Numbers, Real Numbers, Order of Operations and Core Properties (Study Notes)

Number Sets and Real Numbers

• Natural numbers (counting numbers): $\mathbb{N}={1,2,3,\ldots}$
  • Also described as counting items with a next item always known (1, 2, 3, …).
  • There is no zero in natural numbers.
• Whole numbers: $\mathbb{W}=\mathbb{N}\cup{0}={0,1,2,3,\ldots}$
  • All positive whole numbers plus zero.
• Integers: $\mathbb{Z}={\ldots,-2,-1,0,1,2,\ldots}$
  • Includes positive and negative whole numbers and zero; no decimals or fractions.
• Rational numbers: \mathbb{Q}=\left{\frac{p}{q}\;\middle|\; p,q\in\mathbb{Z}, q\neq 0\right}
  • A number that can be expressed as a fraction.
  • Root word is "ratio"; a ratio of two integers.
• Irrational numbers:
  • Not expressible as a fraction; decimals do not terminate or repeat.
  • Examples: $\sqrt{3}$, $\pi$, $e$, and other non-simplifiable square roots.
  • The square root of 3 is irrational because 3 has no perfect square factor; its decimal expansion is nonrepeating and nonterminating.
  • Pi is irrational (and transcendental in a deeper sense; goes on forever without repeating).
  • Euler's number $e$ is irrational.
• Real numbers: $\mathbb{R}$
  • All numbers that are not the square roots of negative numbers.
  • The square roots of negatives are nonreal/imaginary, which will be discussed later.
  • Real numbers are either rational or irrational.
• Imaginary and complex numbers (brief note): contain an $i$ component; these are not real numbers and will be discussed later.

Principal square roots and key examples

• The square root symbol denotes the principal (nonnegative) root for real inputs.
  • Example: $\sqrt{25}=5$ (positive whole number).
  • The principal square root of a real number is nonnegative.
• If a number on the right side is a perfect square, the sqrt may be an integer.
  • Example: $\sqrt{25}=5\in\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$
• For a square root with a variable inside (e.g., $\sqrt{x^2}$), be mindful:
  • In many contexts, $\sqrt{x^2}=|x|$ (the nonnegative value).
  • If you’re just taking the square root of a fixed number like 25, you report the nonnegative root (here, 5).
• Decimal representations and rationality:
  • Terminating decimals are rational (e.g., $0.375=\frac{375}{1000}=\frac{3}{8}$).
  • Repeating decimals are rational (e.g., $0.\overline{3}=\frac{1}{3}$, or 0.3838… with a repeating block).
  • Nonterminating, nonrepeating decimals are irrational (e.g., $\sqrt{3}=1.7320\ldots$, $\pi=3.14159\ldots$).
• Examples from the transcript:
  • $\sqrt{3}$ is irrational and real (not an integer, not rational).
  • $\sqrt{25}=5$ is rational and a real number; the square root symbol yields the principal (positive) root for real numbers.
  • $\pi$ and $e$ are irrational numbers.
  • $\sqrt{71}$ is irrational (no perfect square factor to simplify).
  • $\sqrt{0.09}=0.3=\frac{3}{10}$ is rational (terminating decimal).
  • $\pi$ is irrational (goes on forever without repetition).
• Real numbers exclude imaginary numbers; imaginary numbers involve $i$ (future topics).

Are all numbers rational or irrational?

• Statement: All real numbers are either rational or irrational. There is no overlap, and every real number falls into exactly one of these two categories.
• Practical takeaway: If you can express a number as a fraction, it’s rational; otherwise it’s irrational.

Order of Operations (PEMDAS) and practical rules

• PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Important nuance: Multiplication and Division are on the same level (do from left to right as they appear). Likewise, Addition and Subtraction are on the same level (left to right).
• Example problem 1: Evaluate
  • Expression: $5\,[6+2(4\times 6-7)]$
  • Inside the inner parentheses: $4\times 6=24\;\Rightarrow\;24-7=17$
  • Then multiply by 2: $2\times 17=34$
  • Then add 6: $6+34=40$
  • Finally multiply by 5: $5\times 40=200$
  • Answer: $200$
• Example problem 2:
  • Expression: $\frac{(9+4)\times(15-7)}{7^3-317}$
  • Numerator: $9+4=13\quad\Rightarrow\quad 13\times8=104$
  • Denominator: $7^3=343\;\Rightarrow\;343-317=26$
  • Division: $\frac{104}{26}=4$
  • Answer: $4$
• Example problem 3 (involving exponents and negatives):
  • Expression: $\frac{10-(2-8)^2}{3^2+2^2}$
  • Inner: $2-8=-6\;\Rightarrow\;(-6)^2=36$
  • Numerator: $10-36=-26$
  • Denominator: $3^2+2^2=9+4=13$
  • Result: $-26/13=-2$
• Important difference discussion:
  • Compare $-3^2$ vs $(-3)^2$:
  • $(-3)^2=9$ (negative inside, squared becomes positive).
  • $-3^2=-(3^2)=-9$ (exponent applies to 3 first, then the negative is applied).
  • This distinction highlights that the exponent applies to the base immediately in front of it; parentheses change what is being squared.
• Fraction operations primer (a typical classroom workflow):
  • To add/subtract fractions, use a common denominator (LCD).
  • Example: compute $\left(\frac{1}{3}-\frac{1}{5}\right)\Big/\left(\frac{7}{15}\right)$ with LCD = 15:
  • Convert: $\frac{1}{3}=\frac{5}{15},\; \frac{1}{5}=\frac{3}{15}$
  • Subtract: $\frac{5}{15}-\frac{3}{15}=\frac{2}{15}$
  • Divide by $\frac{7}{15}$ means multiply by the reciprocal: $\frac{2}{15}\times\frac{15}{7}$
  • Cancel the shared factor 15: $\frac{2}{7}$
  • Key rule: dividing by a fraction is equivalent to multiplying by its reciprocal; cancel common factors when possible.
• Note on math entry in online homework: a math editor enables fractions; place the cursor in the answer box to reveal tools for fractions.

