1.7.2: Key Concepts

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
  $a=b$ is read “a is equal to b.”
  The symbol “=” is called the equal sign.
• Inequality
  
• Inequality Symbols

  Inequality Symbols	Words
  $a\ne b$	a is not equal to b.
  $a<b$	a is less than b.
  $a\le b$	a is less than or equal to b.
  $a>b$	a is greater than b.
  $a\ge b$	a is greater than or equal to b.

  Table 1.4
• Grouping Symbols
  $\begin{array}{cccccc}\text{Parentheses}\hfill & & & & & \left(\right)\hfill \\ \text{Brackets}\hfill & & & & & \left[\right]\hfill \\ \text{Braces}\hfill & & & & & \left\{\right\}\hfill \end{array}$
• Exponential Notation
  $a^n$ means multiply a by itself, n times.
  The expression $a^n$ is read a to the $n^{th}$ 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.

  Operation	Phrase	Expression
  Addition	a plus b the sum of $a$ and b a increased by b b more than a the total of a and b b added to a	$a+b$
  Subtraction	a minus $b$ the difference of a and b a decreased by b b less than a b subtracted from a	$a-b$
  Multiplication	a times b the product of $a$ and $b$ twice a	$a·b,ab,a(b),(a)(b)$ $2a$
  Division	a divided by b the quotient of a and b the ratio of a and b b divided into a	$a÷b,a\text{/}b,\frac{a}{b},b\overline{)a}$

  Table 1.5

1.2 Integers

• Opposite Notation
  $$\begin{array}{c}\text{−}a\text{means the opposite of the number}a\hfill \\ \text{The notation}\text{−}a\text{is read as “the opposite of}a\text{.”}\hfill \end{array}$$

• Absolute Value
  The absolute value of a number is its distance from 0 on the number line.
  The absolute value of a number n is written as $\left|n\right|$ and $\left|n\right|\ge 0$ for all numbers.
  Absolute values are always greater than or equal to zero.
• Grouping Symbols
  $$\begin{array}{cccccccccc}\text{Parentheses}\hfill & & & \left(\right)\hfill & & & \text{Braces}\hfill & & & \left\{\right\}\hfill \\ \text{Brackets}\hfill & & & \left[\right]\hfill & & & \text{Absolute value}\hfill & & & \left|\right|\hfill \end{array}$$

• Subtraction Property
  $a-b=a+\left(\text{−}b\right)$
  Subtracting a number is the same as adding its opposite.
• Multiplication and Division of Signed Numbers
  For multiplication and division of two signed numbers:

  Same signs	Result
  • Two positives	Positive
  • Two negatives	Positive

  
  If the signs are the same, the result is positive.

  Different signs	Result
  • Positive and negative	Negative
  • Negative and positive	Negative

  
  If the signs are different, the result is negative.
• Multiplication by $\mathrm{-1}$
  $\mathrm{-1}a=\text{−}a$
  Multiplying a number by $\mathrm{-1}$ gives its opposite.
• How to Use Integers in Applications.
  1. Step 1. Read the problem. Make sure all the words and ideas are understood
  2. Step 2. Identify what we are asked to find.
  3. Step 3. Write a phrase that gives the information to find it.
  4. Step 4. Translate the phrase to an expression.
  5. Step 5. Simplify the expression.
  6. Step 6. Answer the question with a complete sentence.

1.3 Fractions

• Equivalent Fractions Property
  If a, b, and c are numbers where $b\ne 0,c\ne 0,$ then
  $\frac{a}{b}=\frac{a·c}{b·c}\text{and}\frac{a·c}{b·c}=\frac{a}{b}.$
• How to simplify a fraction.
  1. Step 1. Rewrite the numerator and denominator to show the common factors.
     If needed, factor the numerator and denominator into prime numbers first.
  2. Step 2. Simplify using the Equivalent Fractions Property by dividing out common factors.
  3. Step 3. Multiply any remaining factors.
• Fraction Multiplication
  If a, b, c, and d are numbers where $b\ne 0,$ and $d\ne 0,$ then
  $\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}.$
  To multiply fractions, multiply the numerators and multiply the denominators.
• Fraction Division
  If a, b, c, and d are numbers where $b\ne 0,c\ne 0,$ and $d\ne 0,$ then
  $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}.$
  To divide fractions, we multiply the first fraction by the reciprocal of the second.
• Fraction Addition and Subtraction
  If a, b, and c are numbers where $c\ne 0,$ then
  $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\text{and}\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}.$
  To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
• How to add or subtract fractions.
  1. Step 1. Do they have a common denominator?
     • Yes—go to step 2.
     • No—rewrite each fraction with the LCD (least common denominator).
       • Find the LCD.
       • Change each fraction into an equivalent fraction with the LCD as its denominator.
  2. Step 2. Add or subtract the fractions.
  3. Step 3. Simplify, if possible.
• How to simplify an expression with a fraction bar.
  1. Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
  2. Step 2. Simplify the fraction.
• Placement of Negative Sign in a Fraction
  For any positive numbers a and b,
  $\frac{\text{−}a}{b}=\frac{a}{\text{−}b}=-\frac{a}{b}.$
• How to simplify complex fractions.
  1. Step 1. Simplify the numerator.
  2. Step 2. Simplify the denominator.
  3. Step 3. Divide the numerator by the denominator. Simplify if possible.

