Üsküdar American Academy Prep Mathematics Glossary 2023-2024 PDF

Üsküdar American Academy Prep Mathematics 2023-2024 Summer Glossary 1- Numbers and Counting Cardinal Numbers: The numbers 1,2,3,4,5,…… are used to describe the number of elements in either finite or infinite sets. Composite Number: A positive integer that has factors other than just 1 an...

Üsküdar American Academy Prep Mathematics 2023-2024 Summer Glossary 1- Numbers and Counting Cardinal Numbers: The numbers 1,2,3,4,5,…… are used to describe the number of elements in either finite or infinite sets. Composite Number: A positive integer that has factors other than just 1 and the number itself. For example, 4, 6, 8, 9, 10, 12, etc. are all composite numbers. The number 1 is not composite. Decimal Number: A decimal number can be defined as a number whose whole number part and the fractional part is separated by a decimal point. The dot in a decimal number is called a decimal point. The digits following the decimal point show a value smaller than one. Digit (or Numeral): Any of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 used to write numbers. Even Number: An integer that is a multiple of 2. The even numbers are {... , –4, –2, 0, 2, 4, 6,... }. Infinity: The concept of something that is unlimited, endless, without bound. The common symbol for infinity, ∞. There are two types of infinity such that positive infinity, ∞ and negative infinity , −∞. Integers: All positive and negative whole numbers (including zero). That is, the set {... , –3, –2, –1, 0, 1, 2, 3,...}. Integers are indicated by either. Negative Number: A real number less than zero. Zero itself is neither negative nor positive. Number Line: A line representing the set of all real numbers. The number line is typically marked showing integer values. Ordinal Numbers: Numerical words that indicate order. The ordinal numbers are: first, second, third, fourth, etc. Positive Number: A real number greater than zero. Zero itself is not positive. Prime Number: A positive integer which has only 1 and the number itself as factors. Rational Numbers: All positive and negative fractions, including integers and so-called improper fractions. Formally, rational numbers are the set of all real numbers that can be written as a ratio of integers with nonzero denominator. Rational numbers are indicated by the symbol. Real Numbers: All numbers on the number line. This includes (but is not limited to) positives and negatives, integers and rational numbers, square roots, cube roots , π (pi), etc. Real numbers are indicated by either Zero: The number which indicates no quantity, size, or magnitude. Zero is neither negative nor positive. Note: Zero is the additive identity. 2- Basic Terminology Algebra: The mathematics of working with variables. Base: In plane geometry or solid geometry, the bottom of a figure. Brackets: The symbols [ and ]. Circumference: A complete circular arc. Circumference also means the distance around the the outside of a circle. Coordinate Plane (or Cartesian Plane): The plane formed by a horizontal axis and a vertical axis, often labeled the x-axis and y-axis, respectively. Compute: To figure out or evaluate. Constant: 1) As a noun, a term or expression with no variables. 2) As an adjective, constant means the same as fixed. That is, not changing or moving. Curly Brackets: The symbols { and }. Denominator: The bottom part of a fraction. Diameter of a Circle: A line segment between two points on the circle which passes through the center. Diagonal (of a Polygon): A line segment connecting non-adjacent vertices of a polygon. Equation: A mathematical sentence built from expressions using one or more equal signs (=). Evaluate: To figure out or compute. Expression: Numbers, symbols and operators (such as + and ×) grouped together that show the value of something. Formula: An expression used to calculate a desired result. Fraction: A ratio of numbers or variables. Fractions may not have denominator 0. Height (or Altitude): The shortest distance between the base of a geometric figure and its top. Horizontal: Perfectly flat and level. !" Improper Fraction: A fraction which has a larger numerator than denominator. For example, # is an improper fraction. Inequality: A mathematical sentence built from expressions using one or more of the symbols , ≤, or ≥. Mixed Fraction: A combination of a whole number and a proper fraction. Numerator: The top part of a fraction. Origin: On the coordinate plane, the point (0, 0). Parentheses: The symbols ( and ). Singular: parenthesis. Perimeter: The distance around the outside of a plane figure. For a polygon, the perimeter is the sum of the lengths of the sides. Polygon: A closed plane figure for which all sides are line segments. The name of a polygon describes the number of sides. A polygon which has all sides mutually congruent and all angles mutually congruent is called a regular polygon. $ Proper Fraction: A fraction with a smaller numerator than denominator. For example, % is a proper fraction. Radius (plural- Radii): A line segment between the center and a point on the circle or sphere. The word