Unit 1 Notes: The Real Numbers and Introduction to Algebra

Whole Numbers (Section 1.1)

• Place Values

  • The number system is a place-value system: the value of a digit depends on its position in the number.

  • Example reference: Figure 1.1 (conceptual in the transcript).

• Introduction to Rounding

  • How to round to a given place value:

  • Locate the place value and mark it with an arrow.

  • All digits to the left of the arrow do not change.

  • Underline the digit to the right of the given place value.

  • If this digit is greater than or equal to 5, add 1 to the digit in the given place value; otherwise, keep it the same.

  • Replace all digits to the right of the given place value with zeros.

  • Example: Rounding $273,185,249$ to the nearest million, ten-thousand, and hundred (as per the slide).

  • Note: Rounding rules are covered more deeply in Unit 2.

• Identify Multiples

  • A number m is a multiple of n if m = k·n for some counting number k ∈ ℕ.

  • The transcript provides a rowed table showing multiples for n = 2, 3, 4, …, 13:

  • Multiples of 2: $2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, \dots$

  • Multiples of 3: $3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, \dots$

  • Multiples of 4: $4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, \dots$

  • Multiples of 5: $5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, \dots$

  • Multiples of 6: $6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, \dots$

  • Multiples of 7: $7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, \dots$

  • Multiples of 8: $8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, \dots$

  • Multiples of 9: $9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, \dots$

  • Multiples of 10: $10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, \dots$

  • Multiples of 11: $11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, \dots$

  • Multiples of 12: $12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, \dots$

  • Multiples of 13: $13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, \dots$

  • Tip: The slide recommends spending time to rewrite this table (at least 12×12) to memorize it.

• Divisibility Tests

  Divisibility Tests

  If a number is a multiple of n, it means that when you divide that number by n, the remainder is 0. Divisibility tests are quick rules or shortcuts to determine if one whole number can be divided by another without leaving a remainder. These tests are especially useful for simplifying fractions, finding common denominators, and factorization.

  Rules listed in the transcript (selected):

  Divisibility by 2

  • Rule: A number is divisible by 2 if its last digit is an even number.

  • Explanation: Even numbers are $0, 2, 4, 6, 8$. Any whole number ending with one of these digits can be perfectly divided by 2.

  • Example: $102,948$ is divisible by 2 because its last digit, 8, is an even number.

Divisibility by 3

• Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.

• Explanation: To apply this, you add up all the individual digits of the number. If the resulting sum is itself divisible by 3, then the original number is also divisible by 3. If the sum is still a large number, you can repeat the process of summing its digits.

• Example: For $13,593$, the sum of the digits is $1+3+5+9+3 = 21$. Since 21 is divisible by 3 ($21 \div 3 = 7$), then $13,593$ is divisible by 3.

Divisibility by 4

• Rule: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

• Explanation: This rule works because $100$ is divisible by 4. So, any number can be thought of as a part that is a multiple of $100$ (which is always divisible by 4) plus its last two digits. Therefore, only the last two digits need to be checked.

• Example: For $1,224$, the number formed by the last two digits is $24$. Since $24$ is divisible by 4 ($24 \div 4 = 6$), then $1,224$ is divisible by 4.

Divisibility by 5

• Rule: A number is divisible by 5 if its last digit is either 0 or 5.

• Explanation: This is a straightforward rule based on the base-10 number system. Multiples of 5 always end in 0 or 5.

• Example: $33,995$ is divisible by 5 because it ends with 5.

Divisibility by 6

• Rule: A number is divisible by 6 if it is divisible by both 2 and 3.

• Explanation: This rule is based on the fact that $6 = 2 \times 3$, and 2 and 3 are prime numbers (and therefore relatively prime). If a number passes both the divisibility test for 2 and the divisibility test for 3, it will be divisible by 6.

• Example: For $3,324$:

• It is divisible by 2 because its last digit (4) is even.

• It is divisible by 3 because the sum of its digits ($3+3+2+4 = 12$) is divisible by 3.

