Categorie: Tutti - division - operations - exponents - grouping

da David Kedrowski mancano 14 anni

249

MAT.105X A1

The fundamental principles of mathematics include the order of operations, which dictate the sequence in which mathematical expressions should be simplified. The standard hierarchy begins with simplifying any expressions within grouping symbols, such as parentheses.

MAT.105X A1

MAT.105X A1

Define and Identify Sets of Numbers

Real Numbers

All rational and irrational numbers taken together.

Any number you can find on a number line.

Irrational Numbers

Numbers that cannot be written as a ratio of two integers.

Includes all nonrepeating, nonterminating decimals

Rational Numbers

Any number of the form p/q, where p and q are integers and q is not zero

Includes all terminating decimals and all nonterminating repeating decimals

Integers

{ . . . , -3, -2, -1, 0, 1, 2, 3, . . . }

Whole Numbers

{ 0, 1, 2, 3, . . . }

Natural Numbers

{ 1, 2, 3, . . . }

Define Absolute Value and Perform Operations on Real Numbers

When subtracting two real numbers:

  • convert the problem to addition and then follow those rules

  • Remember, to convert a subtraction problem to addition, you add the opposite of the value between subtracted

    When adding two real numbers:

  • if both numbers are positive, add them to get a positive answer
  • if both numbers are negative, add their absolute values and then make the result negative
  • if one number is negative and one number is positive, take the difference between their absolute values and apply the sign of the number with the larger absolute value
  • When dividing two real numbers:

  • if both numbers are positive, the answer is positive
  • if both numbers are negative, the answer is positive
  • if one number is negative and one number is positive, the answer is negative

  • Remember:

  • if the numerator is zero, the answer is zero
  • if the denominator is zero, the answer is undefined
  • both results have no sign
  • When multiplying two real numbers:

  • if both numbers are positive, the answer is positive
  • if both numbers are negative, the answer is positive
  • if one number is negative and one number is positive, the answer is negative

  • When multiplying more than two real numbers:

  • if there are an even number of negative factors, the answer is positive
  • if there are an odd number of negative factors, the answer is negative
  • Absolute Value

    The absolute value of a number is the distance between that number and 0 on the number line

    The absolute value of a number is always positive or zero (nonnegative).

    Learn the Vocabulary for Algebraic Expressions

    Evaluate

    Substituting values for the variables in an algebraic expression and simplifying using the order of operations.

    Coefficient

    Each term has a coefficient, or numeric part.

    Term

    A term is a numbers or a variable or a product or quotient of numbers and variables.

    Algebraic Expression

    An algebraic expression is a collection of numbers, variables, and grouping symbols connected by operations symbols.

    Learn the Properties of Real Numbers

    Distributive Properties

    Multiplication over addition: a(b+c) = ab + ac

    Multiplication over subtraction: a(b-c) = ab - ac

    Inverse Properties

    Addition: a+(-a) = 0

    Multiplication: a(1/a) = 1 [a not zero]

    Identity Properties

    Addition: a+0 = a

    Multiplication: 1a = a

    Associative Properties

    Addition: a+(b+c) = (a+b)+c

    Multiplication: a(bc) = (ab)c

    Commutative Properties

    Addition: a+b = b+a

    Multiplication: ab = ba

    Review Concepts from Geometry

    Circle

    The locus (or set) of all points in a plane a given distance (radius) from a given point in the plane (the center of the circle)

    Circumference & Area

    Circumference: 2 x pi x radius, or pi x diameter

    Area: pi x radius squared

    Trapezoid

    Quadrilateral with one pair of opposite sides parallel

    Perimeter: The sum of the length of all four sides

    Area: average length of the bases x height

    Parallelogram

    Quadrilateral with opposite sides parallel

    Perimeter: The sum of the length of all four sides

    Area: base x height

    Triangle

    Three sided enclosed figure

    Perimeter: The sum of the length of all three sides

    Area: 0.5 x base x height

    Rectangle / Square

    Rectangle: Parallelogram with four right angles

    Square: Rectangle with all sides equal

    Perimeter & Area

    Perimeter: The sum of the length of all four sides

    Area: Base x height

    Angles
    Supplementary

    Two angles that add to 180 degrees

    Complementary

    Two angles that add to 90 degrees

    The Order of Operations

    4. Addition and Subtraction

    Perform addition and subtraction from left to right.

    3. Multiplication and Division

    Perform multiplication and division from left to right.

    2. Exponents

    Simplify expressions with exponents

    1. Grouping Symbols

    If parentheses or other grouping symbols appear in an expression, simplify what is in these grouping symbols first.

    Multiply, Divide, Add, and Subtract Fractions

    Subtract

    Put the fractions over a common denominator

    Subtract the second numerator from the first numerator and keep the common denominator

    Write the answer in simplest terms

    Add

    Put all fractions over a common denominator

    Add the numerators and keep the common denominator

    Write the answer in simplest terms

    Divide

    Make sure denominators are not zero at every step

    Multiply the first fraction by the reciprocal of the second fraction

    Write the answer in simplest terms

    Multiply

    Denominators cannot be zero

    Multiply straight across and express the answer in simplest terms