QUARTERLY EXAM 1 Rules & Definitions Shormann Algebra 1

Like signs

In both multiplication and division, an operation between two numbers with the same sign always results in a positive answer.

concrete

In mathematics, this is a word used to describe real objects. Abstract ideas are always based on real things.

Whole numbers, and all negative numbers.

Integers

Relation

Like a function, except the input (x), can map to more than one output (y).

Euclid's 5 axioms, or common notions: A5

The whole is greater than the part.

Euclid's 5 axioms, or common notions: A1

Things that are equal to the same thing are also equal to one another. If a = c and b = c, then a = b

Euclid's 5 axioms, or common notions: A4

Things which coincide with one another are equal to one another. In other words, a = a (reflexive axiom)

Solid

Three dimensional figures such as a polyhedron formed by four or more polygons that intersect only at their edges; a cone, with a circular base and a lateral surface that comes to a point; a cylinder, with two parallel circular bases connected by a lateral surface; or a sphere, which is the set of points a given distance from a given point called the center.

Systems of Equations

To solve any system of equations, the number of equations must equal the number of unknowns (variables).

Parallel

Two lines are considered parallel if they never intersect.

Angle

Two rays in the same plane that share a common starting point, and do not overlap.

Absolute value symbol

Two vertical bars enclosing a number, like this: |a|.

Euclid's 5 postulates: P1

Two, and only two points determine one unique straight line.

Indeterminate

When both the numerator and denominator equal zero.

Analytical Geometry

Where algebra and geometry meet on the coordinate plane.

Vertex

Where two sides of a polygon, or two rays of an angle, meet. Plural is "vertices"

Additive identity

a+0 = a. Any number plus zero equals that number.

Multiplicative identity

a·1=a. Any number times 1 equals that number.

The standard form for the equation of a line is

y = mx + b Where m is the slope and b is the y-intercept. Thinking of this as a function, y is the dependent variable, and x is the independent variable.

Linear Inequalities:

y > mx + b, and y < mx + b are represented by dashed lines. y ≥ mx + b, and y ≤ mx + b are represented by solid lines.

Euclid's 5 postulates: P5

(Parallel postulate) Given a line N and a point P not on that line, there exists in the plane of P and N and through P only one line M, which does not intersect line N.

Euclid's 5 postulates: P3

A circle may be drawn with any given center and any given radius.

Data

A collection of facts and information recorded during an experiment, from which conclusions and decisions can be made

Mathematical Proof

A deductive argument where rules and definitions are applied to reach a logical conclusion.

Plane

A flat surface having length and width only.

Undefined

A function is said to be undefined when a value is not part of it's domain. For example, when x equals 0 in f(x) = 1/x. We would say the function is undefined at x = 0 because it's output equals 1/0 = ∞, which we cannot define.

Diameter

Any line segment drawn through the center of a circle and terminated in both directions by the circumference. The radius equals half the diameter

Postulate

Assumed to be true without proof

You can change the way any fraction looks, without changing its value

By either multiplying the numerator and denominator by the same number, or dividing the numerator and denominator by the same number. This works because any number divided by itself equals 1.

Slope

Change in y over change in x: "Rise over run"

Numbers used to count objects. Ex. 1, 2, 3, ..... - whole numbers: Counting numbers and the number 0.

Counting (natural) numbers:

Commutative property for addition

If a + b = c, then b + a = c Order does not matter. Example: 2+3 = 5, and 3 + 2 = 5

Algebraic subtraction

If a and b are real numbers, then a-b = a + (-b), and -b is the opposite of b. Example: 5-3 = 5 + (-3)

Multiplicative property of equality

If a, b, and c represent real numbers, and if a=b, then ca = cb. Also, ac = bc, and, a/c =b/c

Additive property of equality

If a, b, and c represent real numbers, and if a=b, then a + c = b + c. Also, c + a = c + b

Commutative property for multiplication

If ab = c, then ba = c Order does not matter. Example: 2·3 = 6, and 3·2 = 6

Euclid's 5 axioms, or common notions: A2

If equals be added to equals, the wholes are equal. If a = b and c = d, then a + c = b + d

Euclid's 5 axioms, or common notions: A3

If equals be subtracted from equals, the remainders are equal. If a = b and c = d, then a - c = b - d

The result of taking the square root of a negative number. Normally, the −1 is factored out and exchanged for the symbol "i". i^2 = -1.

Imaginary numbers

Hypotenuse

In a right triangle, it is the side opposite the right angle.

Triangles

In any triangle, the sum of the measures of the three angles equals 180°. In any triangle, the angles opposite sides of equal lengths have equal measures, and vice-versa.

Unlike signs

In both multiplication and division, an operation between two numbers with different signs always results in a negative answer.