• Since it passes both tests, $3,324$ is divisible by 6.

Divisibility by 7

• Rule: Double the last digit of the number and subtract it from the number formed by the remaining digits. If the result is divisible by 7, then the original number is also divisible by 7. You can repeat this process if the result is still a large number.

• Example: For $798$:

• The last digit is 8. Double it: $2 \times 8 = 16$.

• The remaining digits form the number 79. Subtract the doubled digit: $79 - 16 = 63$.

• Since $63$ is divisible by 7 ($63 \div 7 = 9$), then $798$ is divisible by 7.

Divisibility by 8

• Rule: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

• Explanation: Similar to the rule for 4, this works because $1,000$ is divisible by 8. So, we only need to check the hundreds, tens, and units digits.

• Example: For $1,232,016$, consider the last three digits: $016$ (which is 16). Since 16 is divisible by 8 ($16 \div 8 = 2$), then $1,232,016$ is divisible by 8.

Divisibility by 9

• Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.

• Explanation: This rule is very similar to the divisibility rule for 3. You can sum the digits repeatedly until you get a single digit. If that single digit is 9, the number is divisible by 9.

• Example: For $189$, the sum of the digits is $1+8+9 = 18$. Since 18 is divisible by 9 ($18 \div 9 = 2$), then $189$ is divisible by 9.

• ### Divisibility by 10

  • Rule: A number is divisible by 10 if its last digit is 0.

  • Explanation: Any multiple of 10 always ends in 0.

  • Example: $1,028,891,940$ is divisible by 10 because it ends with 0.

Note: These rules work for numbers of any size, not just small ones. They provide efficient ways to check divisibility without performing long division, aiding in mental math and quick calculations.

• • Rules listed in the transcript (selected):

  • 2: The last digit is even. Numbers with last digit ∈ {0,2,4,6,8} are divisible by 2.

    • Example: 102,948 is divisible by 2.

  • 3: The sum of the digits is divisible by 3.

    • Example: 13,593 has sum 1+3+5+9+3 = 21, which is divisible by 3.

  • 4: The last 2 digits are divisible by 4.

    • Example: 1,224 has 24 divisible by 4.

  • 5: The last digit is 0 or 5.

    • Example: 33,995 ends with 5, so divisible by 5.

  • 6: The rules of 2 and 3 are both true.

    • Example: 3,324 is divisible by both 2 and 3, hence by 6.

  • 7: A trick shown in the transcript: Subtract 2×(last digit) from the rest. If the result is divisible by 7, so is the original number.

    • Example: For 798, 79 − 2×8 = 63, and 63 is divisible by 7, so 798 is divisible by 7.

  • 8: A rule based on the hundreds digit (even/odd) and the last two digits; the transcript gives two examples:

    • If the hundreds digit is even, the last two digits must be divisible by 8.

    • If the hundreds digit is odd, the last two digits must yield a number that is divisible by 4 when divided (the example used 36 ÷ 4 = 9).

  • 9: The sum of the digits is divisible by 9.

    • Example: 189 -> 1+8+9 = 18, divisible by 9.

  • 10: The last digit is 0.

    • Example: 1,028,891,940 ends with 0, so divisible by 10.

  • Practice: Divisibility Tests example: determine if 84 and 6,480 are divisible by the numbers 2, 3, 5, 6, 9 and 10.

  • Note: The transcript emphasizes that these rules work for numbers of any size, not just small ones.

• Practice and Application

  • The section includes explicit practice problems to apply divisibility rules to sample numbers (e.g., 84 and 6,480).

Language of Algebra (Section 1.2)

• Learning Outcomes

  • By the end of this section: Use variables and algebraic symbols; simplify expressions using the order of operations; evaluate expressions; identify and combine like terms; translate English phrases to algebraic expressions.

• Algebra: Terminology

  • Variable: a letter representing a number whose value may change.

  • Constant: a number whose value never changes.