1.4 Decimals

• How to round decimals.
  1. Step 1. Locate the given place value and mark it with an arrow.
  2. Step 2. Underline the digit to the right of the place value.
  3. Step 3. Is the underlined digit greater than or equal to $5?$
     • Yes: add 1 to the digit in the given place value.
     • No: do not change the digit in the given place value
  4. Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.
• How to add or subtract decimals.
  1. Step 1. Determine the sign of the sum or difference.
  2. Step 2. Write the numbers so the decimal points line up vertically.
  3. Step 3. Use zeros as placeholders, as needed.
  4. Step 4. Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.
  5. Step 5. Write the sum or difference with the appropriate sign
• How to multiply decimals.
  1. Step 1. Determine the sign of the product.
  2. Step 2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
  3. Step 3. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
  4. Step 4. Write the product with the appropriate sign.
• How to multiply a decimal by a power of ten.
  1. Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
  2. Step 2. Add zeros at the end of the number as needed.
• How to divide decimals.
  1. Step 1. Determine the sign of the quotient.
  2. Step 2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places—adding zeros as needed.
  3. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
  4. Step 4. Write the quotient with the appropriate sign.
• How to convert a decimal to a proper fraction and a fraction to a decimal.
  1. Step 1. To convert a decimal to a proper fraction, determine the place value of the final digit.
  2. Step 2. Write the fraction.
     • numerator—the “numbers” to the right of the decimal point
     • denominator—the place value corresponding to the final digit
  3. Step 3. To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.
• How to convert a percent to a decimal and a decimal to a percent.
  1. Step 1. To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.
  2. Step 2. To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.
• Square Root Notation
  $\sqrt{m}$ is read “the square root of m.”
  If $m=n^2,$ then $\sqrt{m}=n,$ for $n\ge 0.$
  The square root of m, $\sqrt{m},$ is the positive number whose square is m.
• Rational or Irrational
  If the decimal form of a number
  • repeats or stops, the number is a rational number.
  • does not repeat and does not stop, the number is an irrational number.
• Real Numbers

  

  Figure 1.9

1.5 Properties of Real Numbers

Commutative Property When adding or multiplying, changing the order gives the same result $\begin{array}{cccccc}\text{of addition}\text{If}a,b\text{are real numbers, then}\hfill & & & \hfill a+b& =\hfill & b+a\hfill \\ \text{of multiplication}\text{If}a,b\text{are real numbers, then}\hfill & & & \hfill a·b& =\hfill & b·a\hfill \end{array}$

Associative Property When adding or multiplying, changing the grouping gives the same result. $\begin{array}{cccccc}\text{of addition}\text{If}a,b,\text{and}c\text{are real numbers, then}\hfill & & & \hfill \left(a+b\right)+c& =\hfill & a+\left(b+c\right)\hfill \\ \text{of multiplication}\text{If}a,b,\text{and}c\text{are real numbers, then}\hfill & & & \hfill \left(a·b\right)·c& =\hfill & a·\left(b·c\right)\hfill \end{array}$

Distributive Property $\begin{array}{cccccc}\text{If}a,b,\text{and}c\text{are real numbers, then}\hfill & & & \hfill a\left(b+c\right)& =\hfill & ab+ac\hfill \\ \\ & & & \hfill \left(b+c\right)a& =\hfill & ba+ca\hfill \\ \\ & & & \hfill a\left(b-c\right)& =\hfill & ab-ac\hfill \\ \\ & & & \hfill \left(b-c\right)a& =\hfill & ba-ca\hfill \end{array}$

Identity Property $\begin{array}{cccc}\text{of addition}\text{For any real number}a\text{:}\hfill & & & a+0=a\hfill \\ 0\text{is the}\text{additive identity}\hfill & & & 0+a=a\hfill \\ \text{of multiplication}\text{For any real number}a\text{:}\hfill & & & a·1=a\hfill \\ 1\text{is the}\text{multiplicative identity}\hfill & & & 1·a=a\hfill \end{array}$

Inverse Property $\begin{array}{cccc}\text{of addition}\text{For any real number}a,\hfill & & & \hfill a+\left(\text{−}a\right)=0\\ \text{−}a\text{is the}\text{additive inverse}\text{of}a\hfill & & & \\ \text{A number and its}opposite\text{add to zero.}\hfill & & & \\ \text{of multiplication}\text{For any real number}a,a\ne 0\hfill & & & \hfill a·\frac{1}{a}=1\\ \\ \frac{1}{a}\text{is the}\text{multiplicative inverse}\text{of}a\hfill & & & \\ \text{A number and its}reciprocal\text{multiply to one.}\hfill & & & \end{array}$

Properties of Zero $\begin{array}{cccc}\text{For any real number}a,\hfill & & & a·0=0\hfill \\ & & & 0·a=0\hfill \\ \text{For any real number}a,a\ne 0,\hfill & & & \frac{0}{a}=0\hfill \\ \text{For any real number}a,\hfill & & & \frac{a}{0}\text{is undefined}\hfill \end{array}$