radius also refers to the length of this segment. Rectangle: A box shape on a plane. Formally, a rectangle is a quadrilateral with four congruent angles (all 90°). Reduce a Fraction: Simplify. That is, cancel out all common factors in the numerator and denominator until no common factors remain. Side of a Polygon: Any of the line segments that make up a polygon. For example, a triangle has three sides. Simplify: To use the rules of arithmetic and algebra to rewrite an expression in the simplest form. Square: A rectangle with all four sides of equal length. Formally, a square is a quadrilateral with four congruent sides and four congruent angles (all 90°). Triple: Multiply by three. Variable: A quantity that can change or that may take on different values. Vertex (plural- Vertices): 1) The common endpoint of two or more rays or line segments. Vertex typically means a corner or a point where lines meet. The word vertex is most commonly used to denote the corners of a polygon. 2) Vertex is also sometimes used to indicate the 'top' or high point of something, such as the vertex of an isosceles triangle , which is the 'top' corner opposite its base. Vertical: Straight up and down. For example, a wall is vertical. Volume: The total amount of space enclosed in a solid. 3- Basic Operations Addition: Addend: can be defined as the numbers or terms added together to form the sum. Sum (Total): The result of adding a set of numbers or algebraic expressions. Division: Dividend: a number to be divided by another number. Divisor: a number that divides into another without a remainder. Quotient: The result of dividing two numbers or expressions. For example, the 40 divided by has a quotient of 8. Note: 43 divided by 5 has a quotient of 8 and a remainder of 3. Remainder: The part left over after long division. Multiplication: Factors: The numbers that we multiply are the factors of the product. Product: The result of multiplying a set of numbers or expressions. Subtraction: Difference: The result of subtracting two numbers or expressions. Minuend: The number in a subtraction sentence from which we subtract another number is called a minuend. Subtrahend: The number which we subtract from another number in a subtraction sentence is called a subtrahend. 4- Place Value and Rounding Place Value: In math, every digit in a number has a place value. Place value can be defined as the value represented by a digit in a number on the basis of its position in the number. Example: Rounding a Number: A method of approximating a number using a nearby number at a given degree of accuracy. The rounding rules are below: Terminating Decimal Numbers: Decimal numbers have a finite number of digits after the decimal point. Non-Terminating Decimal Numbers: The digits after the decimal point of non - terminating decimals repeat endlessly. In other words, we can say that these decimal numbers have an infinite number of digits after the decimal point. These decimal numbers are further divided into recurring and non - recurring decimal numbers. 1) Recurring (Repeating) Decimal Numbers: These decimal numbers have an infinite number of digits after the decimal point, however, these digits are repeated at regular intervals. Examples: 120.353535.... ; 2.22222.... 2) Non-Recurring (Non-Repeating) Decimal Numbers: These are non -terminating, non -repeating decimal numbers. These decimal numbers have not only an infinite number of digits at their decimal places, but their decimal place digits do not follow a specific order. Examples: 1.3687493043.... ; 456.789321633894... ; 1.2376894.... 5- Exponents and Scientific Notation Base: x in the expression ax. For example, 3 is the exponent in 23. Exponent (or Power): x in the expression ax. Scientific Notation: A standardized way of writing real numbers. In scientific notation, all real numbers are written in the form a·10b, where 1 ≤ a < 10 and b is an integer. For example, 351 is written 3.51·102 in scientific notation. Power: The result of raising a base to an exponent. 6- Properties of Number Systems Additive Inverse (or Opposite): Two numbers are additive inverses if they add to give a sum of zero. Associative Property: To “associate” means to connect or join with something. According to the associative property of addition, the sum of three or more numbers remains the same regardless of how the numbers are grouped. Let’s sing: When adding numbers three or more, Group it anyway, you will still score! Don’t let adding in order be the aim, Because the sum will just be the same! Commutative Property: states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. Distributive Property: To “distribute” means to divide something or give a share or part of something. According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together. Let’s sing: To multiply big numbers, break one apart, Multiply with the addends, right at the start, Now, add the products, part by part. The answer is right! You’re so smart! ! Reciprocal (or Multiplicative Inverse): The reciprocal of x is &. In other words, a reciprocal is a fraction flipped upside down. Multiplicative inverse means the same thing as reciprocal. Zero-product Property: simply states that if ab=0, then either a=0 or b=0. 