  • The four basic arithmetic operations form the basis of algebraic expressions: Addition (+), Subtraction (−), Multiplication (× or ∙), Division (÷ or /).

  • When translating between English and algebra, pay attention to words like “of” and “and.”

  • Operation symbol and common phrases:

  • Addition: +, the sum of A and B

  • Subtraction: −, the difference of A and B

  • Multiplication: × or ∙, the product of A and B

  • Division: ÷ or divided by, the quotient of A and B; dividend and divisor definitions are given.

• Equalities and Inequalities

  • The symbol = is the equal sign; A = B reads “A is equal to B.”

  • Inequalities use <, ≤, >, ≥ and the words that correspond:

  • < means is less than; ≤ means is less than or equal to; > means is greater than; ≥ means is greater than or equal to.

  • A number line visualization: numbers on the left are smaller than numbers on the right.

• Translating from Algebra to English

  • Examples of translating expressions with inequalities or equations.

• Algebraic Expressions vs Equations

  • An expression is a number, a variable, or a combination of numbers and variables with operation symbols.

  • An equation is two expressions connected by an equal sign.

  • Examples illustrate distinctions between expressions and equations.

• Exponential Notation

  • a^n means a multiplied by itself n times; base a, exponent n.

  • Special names: 2^2 = 4 (squared), 3^3 = 27 (cubed).

  • Simplify: 2^4, 3^3, 7^2, etc. (examples in transcript: 2^4, 3^3, 7^2).

• Order of Operations (with GEDMAS/PEMDAS)

  • Rules: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

  • Mnemonic: Please Excuse My Dear Aunt Sally (or GEDMAS).

• Simplify Expressions Using the Order of Operations

  • Example problems provided to apply the order of operations.

• Evaluating Expressions

  • Evaluating: substitute a given number for the variable and simplify.

  • Example: Evaluate an expression when a certain value is substituted for the variable(s).

• Algebra Vocabulary and Like Terms

  • A term can be a constant or the product of a constant and one or more variables.

  • Coefficient: the constant multiplying the variable in a term.

  • Variable expressions consist of variable terms, coefficients, variable parts, and constant terms.

• Identify and Combine Like Terms

  • Like terms: constants or terms with the same variables raised to the same powers.

  • Simplification: group like terms together, then add coefficients while keeping the variable part the same.

  • Examples demonstrate identifying like terms in expressions.

• Translating English Phrases to Algebraic Expressions

  • Key words: sum, product, quotient, difference, etc., map to +, ×, ÷, − respectively.

  • Examples include translating phrases like "The sum of A and B" and "The quotient of A and B" into algebraic expressions.

• Translating and Solving Word Problems

  • Emphasizes the use of translating word problems to algebraic expressions in order to solve.

• Sample Problem

  • Example: A child’s dose is three-quarters of half of an adult’s dose. Write an expression for the child’s dose in terms of the adult’s dose D: the child’s dose is $frac{3}{4} imes frac{1}{2} imes D = frac{3}{8}D.$

Integers (Section 1.3)

• Learning Outcomes

  • By the end of this section: Use negatives and opposites; simplify expressions with absolute value; perform integer addition, subtraction, multiplication, and division; simplify expressions with integers; evaluate variable expressions with integers; translate phrases to algebraic expressions; apply integers in real-world contexts.

• Using Inequalities with Negative and Positive Integers

  • On a number line, integers are marked; the line extends indefinitely in both directions.

  • Inequalities are visualized on the number line to compare numbers; smaller numbers lie to the left.

  • Examples:

  • -20 < -15

  • -5 < 9

  • -10 < -5

• Order of Pairs with Inequalities

  • Practice: ordering pairs of integers using < or > notation (examples shown in transcript).

• Opposites (Additive Inverses)

  • The opposite of a number a is −a; additive inverse; a + (−a) = 0.

  • Opposite notation: −a means the opposite of a.

  • Examples: Opposite of −16 is 16; opposite of 10 is −10; opposite of 0 is 0.

