Notes on Real Numbers and Order of Operations

Real Numbers

• Real numbers are the numbers on the number line. They include: integers, fractions, decimals, and also special cases like π. They are the set of numbers you can plot on the number line. There are numbers not on the number line, but we won’t cover those yet this semester.

• Examples on the number line: 0 (origin), positive numbers to the right, negative numbers to the left.

• Real numbers include: integers, fractions, decimals, and pi-type quantities; they are all part of the real numbers.

Order of Operations

• The classic rule is often memorized with the phrase “Please Excuse My Dear Aunt Sally” (PEMDAS):

  • P = Parentheses

  • E = Exponents

  • MD = Multiplication and Division from left to right

  • AS = Addition and Subtraction from left to right

• In practice, you perform operations in this order, including what is inside any parentheses or absolute value signs first.

• A note on implied parentheses: sometimes parentheses are implied by the notation; treat what’s inside them first.

• You should also remember that exponents include square roots as a form of exponentiation.

Absolute Value

• Absolute value |x| is the distance from 0 on the number line; it is always nonnegative.

• Common misconceptions: an absolute value does not turn every number positive; the sign outside the absolute value can still determine the final sign.

• Examples:

  • $|2-13| = |-11| = 11$

  • If you have $- |4|$, then the result is $-4$, because the absolute value portion evaluates to 4, and the negative sign outside stays.

• Important rule: compute inside the absolute value first (the expression inside the bars or parentheses), then apply the outer absolute value.

• A typical pitfall: treating |2-13| as |2| - 13 or similar; you must follow the order of operations inside the absolute value.

Exponents

• An exponent notation uses a base and an exponent: the base is the number being raised, the exponent is how many times to multiply the base by itself.

• Notation:

  • The base is written first, the exponent second: $a^b$ where a is the base and b is the exponent.

• Examples and explanations:

  • A simple example: $4^5$ (four raised to the fifth power).

  • Another example: $5^6$ (five raised to the sixth power).

• Important distinction with negative bases and exponents:

  • $(-5)^2 = 25$ (the negative sign is inside the base, squared, becomes positive).

  • $-(5^2) = -25$ (the exponent applies to the 5 first, giving 25, then the negative outside is applied).

  • Similarly, $(-7)^2 = 49$, while $-(7^2) = -49$.

• Takeaway: a negative sign outside the exponent is different from having the negative inside the base with the exponent.

Order of Operations (Deeper view)

• The formal sequence (as above) is reiterated: Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right).

• There are many example problems that reinforce this order; when evaluating, start from the innermost parentheses and work outward, applying exponents and then multiplication/division and addition/subtraction in the specified order.

• If you’re confused, it can help to break into smaller steps and follow PEMDAS strictly.

Fractions and Decimals

• Fractions: a/b where a is the numerator and b is the denominator. The denominator must be nonzero and typically a positive integer; negative denominators are handled by moving the sign to the numerator.

• Fractions are rational numbers (ratio of two integers).

• Decimals can be written as fractions; if the decimal terminates, the fraction will terminate; if it repeats, you usually keep it as a repeating decimal or convert to fraction if needed.

• Converting a decimal to a fraction:

  • Example: 0.24 = $rac{24}{100} = rac{6}{25}$ when reduced by factoring (24 = 2^3·3, 100 = 2^2·5^2; cancel common factors).

• A note on decimals from an example: converting 0.333… to a fraction is 1/3, illustrating that some decimals are repeating and may not terminate.

• Writing decimals from fractions: if a decimal does not terminate, keep as a decimal; if it terminates, you can also write it as a reduced fraction.

• Rounding and precision in decimals: rounding rules apply to the digits to the right of the place you’re rounding to (e.g., nearest hundredth, thousandth).

Writing Fractions in Lowest Terms

• To reduce fractions, factor the numerator and denominator and cancel common primes.

• Example method (one approach):

  • For $rac{36}{42}$, prime factor: 36 = 2^2 · 3^2, 42 = 2 · 3 · 7. Cancel common factors (2 and 3) to obtain $rac{6}{7}$ or equivalently $rac{2 imes 3}{7} imes 1$ depending on the factoring path.

• Another example: if instructions say leave as a product of primes, you can present as $rac{2 imes 3}{7}$ or keep as 6/7 depending on the prompt.

• Key idea: factor everything into primes, cancel common factors, then present the simplified fraction.

Writing Decimals as Fractions (and vice versa)

• A decimal with digits after the decimal point corresponds to a fraction with denominator a power of 10.

• Example process:

  • 0.24 has two digits after the decimal, so it corresponds to $rac{24}{100}$, which simplifies to $rac{6}{25}$ via prime factorization.

• The general idea is to count how many digits are to the right of the decimal and use the appropriate power of 10 as the denominator.

Rounding

• Rounding is a rule-based operation, not estimation, and should be applied consistently.

• Rule: look at the digit to the right of the place you are rounding to (call it the first digit dropped). If it is 0–4, round down; if it is 5–9, round up.

• A practical real-world note: rounding can resemble real-life cash rounding in some places (e.g., rounding the cents when paying with cash in countries that eliminated 1¢/2¢ coins).

• The concept of rounding to a specified place (e.g., nearest thousandth, nearest hundredth) is used to determine the final value based on the digits you’re discarding.

Converting Percent to Decimal

• Percent means per hundred, so 1% = 0.01.

• Procedure:

  • Remove the percent sign and move the decimal two places to the left.

  • Example: 3% → 0.03; 16.7% → 0.167.

• Important note: errors can arise if the decimal point is misread or mispositioned; ensure the placement is correct.

Multiplying Fractions (and Reducing via Factoring)

• Always factor first to simplify before multiplying.

