Key Concepts

1.1 Use the Language of Algebra

  • Divisibility Tests
    A number is divisible by:
      2 if the last digit is 0, 2, 4, 6, or 8.
      3 if the sum of the digits is divisible by 3.
      5 if the last digit is 5 or 0.
      6 if it is divisible by both 2 and 3.
      10 if it ends with 0.
  • How to find the prime factorization of a composite number.
    1. Step 1. Find two factors whose product is the given number, and use these numbers to create two branches.
    2. Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
    3. Step 3. If a factor is not prime, write it as the product of two factors and continue the process.
    4. Step 4. Write the composite number as the product of all the circled primes.
  • How To Find the least common multiple using the prime factors method.
    1. Step 1. Write each number as a product of primes.
    2. Step 2. List the primes of each number. Match primes vertically when possible.
    3. Step 3. Bring down the columns.
    4. Step 4. Multiply the factors.
  • Equality Symbol
    a=ba=b is read “a is equal to b.”
    The symbol “=” is called the equal sign.
  • Inequality
    For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.
  • Inequality Symbols
  • G

  • anan means multiply a by itself, n times.
    The expression anan is read a to the nthnth power.
  • Simplify an Expression
    To simplify an expression, do all operations in the expression.
  • How to use the order of operations.
    1. Step 1. Parentheses and Other Grouping Symbols
      • Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
    2. Step 2. Exponents
      • Simplify all expressions with exponents.
    3. Step 3. Multiplication and Division
      • Perform all multiplication and division in order from left to right. These operations have equal priority.
    4. Step 4. Addition and Subtraction
      • Perform all addition and subtraction in order from left to right. These operations have equal priority.
  • How to combine like terms.
    1. Step 1. Identify like terms.
    2. Step 2. Rearrange the expression so like terms are together.
    3. Step 3. Add or subtract the coefficients and keep the same variable for each group of like terms.
  • A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.
  • Figure 1.9

1.5 Properties of Real Numbers