  • Example: compute −(−15) = 15.

• Absolute Value

  • Definition: |a| is the distance of a from 0 on the number line; |a| ≥ 0.

  • Examples: |−16| = 16, |10| = 10, |0| = 0.

  • Absolute values and inequalities: Compare absolute values (e.g., −16, −17, 20, etc.) using <, >, = relations.

  • Absolute value as a grouping symbol: treat |…| as a grouping device during simplification (e.g., |3−2(6−3)| + 22|10−4|).

• Adding Integers

  • If signs are the same, add absolute values; sign of the sum is the sign of the numbers.

  • Example: $-39 + ( -835 ) = -874.$

  • If signs differ, subtract absolute values; sign of the sum is the sign of the number with the larger absolute value.

  • Example: $-39 + 835 = 796.$

• Subtracting Integers

  • Subtracting a number is the same as adding its opposite: $a - b = a + (−b).$

  • Example: $-227 - (−98) = -227 + 98 = -129.$

• Adding and Subtracting Integers (Practice)

  • Example expression: $-28 + (-38) - (-110) + 155.$

• Multiplying Integers

  • Sign rules for multiplication of signed numbers:

  • (+)×(+) = (+)

  • (−)×(−) = (+)

  • (+)×(−) = (−)

  • Examples shown: a) $227 × 98$, b) $35(−734)$.

• Dividing Integers

  • Sign rules for division of signed numbers:

  • (+) ÷ (+) = (+)

  • (−) ÷ (−) = (+)

  • (+) ÷ (−) = (−)

  • Methods: prime factoring or long division.

  • Examples shown: a) $224 ÷ 16$, b) $784 ÷ (−4)$.

• Order of Operations with Signed Numbers

  • Example: evaluate $[103 - (−300 - (−2) - 25)] ÷ (−25).$

• Evaluating Expressions with Integers

  • Evaluate expressions by substituting given values for variables and simplifying.

  • Example pattern: replacing variables with numbers and applying rules for signed numbers.

• Application Problems with Integers

  • Strategy: read problem, identify what to find, translate the phrase to an algebraic expression, then simplify and answer.

  • Real-world example: body temperature ranges; a patient example discusses a temperature change to reach a hypothermia target.

  • Example task: given a patient baseline temperature and a target, find the integer change necessary to reach the target.

Fractions (Section 1.4)

• Learning Outcomes

  • By the end of this section: Find equivalent fractions; simplify fractions; multiply and divide fractions; simplify expressions with a fraction bar; translate phrases to expressions with fractions; add/subtract fractions with common and different denominators; use the order of operations to simplify complex fractions; evaluate variable expressions with fractions.

• Fractions: Definition and Parts

  • A fraction represents parts of a whole with a nonzero denominator.

  • Numerator: top part (how many parts are included).

  • Denominator: bottom part (into how many equal parts the whole is divided).

  • Property of One: any nonzero number divided by itself equals 1: $\frac{a}{a} = 1, \; a ≠ 0.$

• Types of Fractions

  • Proper: numerator < denominator (e.g., $\frac{1}{4}$).

  • Improper: numerator ≥ denominator (e.g., $\frac{5}{4}$).

  • Mixed fraction (mixed number): whole number and a proper fraction (e.g., $1\frac{1}{4}$).

• Equivalent Fractions

  • Fractions with the same value; cross-multiplication confirms equality: for fractions $\frac{a}{b} = \frac{c}{d}$ with $b ≠ 0, d ≠ 0$, then $ad = bc.$ Example tasks show finding four fractions equivalent to a given value.

• Simplify Fractions

  • Steps:

  • Factor numerator and denominator to reveal common factors.

  • Cancel common factors to reduce to lowest terms.

  • If needed, factor further into primes before cancellation.

  • Examples: $\frac{10}{35} = \frac{2}{7}$; $\frac{120}{450} = \frac{4}{15}$.