The reciprocal of x is

1/X

Perimeter of a Rectangle

2(l + w)

Perimeter of a Square

4(s)

Associative property for addition

(a + b) + c = a + (b + c) When you have more than two numbers to add, the way you group them in pairs does not matter. Example: (2+3)+4 = (5) + 4 = 9, and 2+(3+4) = 2+(7) = 9

Associative property for multiplication

(a·b)c = a(b·c) When you have more than two numbers to multiply, the way you group them in pairs does not matter. Example: (2·3)·4 = (6)·4 = 24, and 2·(3·4) = 2·(12) = 24

Logic

(~ = "not") Conditional Statement; If p, then q Converse; If q, then p Inverse; If ~p, then ~q Contrapositive; If ~q, then ~p

Algebra

A generalization of arithmetic, where letters representing numbers are combined according to the rules of arithmetic, often to solve for an unknown value.

Transversal

A line that intersects two or more parallel lines, and is not perpendicular to those lines.

Line Segment

A line with a start point and end point.

Ray

A line with a starting point but no end point.

Prime number

A number that is only divisible by itself and 1, such as 2,3, 5, 7, 11, etc.

Absolute value

A number which describes the distance from zero. It is always positive.

Ordered Pair

A pair of numbers, written in a specific order. On a coordinate plane, an ordered pair is used to identify the location of a point, and has the form (x,y).

Quadrilaterals

A parallelogram has two pairs of parallel sides; a trapezoid has exactly one pair of parallel sides; a rectangle is a parallelogram with four right angles; a rhombus is an equilateral parallelogram; a square is a rhombus with four right angles.

Concave Polygon

A polygon that has at least one indentation

Convex Polygon

A polygon that has no indentations.

Function

A relationship where the output (y) depends on the input (x). Each input, or domain value, maps to one, and only one value in the output, or range.

Tesselation

A repeating pattern of shapes that covers a plane with no gaps or overlaps

Triangles by Angles

A right triangle has one right angle, an obtuse triangle has one angle greater than a right angle, and an acute triangle has three acute angles.

Side

A segment of a polygon.

Axiom

A self-evident statement about something obvious, normally common to all sciences.

Euclid's 5 postulates: P2

A straight line extends an indefinite length in either direction.

numeral

A symbol used to express the idea of a number.

Line

A widthless length. Its location is represented on paper by using a pencil and straight edge.

Circle

All points drawn in a plane that are equidistant from one center point.

Euclid's 5 postulates: P4

All right angles are equal to one another

Angles

All right angles equal 90° All straight angles equal 180° (two right angles)

Cartesian Coordinate System

Also called a coordinate plane, it is a plane containing a horizontal,"x"-axis and vertical,"y"-axis. It is used to graph ordered pairs, functions, experimental data, etc

Theorem

Also called propositions, these are true mathematical statements requiring proof.

Obtuse Angle

An angle greater than a right angle

Acute Angle

An angle less than a right angle

Triangles by Sides

An equilateral triangle has all sides congruent (equal), an isosceles triangle has two sides congruent, and a scalene triangle has no sides congruent.

Fractal

An object or quantity that displays self similarity on all scales

abstract

Dealing with properties and ideas of things, just because they are things regardless of feelings, emotions and sensations we might connect with them. An idea that can describe an infinite number of concrete objects. ie: The number 3 is an abstract idea that can describe 3 bears, 3 cars...ect.

Any of the Hindu-Arabic numerals 1 through 9, and 0.

Digit

Polygons

For any polygon, the number of sides equals the number of vertices. For any convex polygon of N sides, the sum of the measures of the interior angles equals (N - 2)180°.

Right Angle

Formed when two lines or lines segments intersect, forming 4 adjacent angles that are equal to each other. Lines or line segments forming right angles are perpendicular to each other.

Memorize these Fraction- Decimal- Percent Conversions

Fraction Decimal (0.xx) Percent (xx%) 1/2 0.50 50% 1/4 0.25 25% 1/8 0.125 12.5% 1/3 0.3 33.3% 1/6 0.16 16.6% 1/5 0.2 20% 1/10 0.1 10%

Proportional

Having a constant ratio.

Any number used to describe a positive or negative number. Includes all integers, and all decimal numbers and fractions.

Real numbers:

Analogy

Resemblance in some particulars between things otherwise unlike; similar.

Polygon

Simple, closed, coplanar geometric figures whose sides are straight lines.

%

Symbol for percent, which is an abbreviation for "per 100". For example 50% is the same thing as writing 50/100 , it's just faster to write or type 50%

Point

That which has no part. Its location is represented by a dot.

Denominator

The bottom value in a fraction.

Y-intercept

The location where a function crosses the y-axis.

deductive reasoning

The process of applying rules.

inductive reasoning

The process of discovering rules.

Ratio

The size of one thing relative to another.

Numerator

The top value in a fraction.

Distributive property

property: a(b+c) = a·b + a·c "Distribute" the a over the b and the c. Example: 3(4+5) = 3(4)+3(5) = 12+15=27.