• For simple problems, direct multiplication is fine, but factoring makes larger problems manageable.

• Example factor-and-cancel workflow:

  • Given two fractions to multiply, factor all numbers into primes and cancel common primes across numerator and denominator before multiplying.

• Example (an explicit cancellation workflow):

  • Consider the product with numbers 33, 14, 27 in the numerator and 49, 45, 22 in the denominator. Factor:

  • Numerator: $33=3\cdot 11,\;14=2\cdot 7,\;27=3^3$

  • Denominator: $49=7^2,\;45=3^2\cdot 5,\;22=2\cdot 11$

  • Cancel common factors: cancel a 3 with a 3, another 3 with a 3, a 11 with a 11, a 2 with a 2, and a 7 with a 7, leaving $\frac{3\cdot 3}{7\cdot 5}=\frac{9}{35}$.

• Another method: multiply numerators and denominators first, then simplify; however, factoring first is generally more efficient and reduces risk of arithmetic errors.

• Division of fractions: divide by a fraction by multiplying by its reciprocal:

  • $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$ (provided $c\neq 0$).

• Always try to factor and cancel when dividing fractions as well.

Adding and Subtracting Fractions (Common Denominators)

• To add or subtract fractions, you must have a common denominator (apples to apples).

• Find the least common denominator (LCD) using prime factorization of the denominators.

• Example: Combine $\frac{1}{10}, \frac{1}{8}, \frac{1}{4}$

  • Factor denominators: $10=2\cdot 5,\; 8=2^3,\; 4=2^2$

  • LCD must include all prime factors with the highest powers: $2^3 \cdot 5 = 40$

  • Convert each fraction to denominator 40: $\frac{1}{10} = \frac{4}{40}, \; \frac{1}{8} = \frac{5}{40}, \; \frac{1}{4} = \frac{10}{40}$

  • Sum: $\frac{4}{40}+\frac{5}{40}+\frac{10}{40}=\frac{19}{40}$

• The process is similar for subtraction; you still use a common denominator.

Distributive Property

• The distributive property states: $a\cdot(b+c)=a\cdot b + a\cdot c$

• Example: Suppose you have (5\cdot m + 15) as an expanded form; distributive property shows how to apply multiplication to a sum.

• Example in the transcript: (\frac{1}{4} \cdot (12x) = \frac{12x}{4} = 3x) and (8 \cdot \frac{1}{4} = \frac{8}{4} = 2).

• When expressions get more complicated, you can break them apart step by step using the distributive property.

Translating Expressions into English

• Read the expressions carefully and translate them into plain-English terms.

• Example: Combining like terms, exponents, and the distributive property are all core building blocks for translating algebra into words and then back into algebraic form.

Exponents (Review and Language)

• Revisit the idea that exponents are repeated multiplication and that negative signs interact with exponents in nuanced ways.

• When expressing expressions like (-2)^4, you’re multiplying -2 by itself four times, giving a positive result; when you have -2^4, the exponent applies to 2 first, giving -(2^4) = -16.

Terminology in Algebra (Terms, Coefficients, Like Terms)

• Term: an expression that can be added or subtracted with other terms; a product of numbers and variables is a term.

• Variable: a letter representing an unknown value (e.g., x, y, m).

• Coefficient: the numerical factor of a term; for example in 27x, the coefficient is 27.

• Constant: a term with no variable (e.g., 3).

• Like terms: terms that have the same variable raised to the same power; they can be combined.

• Examples:

  • In 5m + 8, there are two terms; the coefficient of m is 5; the constant is 8.

  • In 2y + 6 + 5x, there are three terms; coefficients are 2 for y and 5 for x; the constant is 6; no like terms to combine here.

Combining Like Terms

• Like terms can be added or subtracted by combining their coefficients:

  • Example: 6x^3 - 4x^3 = 2x^3

  • Example: 8y + 2 (only the y-terms can combine; constants stay separate)

• When exponents differ, the terms are not like terms and cannot be combined directly.

Distributive Property (Revisited)

• The distributive property is used to expand products across sums or to simplify expressions:

  • Example: 5(m + 3) = 5m + 15

  • Example: (12x) ÷ 4 + 8 × (1/4) simplifies to 3x + 2 as shown earlier.

Summary of the Session (Instructor Tips)

• If you’re stuck, break problems into smaller parts and apply the order of operations step by step.

• When uncertain, consult a tutor or instructor for guided practice and feedback.

• The instructor emphasizes practice with factoring, evaluating step-by-step, and ensuring you understand each operation’s role.

Quick Reference: MyOpenMath Access (Course Logistics)

• Course ID: $292298$

• Enrollment Key: The key is described as capital M, followed by lowercase a t h (i.e., “Math” with M capital and a t h lowercase).

• When signing in: enter your name exactly as registered in CSN (matching capitalization and spelling).

• Important: Name must match exactly with the CSN registration to avoid account mismatches (e.g., Mary Jones vs Mary Jones with different spellings).

• If you have name changes (marriage/divorce), inform the instructor so records stay aligned.

• The instructor notes it’s essential to use the correct name to avoid administrative mix-ups.

Break Schedule (Class Logistics)

• A break is scheduled: at 12:11 PM, a ten-minute break, returning around 12:23 PM.

• The instructor mentions trying to improve the sign-in system and states a commitment to returning promptly after breaks.

• Assignments referenced: One, two, three (1, 2, 3) are listed in the syllabus; details not repeated here, but keep an eye on the syllabus for due dates and content.

Final Note

• The transcript ends mid-session with an announcement about attendance tracking: “And you’ve got” which implies continuing activity beyond the transcript provided. This section is left as a placeholder for the next portion of the lesson or administrative tasks.