• Multiply Fractions

  • Rule: multiply numerators together and denominators together; simplify if possible.

  • Example: $\frac{120}{450} \times \frac{500}{240}$, simplify along the way.

• Reciprocals

  • The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ (provided a ≠ 0).

  • They multiply to 1: $\frac{a}{b} \times \frac{b}{a} = 1.$ Example: reciprocal of $\frac{5}{4}$ is $\frac{4}{5}$; reciprocal of $500$ is $\frac{1}{500}$.

• Divide Fractions

  • To divide by a fraction, multiply by its reciprocal: $\frac{a}{b} ÷ \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ (with c ≠ 0).

  • Examples shown: e.g., $\frac{1}{35} ÷ \frac{10}{7}$ and $\frac{1000}{13} ÷ \frac{500}{65}$.

• Complex Fractions

  • A complex fraction has a fraction bar where either the numerator or the denominator contains a fraction.

  • Approach: treat the overall fraction bar as division: (numerator) ÷ (denominator) and simplify stepwise.

  • Examples illustrate evaluating complex fractions to simpler forms.

• Placement of a Negative Sign in Fractions

  • For positive numbers a, b, c, −a ÷ b = − (a/b) and a ÷ (−b) = − (a/b), etc.; move negatives to the front when needed.

  • Examples show rewriting fractions with the sign in front: e.g., −2/11, 3/−7, −5/−9, −11/−13.

• Simplify Expressions with a Fraction Bar

  • Steps: simplify the numerator and denominator separately, then simplify the resulting fraction.

  • Examples: simplify expressions like |6−19|^2−(10−20) and 6+(8^2−5×2).

• Add or Subtract Fractions

  • If fractions share a common denominator, add/subtract numerators and keep the common denominator.

  • If not, rewrite with the LCD (least common denominator), then perform addition/subtraction and simplify.

  • Examples: $\frac{3}{5} - \frac{1}{5} = \frac{2}{5}$; $\frac{15}{16} - \frac{3}{16} = \frac{12}{16} = \frac{3}{4}$.

• Add or Subtract Fractions with Different Denominators

  • Use LCD of denominators; convert each fraction to an equivalent fraction with the LCD, then sum.

  • Note: LCD = LCM of denominators.

  • Examples provided illustrate the process.

• Converting Between Improper Fractions and Mixed Numbers

  • To convert an improper fraction to a mixed number: divide the numerator by the denominator to determine wholes; remainder becomes the new numerator.

  • To convert a mixed number to an improper fraction: multiply the whole number by the denominator and add the numerator.

  • Examples given for conversions.

• Order of Operations and Complex Fractions (continued)

  • Simplify numerator, then denominator, then divide; apply LCD when needed.

  • Example: a complex fraction simplification.

• Evaluating Expressions with Fractions

  • Evaluate expressions with given variable values and fractions; examples show the process.

Decimals (Section 1.5)

• Learning Outcomes

  • By the end of this section: Name and write decimals; round decimals; add and subtract decimals; multiply and divide decimals; convert decimals, fractions, and percents.

• Naming Decimals

  • Decimals are fractions with denominators that are powers of 10.

  • Naming rule:

  • Name the number left of the decimal point.

  • Use "and" for the decimal point.

  • Name the digits to the right of the decimal point as a whole-number sequence, then state the decimal place of the last digit.

  • Examples given: −12.23 and 0.00027.

• Writing Decimals

  • Write a decimal from words by locating the decimal point with the word "and"; place a decimal point accordingly.

  • Determine number of decimal places from the last word; fill with zeros as needed.

  • Examples: write decimals for phrases such as "Twenty-three and fifty-seven thousandths" and "Negative ten and two tenths".

• Rounding Decimals

  • Steps align with rounding for whole numbers: locate place value, look at the next digit, round up or down, then remove the digits to the right of the rounding place.

  • Example: round $18.28435$ to the nearest tenths, thousandths, ten-thousandth.

