each member of a randomly chosen group of a population is selected. (every student from a random class is selected)
- Self-selected or voluntary sampling has only willing and interested members of a population participate. (any student who wants to do a survey in a school can do it)
- With convenience sampling, only members of a population who are convenient to include are selected. (surveying the students who sit next to you in class and are near your locker)
- Systematic or interval sampling has every nth member of a population selected. (sort the school’s students by last name and select every 30th student)