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1.1 – Types of Numbers

Section 1 (Apex Pg 1 – 8)

Objectives

  • Organize collections of numbers into mathematical sets.
Keywords
  • Set – a collection of objects.
  • Subset – a set that is part of a larger set.
  • Element – an element is an individual item of a set.
  • Counting Numbers / Natural Numbers – the numbers used to count: {1, 2, 3, 4, 5, …}.
    • They go on forever; they are infinite, positive integers.
  • Whole Numbers – the counting numbers (positive integers) and also zero:  {0, 1, 2, 3, 4, 5, …}.
    • They go on forever.
    • WhOle numbers have a zerO in them!
  • Integers – whole numbers and their opposites:  {…, –3, –2, –1, 0, 1, 2, 3, …}.
  • Positive Integers – these are the counting numbers / natural numbers.
    • They are greater than zero:  {1, 2, 3, 4, 5,…}.
  • Negative Integers – the integers that are less than zero.
    • They are the opposites of the positive integers  {…, –3, –2, –1}.
  • Opposites – two numbers that are equally far from zero in opposite directions along the number line.
    • Adding a minus (or negative) sign to a positive number creates its opposite.
    • Removing the minus (or negative) sign from a negative number creates its opposite.
Examples
1. Months
  • Set – all of the months of the year
  • Subset – winter months { December, January, February ]
  • Element – December
2. Numbers
  • Set – all numbers
  • Subset – positive, even integers: {2, 4, 6, 8, 10,…}
  • Element – 4

Notes

  • A set is often shown by listing its elements within curly braces, sometimes called brackets { }.
  • The ellipses (…) show that these numbers go on forever.
sets

 


Section 2 (Apex Pg 9 – 19)

Objectives

  • Convert fractions to decimals to percent.
Keywords
  • Fraction – a part, or parts, of the whole.
  • Numerator – the number of parts being considered.
  • Denominator – the total number of parts in the whole item.
  • Decimal – you can write every fraction as a decimal by dividing the numerator by the denominator.
  • Terminating Decimal – a number that ends
  • Repeating Decimal – a number that keeps on going forever
  • Percent – “per one hundred”
    • The symbol that represents percent is %.
    • Convert a decimal to a percentage by moving the decimal point two places to the right and putting % after it.
Notes
  • The symbol ≠ means “is not equal to.”
fraction decimal percentage

 

 

 

 dollars

decimals01

 


Section 3 (Apex Pg 20 – 32)

Objectives

  • Define rational and irrational numbers, and explore their properties.
  • Characterize numbers as fractions, exponents, and square roots.
  • Show the relationships between numbers with a number line and inequality symbols.

Key Words

  • Real Numbers – a set made up of all the numbers on the number line.
    • Any real number is either a rational or irrational number.
  • Rational Numbers – the set of all numbers that can be written as a fraction (which is a ratio of two integers, where the denominator is not zero).
    • The rational numbers are a subset of the real numbers.
  • Irrational Numbers – numbers that cannot be written in the form , where a and b are integers.
    • Irrational numbers cannot be written as terminating decimals or repeating decimals.
  • Proof by Contradiction – a method of proving that a proposition is true by showing that an assumption is false.
  • Proposition – a sentence that is either true or false.
  • Assumption – a statement that is accepted as true at the beginning of a proof.
  • Exponent – a small raised number that tells you how many times to multiply the base number by itself.
  • Base – a number in exponential form. It appears beneath the exponent.
Notes
  • Squares and Square Roots are inverse (opposite) operations
  • Rational Numbers
    • Terminating (ending) numbers such as 4.5, 3.75, 2, 1/4, 2
    • or Repeating numbers such as 0.222…, 3.876587658765…, 0.7777…
  • Irrational Numbers
    • They go on forever
    • and they do not have any pattern
    • Examples
      • π
      • √2, √3, √5, √7, etc.
      • e
  • Ratios make up Rational Numbers
    • Examples
      • 3/5 = 3:5 = 3 to 5
      • 2/7 = 2:7 = 2 to 7

 

Alg 1A 1.01 - Exponential Expressions

 

 

 

 

 

Alg 1A 1.01 - Exponents

 

 

 

 

 

 

 

 


Section 4 (Apex Pg 33 – 39)

Objectives

  • Show the relationships between numbers with a number line and inequality symbols.

 

Key Words

  • Number Line – a line that uses equally spaced marks to represent numbers.
    • Numbers to the right of zero are positive, and numbers to the left of zero are negative.
    • The greater a number is, the farther to the right it is placed on the number line.
  • Comparison Symbols – Symbols used to compare two numbers or expressions.
    • Comparison symbols include:

> (greater than)
< (less than)
≥ (greater than or equal to, at least)
≤ (less than or equal to, at most)
= (equal to)
 (not equal to)

  • at least : a number or more
  • at most: a number or smaller

numberline

Alg1A Inequalities

Alg1A 1.01 Comparison Symbols

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