• Add and Subtract Decimals

  • Align decimal points; use zeros as placeholders; compute as integers; place decimal point in the result.

  • Examples: $3.27 + (−4.199)$ and $−0.39 + (−9)$.

• Multiply Decimals

  • Steps:

  • Determine the sign of the product.

  • Multiply as if integers; ignore decimals during multiplication.

  • Place decimal point in the product: the number of decimal places equals the sum of decimal places in the factors.

  • Apply the correct sign.

  • Examples: $1.7(−4.192)$ and $−0.9(−9.22)$.

• Multiply Decimals by a Power of 10

  • Move decimal point to the right by the number of zeros in the power of 10.

  • Example: multiply by 10, by 1000, etc.

• Divide Decimals

  • Steps:

  • Determine the sign of the quotient.

  • Rewrite as a fraction (dividend/divisor).

  • Multiply by a power of 10 to remove decimals in the divisor.

  • Perform the division and assign the correct sign.

  • Example: $38.7 ÷ 0.9$.

• Convert Decimals to Proper Fractions

  • Determine the place value of the final digit; write numerator as the digits to the right of the decimal point; denominator as the corresponding power of 10; simplify.

  • Example: Convert $0.016$ to a fraction.

• Convert Fractions to Decimals

  • Divide the numerator by the denominator; produce decimal to the required places (three decimal places in example).

  • Example: convert $7/8$ to a decimal.

• Repeating Decimals

  • A repeating decimal has a bar notation over the repeating part; convert the fraction to a decimal and indicate the repeating bar.

  • Example: convert $7/11$ to a decimal with a repeating bar.

• Percents and Decimals

  • Percents: a percent is a ratio with denominator 100; to convert a percent to a decimal, move the decimal two places left: e.g., 9.8 ext{%} → $0.098$.

  • Decimal to Percent: move decimal two places right and append %.

  • Examples provided for converting 9.8 ext{%} and 156% to decimals, and conversely for decimals to percents.

The Real Numbers (Section 1.6)

• Learning Outcomes

  • By the end of this section: Simplify expressions with square roots; identify integers, rational numbers, irrational numbers, and real numbers.

• Squares and Square Roots

  • The square of a number n is n^2; table of squares for n = 1, 2, 3, …, 15 is listed (e.g., 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, etc.).

  • The square root of a number n is the nonnegative number m such that m^2 = n; denoted $\sqrt{n}$.

  • Every positive number has two real square roots: ±√n.

• Square Root Notation

  • The radical sign √ denotes the principal (nonnegative) square root: $\sqrt{n}$.

  • For a negative radicand (e.g., $-6$), the square root is not a real number.

  • Examples: simplify √169, and note that −√441 is not a real principal value for a negative radicand; the transcript treats −√441 as an expression, not a real number.

• Rational vs Irrational Numbers

  • Rational numbers are of the form a/b with a, b ∈ ℤ and b ≠ 0; their decimal expansion terminates or repeats.

  • Irrational numbers cannot be written as a ratio of integers; their decimal expansion neither terminates nor repeats.

  • Example identifications: 15.25 is rational; −√3 is irrational; −5/7 is rational; π is irrational.

  • Note: The square root of a negative number is not a real number (e.g., √−6 is not real).

• Identifying Types of Numbers

  • Given a list (e.g., 0, −11, 11/28, 9, √5, √25, √−2, 1.23, 10.287408…, …), classify into: Whole numbers, Integers, Rational numbers, Irrational numbers, Real numbers.

Properties of Real Numbers (Section 1.7)

• Learning Outcomes

  • By the end of this section: Use the commutative, associative, identity, and inverse properties of addition and multiplication; use zero properties; apply the distributive property to simplify expressions.

• Commutative Property

  • Addition: a + b = b + a for all real a, b.

  • Multiplication: a × b = b × a for all real a, b.

  • Note: Subtraction and division are not commutative (a − b ≠ b − a in general; a ÷ b ≠ b ÷ a).