7- Rational Expressions Ratio: The result of dividing one number or expression by another. 8- Factoring Expand (Distribute): To multiply an algebraic expression by a one-term expression, the distributive law is applied: a(b+c)= ab + ac. This process is called expanding. Factoring (Factorizing): Finding what to multiply to get an expression. Factor of an Integer: Any integer which divides evenly into a given integer. For example, 8 is a factor of 24. Factor Tree: A structure used to find the prime factorization of a positive integer. Perfect Square: Any number that is the square of a rational number. For example, 0, 1, 4, 9, 16, 25, ! ' etc. are all perfect squares. So are "% and (. 9- Solution Sets Absolute Value: The absolute value of a number is the distance between the number and the origin. , shown by | |. Closed Interval: An interval that contains its endpoints. Interval: The set of all real numbers between two given numbers. Open Interval: An interval that does not contain its endpoints. Ordered pair: Two numbers written in the form (x, y). Ordered triple: Three numbers written in the form (x, y, z). Solution (Solution Set): Any and all value(s) of the variable(s) that satisfies an equation, inequality, system of equations, or system of inequalities. With a system of equations or system of inequalities, the solution set is the set containing value(s) of the variable(s) that satisfy all equations and/or inequalities in the system. Solve: Find all solutions to an equation, inequality, or a system of equations and/or inequalities. System of Equations: Two or more equations containing common variable(s). System of Inequalities: Two or more inequalities containing common variable(s). 10- Prime Numbers Greatest Common Factor GCF: The largest integer that divides evenly into each of a given set of numbers. Least Common Multiple LCM: The smallest positive integer into which two or more integers divide evenly. Prime Factorization: Writing an integer as a product of powers of prime numbers. Relatively Prime: Describes two numbers for which the only common factor is 1. In other words, relatively prime numbers have a greatest common factor (gcf) of 1. For example, 6 and 35 are relatively prime (gcf = 1). The numbers 6 and 8 are not relatively prime (gcf = 2). 11- Radicals Cube Root: A number that must be multiplied by itself three times to equal a given number. The cube " ! root of x is written √𝑥 or 𝑥 !. Radical Symbol: The symbol, which is used to indicate square roots or nth roots. Radicand: The number under the √ (radical) symbol. Root of a Number: A term that can refer to the square root or nth root of a number. Square Root: A nonnegative number that must be multiplied by itself to equal a given number. The square root of x is written √𝑥 or x½. For example, √9 = 3 since 32 = 9. 12- Basic Geometry Acute Angle: An angle that has measure less than 90°. Adjacent: Next to; neighboring. Adjacent Angles: Two angles in a plane which share a common vertex and a common side but do not overlap. Angles 1 and 2 below are adjacent angles. Alternate Interior Angles: Two interior angles which lie on different parallel lines and on opposite sides of a transversal. Alternate interior angles are congruent. In the drawing below, angles 3 and 6 are alternate interior angles, as are angles 4 and 5. Alternate Exterior Angles: Two exterior angles on opposite sides of a transversal which lie on different parallel lines. Alternate exterior angles are congruent. In the drawing below, angles 1 and 8 are alternate exterior angles, as are angles 2 and 7. Angle: 1) A combination of two rays with a common endpoint. 2) An angle measures how much bending or turning a ray (=one side of the angle) does to get into the position of another ray (=the other side of the angle). The size of an angle is measured in degrees. Acute Triangle: A triangle for which all interior angles are acute (less than 90 degrees). Example: Coincident Lines: Two lines that lie on top of one another. (Coincident lines have all points in common!) Complementary Angles: Two acute angles that add up to 90°. For example, 40° and 50° are complementary. In the diagram below, angles 1 and 2 are complementary. Congruent: Exactly equal in size and shape. Congruent sides or segments have the exact same length. Congruent angles have the exact same measure. Corresponding Angles: When two parallel lines crossed by another line (called Transversal Line), the angles in matching corners are called Corresponding Angles. Degree: A unit of angle measure. Equiangular Triangle: A triangle with three congruent angles. In Euclidean geometry, all equiangular triangles are equilateral and vice-versa. Equilateral Triangle: A triangle with three congruent sides. Full Angle: An angle whose measure is 360 degrees. Hypotenuse: The side of a right triangle opposite side of the right angle. Interior Angle: An angle on the interior of a plane figure. Intersecting Lines: Two lines crossing each other at one point. Isosceles Triangle: A triangle with two