Properties of real numbers (basic algebraic laws)

• Commutative properties:
  • Addition: $a+b=b+a$
  • Multiplication: $a\cdot b=b\cdot a$
• Associative properties:
  • Addition: $(a+b)+c=a+(b+c)$
  • Multiplication: $(a\cdot b)\cdot c=a\cdot(b\cdot c)$
• Distributive property:
  • $a\,(b+c)=a\cdot b+a\cdot c$
• Identity properties:
  • Additive identity: $a+0=a$
  • Multiplicative identity: $a\cdot 1=a$
• Inverse properties:
  • Additive inverse: $a+(-a)=0$
  • Multiplicative inverse: for $a\neq 0,\; a\cdot\frac{1}{a}=1$ (reciprocal)
• These properties are summarized in the classroom as a quick reference for algebraic manipulation.

Variables and constants; evaluating expressions

• Variables vs constants:
  • Constants are the numeric parts (e.g., 15, 76).
  • Variables are letters (e.g., g, b, t) representing unknowns.
  • When an algebraic expression has multiple answers, separate them with commas (e.g., for a listing of constants and variables).
• Example: evaluate algebraic expression with given values:
  • Expression: $-9b+5t$ with $b=-3$ and $t=-1$ (note: the transcript shows a small inconsistency with t; the calculation used t=-1).
  • Substitution: $-9(-3)+5(-1)$
  • Compute: $27-5=22$
  • Answer: $22$
• Practical tip: use a calculator for large multiplications or to verify arithmetic; it’s fine to rely on mental math, but calculators help avoid simple mistakes.

Quick plan for success in the course (strategies mentioned)

• If you have a device, follow along and work problems rather than passively watching.
• If you miss something, go back to the textbook or MyOpenMath assignments to reinforce understanding.
• Seek tutoring or math labs for fractions, order of operations, and foundational concepts to stay on track.
• Don’t hesitate to ask questions in class; feedback and clarification help prevent point losses on tricky items.
• The instructor emphasizes that if a computer or an automated system marks you wrong but you believe your answer is correct, speak up and you’ll likely get the points back after review.

Connections to prior knowledge and real-world relevance

• Sets and number systems underpin all branches of mathematics, from basic arithmetic to calculus and beyond.
• Understanding which numbers are rational vs irrational informs numerical methods, measurement precision, and modeling real-world quantities.
• Order of operations mirrors practical calculation rules used in programming, finance, engineering, and science to ensure consistent results.
• Fractions, decimals, and ratios appear in everyday contexts (recipes, measurements, money, speeds); recognizing when a decimal terminates or repeats helps identify rational numbers quickly.
• Algebraic properties (commutativity, associativity, distributivity, identities, and inverses) are foundational for simplifying expressions and solving equations.

Practical illustrations and formulas (recap)

• Set notations:
  • $\mathbb{N}={1,2,3,\ldots}$
  • $\mathbb{W}={0,1,2,3,\ldots}$
  • $\mathbb{Z}={\ldots,-2,-1,0,1,2,\ldots}$
  • \mathbb{Q}=\left{\frac{p}{q}\;|\; p,q\in\mathbb{Z}, q\neq 0\right}
  • $\mathbb{R}$ (real numbers)
• Key numeric facts:
  • Principal square root: $\sqrt{a}\ge 0$ for real $a\ge 0$; $\sqrt{25}=5$;
  • Rational vs irrational: a number is rational if it can be written as a fraction; otherwise irrational.
  • Examples: $\sqrt{3}\text{ is irrational};\ \pi,\ e\text{ are irrational};\ \sqrt{25}=5\in\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$
• Quick practice questions to test understanding (conceptual):
  • Is $\sqrt{71}$ rational or irrational? (Irrational)
  • Is $\frac{3}{4}$ rational or irrational? (Rational)
  • Is $0.375$ terminating or repeating? (Terminating; rational)
  • Is $\pi$ rational or irrational? (Irrational)
  • When would you use the LCD in fractions? (When adding/subtracting fractions with different denominators)

Summary takeaways

• Real numbers comprise rationals and irrationals; imaginary numbers extend the number system beyond the reals.
• The square root operation has a principal (nonnegative) result for real inputs; caution with signs when squaring negatives or dealing with variables inside the root.
• Decimal representations reveal rationality: terminating or repeating decimals are rational; nonterminating nonrepeating decimals are irrational.
• Order of operations is essential for consistent results; remember M and D share a level, A and S share a level, and compute left-to-right within each level.
• Fractions require common denominators for addition/subtraction and reciprocals for division; algebraic properties provide powerful rules for simplifying expressions.
• Practice with substitution and evaluation of expressions to strengthen fluency in manipulating variables and constants.