• Associative Property

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

  • Multiplication: (a × b) × c = a × (b × c).

  • Example: (4 + 10) + 5 = 4 + (10 + 5) and (4 × 10) × 5 = 4 × (10 × 5).

• Making Work Easy!

  • Using properties to simplify expressions: e.g., (1/2 + 1/7) + 6/7 can be reorganized using associativity/commutativity.

• Identity and Inverse Properties of Addition and Multiplication

  • Additive identity: 0 is the additive identity since a + 0 = a and 0 + a = a.

  • Additive inverse: −a is the additive inverse of a since a + (−a) = 0.

  • Multiplicative identity: 1 is the multiplicative identity since a × 1 = a and 1 × a = a.

  • Multiplicative inverse (reciprocal): For a ≠ 0, a × (1/a) = 1.

• Inverse Property; Additive and Multiplicative Inverses

  • Find the additive inverse (opposite) and the multiplicative inverse (reciprocal) of sample numbers (e.g., 4, −8, 9, 0.2).

• Multiplication and Division with Zero

  • Product with zero is zero: 0 × a = 0 for any a.

  • Zero divided by any nonzero number is zero: $0 ÷ a = 0$ for a ≠ 0.

  • Division by zero is undefined: a ÷ 0 is undefined for any a.

• Distributive Property

  • a × (b + c) = a×b + a×c for all real a, b, c.

  • Examples provided include evaluating expressions using distribution.

• Revisiting Order of Operations

  • Apply order of operations to complex expressions; the example exercises reinforce applying all properties alongside the order of operations.

• Unit 1 - Bringing it all together

  • A set of challenging composite problems intended to combine multiple concepts from Units 1.1–1.7.

  • Example problems include: evaluating complex expressions with mixed operations, fractions, decimals, negatives, and absolute values; translating word problems into algebraic expressions; applying the distributive, associative, and commutative properties.

Connections and Practical Implications (across sections)

• Foundational concepts

  • Place value, rounding, multiples, and divisibility underpin number sense used in higher algebra and number theory.

  • Prime factorization, LCM, and GCF provide tools for simplifying fractions and solving equations.

  • Translating between English phrases and algebraic expressions builds foundational algebraic literacy.

• Real-world relevance

  • Divisibility rules and prime factorization relate to factorizing numbers in cryptography and computer science contexts.

  • Fractions, decimals, and percents are essential for financial calculations, measurements, and data interpretation.

  • Absolute value, integers, and inequalities model real-world quantities that can be negative (e.g., temperature changes, elevations).

• Ethical/philosophical/practical implications

  • Emphasizes careful translation between language and symbols to avoid misinterpretation in problem solving.

  • Highlights the importance of foundational arithmetic rules (order of operations, identity/inverse properties) for logical reasoning and rigorous mathematics.

촉

Let's break down the terminology related to prime factorization:

• Factors: If you have two numbers, say $a$ and $b$, and their product is $m$ (i.e., $a \cdot b = m$), then $a$ and $b$ are considered factors of $m$. For example, for the number 12, its factors include 1, 2, 3, 4, 6, and 12, because $1 \cdot 12 = 12$, $2 \cdot 6 = 12$, and $3 \cdot 4 = 12$.

• Prime Number: A prime number is a counting number (a positive integer) that is greater than 1 and has exactly two distinct positive divisors: 1 and itself. Examples include 2, 3, 5, 7, 11, etc.

• Composite Number: A composite number is a counting number that is not prime. This means it is a number greater than 1 that has factors other than 1 and itself. For instance, 4 is a composite number because its factors are 1, 2, and 4. Another example is 6, with factors 1, 2, 3, and 6.

• Prime Factorization: The prime factorization of a number is expressing that number as a product solely of prime numbers. These prime numbers in the product are called the prime factors. For example, the prime factorization of 12 is $2 \cdot 2 \cdot 3$ (or $2^2 \cdot 3$), where 2 and 3 are the prime factors.