sides that have the same length. Formally, an isosceles triangle is a triangle with at least two congruent sides. Line: An infinite set of points in one dimesion. Line Segment: A section of a line between two points. line segment Measurement: The process of assigning a number to a physical property. Examples of measurement include length, size of an angle, area, volume, mass, time, etc. Midpoint: The point halfway between two given points. Obtuse Angle: An angle that has measure more than 90° and less than 180°. Obtuse Triangle: A triangle which has an obtuse angle as one of its interior angles. Example: Parallel Lines: Two lines in the plane that do not intersect each other at all. Perpendicular: At a 90° angle. Perpendicular Lines: Two lines in the same plane that meet at 90-degree angles (right angles). Plane: A flat surface extending endlessly in all directions. Point: The geometric figure formed at the intersection of two distinct lines. Ray: A part of a line starting at a particular point and extending infinitely in one direction. Reflex Angle: An angle whose measure is greater than 180 degrees and less than 360 degrees. Right Angle: A 90° angle. Right Triangle: A triangle which has a right (90°) interior angle. Scalene Triangle: A triangle for which all three sides have different lengths. Straight Angle: An angle whose measure is exactly 180 degrees. Supplement of an Angle: For any angle A between 0° and 180°, the supplement of A is 180° – A. Example: p and q are supplementary angles. Transversal: A line that cuts across a set of lines or the sides of a plane figure. Triangle: is a closed, two-dimensional shape with three straight sides. Vertical Angles: Vertical angles are angles opposite one another at the intersection of two lines. Vertical angles are congruent. In the diagram below, 2 and 4 are vertical angles. So are 1 and 3. 13- Sets Cardinality of a Set: The number of elements in a set., whether the set is finite or infinite. Note: Not all infinite sets have the same cardinality. Complement of a Set: The elements not contained in a given set. The complement of set A is indicated by AC. Countable: Describes the cardinality of a countably infinite set. Difference of Two Sets: Difference of sets A and B is the set of elements which belong to A, but do not belong to B, denoted by A-B or A\B. A-B = {x: x∈A but x∉B} Disjoint Sets: Two or more sets which have no elements in common. Distinct: Different. Elements (or Members): Different objects forming a set. Elements are denoted by small letters. The symbols are below: ∈ : ‘is an element of’ ∉ : ‘is not element of’ Empty Set (Null Set): The set with no elements. The empty set can be written or {}. Equal Sets: Set A and B are equal sets if they contain identical elements. Equivalent Sets: Sets which contain the same number of elements are called as equivalent. Alternatively, sets whose cardinalities are the same. Finite Set: A set whose elements are countable. Infinite Set: A set whose elements are uncountable, so which describes a set which is not finite. Intersection of Sets: Intersection of two sets A and B is the set of all elements which belong to both set A and set B, denoted by A∩B. A∩B ={x: x∈A and x∈B} Union of Sets: The union of two sets A and B is the set of all elements which belong to A or B, denoted by A∪B. A∪B ={x: x∈A or x∈B} Universal Set: A set containing all elements being discussed is called the universal set labeled by the capital letter U. Venn Diagram: Elements of a set are written in a closed curve or in any geometric shape such as rectangle and square. List Notation (Listing/ Roster Method): Elements of a set are listed by comma inside curly brackets. Set: A well-defined collection of distinct objects. - The word ‘well-defined’ refers to a specific property which eases to identify whether the given object belongs to the set or not. - The word ‘distinct’ means that the objects of a set must be all different. Sets are represented by capital letters such as A and B. Set-builder Notation: Elements of a set are described by a common property. A simple example of set-builder notation is below: Subset of a set: Set A is a subset of set B if all of the elements (if any) of set A are contained in set B. SOME USEFUL VIDEO LINKS: https://edpuzzle.com/media/5e736da8afb4c73dcdc682c3 (Solving Absolute Value Equations) https://edpuzzle.com/media/5e73a3e1d3c9903da2750a31 https://edpuzzle.com/media/5e73b36245f2503d8e071a7f (GCD&LCM) https://edpuzzle.com/media/5e73a8ff338c073e3984cdbb (GCD) (Keyword: monomial- an algebraic expression consisting of one term) https://edpuzzle.com/media/5e60edee726bb240b1416322 (LCM) https://edpuzzle.com/media/5e73e3d09c1cd73ebc34b904 (GCD/GCF and LCM problems) https://drive.google.com/file/d/1P2vzFCuqX0lJ1tToNEZPYMJgVzqFHVHu/view (Video record of Lecture Notes 13: Radicals- Part1) https://drive.google.com/file/d/1b_qF651aBPzOfcJGgCr0kymvWh6bU6-U/view (Video record of Lecture Notes 13: Radicals- Part2) https://edpuzzle.com/media/5e9053d7e957223ef2531c61 (Simplifying Radicals) https://edpuzzle.com/media/5e90573338498c3f15a448e7 (Simplify Radicals With Variable Radicands)