Unit 1: Performing Operations with Real Numbers - Real Numbers - Full Study Guide

Real Number Types and Their History

Historical Timeline of Number Systems:
    - Counting (Ancient Times): Human counting activities began alongside the development of spoken language and trade.
    - Written Numbers (8000 BCE): Farmers, cattle herders, and traders in the Middle East used tokens, stones, or markers to represent quantities (e.g., a head of livestock or a length of cloth).
    - Fractions (Egyptian Era): Introduced by Egyptians to express reciprocals (e.g., sharing loaves of bread) and dividing wholes into equal parts.
    - Irrational Numbers (500 BCE): Recognition that some quantities could not be expressed as fractions. Examples include the diagonal of a square with $1\text{ unit}$ sides (slightly less than $1\frac{1}{2}$ ) and the ratio of circumference to diameter (slightly more than $3$ and less than $3\frac{1}{4}$ ).
    - Zero (500 CE): Formally introduced into the number system in India as a numeral for calculations representing the absence of quantity.
    - Negative Numbers (7th Century): Developed in India to solve mathematical equations and track debts or losses.
The Real Number System:
    - Term to Know: Real Number: Any number that can be represented in decimal form ( $base\,10$ ). The position of each digit indicates its place value.
    - Natural Numbers: Often called "counting numbers." These are the set ${1, 2, 3, …}$ .
    - Whole Numbers: The set of natural numbers plus zero: ${0, 1, 2, 3, …}$
    - Integers: The set containing all natural numbers, their opposites (negatives), and zero: ${…, -3, -2, -1, 0, 1, 2, 3, …}$ .
    - Rational Numbers: Numbers that can be expressed as a ratio of two integers $\frac{a}{b}$ , where $b \neq 0$ (division by zero is undefined). In decimal form, rational numbers either terminate (e.g., $0.75$ ) or repeat (indicated by a bar over the repeating digits).
    - Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers. Their decimal representations are non-terminating and non-repeating. Examples include:
        - $\pi\,(\text{Pi})$ : The ratio of a circle's circumference to its diameter.
        - The Golden Ratio: Found in the Fibonacci sequence (sunflower seed spirals, pinecones).
        - Planck's Constant: Used in quantum mechanics.
        - Non-perfect square roots: Such as $\sqrt{3} \approx 1.7320508$ .

The Real Number System: * Term to Know: Real Number: Any number that can be represented in decimal form ( $base\,10$ ). The position of each digit indicates its place value. * Natural Numbers: Often called "counting numbers." These are the set ${1, 2, 3, …}$ . * Whole Numbers: The set of natural numbers plus zero: ${0, 1, 2, 3, …}$ * Integers: The set containing all natural numbers, their opposites (negatives), and zero: ${…, -3, -2, -1, 0, 1, 2, 3, …}$ . * Rational Numbers: Numbers that can be expressed as a ratio of two integers $\frac{a}{b}$ , where $b \neq 0$ (division by zero is undefined). In decimal form, rational numbers either terminate (e.g., $0.75$ ) or repeat (indicated by a bar over the repeating digits). * Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers. Their decimal representations are non-terminating and non-repeating. Examples include: * $\pi\,(\text{Pi})$ : The ratio of a circle's circumference to its diameter. * The Golden Ratio: Found in the Fibonacci sequence (sunflower seed spirals, pinecones). * Planck's Constant: Used in quantum mechanics. * Non-perfect square roots: Such as $\sqrt{3} \approx 1.7320508$ .

Performing Operations with Positive and Negative Numbers

Adding Integers: * Same Signs: Add the unsigned numbers (absolute values) and keep the original sign. * Example: $8 + 4 = 12$ * Example: $-8 + (-4) = -12$ * Different Signs: Subtract the smaller unsigned number from the larger and keep the sign of the number with the larger absolute value. * Example: $-10 + 3 = -7$ * Example: $10 + (-3) = 7$
Subtracting Integers: * Big Idea: Change the subtraction problem to an addition problem by "adding the opposite." * Example: $-5 - 3 = -5 + (-3) = -8$ * Example: $4 - (-4) = 4 + 4 = 8$
Multiplying and Dividing Integers: * Like Signs: The product or quotient of two numbers with matching signs is always positive. * $(+) \times (+) = (+)$ * $(-) \times (-) = (+)$ * Unlike Signs: The product or quotient of two numbers with opposite signs is always negative. * $(+) \times (-) = (-)$ * $(-) \div (+) = (-)$ * Contextual Note: Parentheses can signify multiplication (e.g., $3(-8) = -24$ ), whereas a sign between values indicates addition or subtraction (e.g., $3 - 8 = -5$ ).

Introduction to Exponents

Definition: Exponents represent repeated multiplication. In the form $b^x$ , $b$ is the base (the number being multiplied) and $x$ is the exponent (how many times the base is used as a factor).
Special Terminology:
* Squared: A base raised to the power of $2$ (e.g., $s^2$ relates to the area of a square).
* Cubed: A base raised to the power of $3$ (e.g., $s^3$ relates to the volume of a cube).
Rules for Exponents of 0 and 1:
* Power of 1: Any value raised to the power of $1$ remains the same ( $a^1 = a$ ).
* Power of 0: Any nonzero value raised to the power of $0$ equals $1$ ( $a^0 = 1$ ).
Historical Note: The word exponent comes from the Latin expo ("out of") and ponere ("place"). Babylonian work with exponents dates back to the 23rd century BC.

Order of Operations (PEMDAS)

The PEMDAS Hierarchy: 1. P (Parentheses/Grouping): Evaluate operations inside parentheses, brackets $[]$ , or braces ${}$ first. Work from the innermost to the outermost. 2. E (Exponents/Radicals): Evaluate powers and roots. 3. MD (Multiply and Divide): Perform multiplication and division from left to right as they appear. 4. AS (Add and Subtract): Perform addition and subtraction from left to right as they appear.
Grouping Symbols Beyond Parentheses: * Fraction Bars: Implies parentheses around the numerator and denominator. Evaluate both fully before dividing ( $\frac{a+b}{c+d} = (a+b) \div (c+d)$ ). * Absolute Value Bars: Acts as a grouping symbol. Simplify the expression inside before applying the absolute value ( $|-7 + 4| = |-3| = 3$ ). * Radical Symbols: The expression under the radical (the radicand) is grouped ( $\sqrt{16 + 9} = \sqrt{25} = 5$ ).

Absolute Value

Definition: The absolute value of a number $x$ , denoted $|x|$ , is its distance from zero on the number line. Distance is always non-negative. * Example: $|7| = 7$ and $|-7| = 7$ .
Arithmetic with Absolute Value: * Evaluate the absolute value of terms first, then perform addition or subtraction across them (unless they are inside the bars). * Negative signs on the outside: If a negative resides outside the bar, it applies after the absolute value is found (e.g., $-|-7| = -7$ ).
Properties: * Product Property: $|a \times b| = |a| \times |b|$ * Quotient Property: $|\frac{a}{b}| = \frac{|a|}{|b|}$ * Note: $|a + b|$ is not necessarily equal to $|a| + |b|$ .

Advanced Exponent Properties

Product Property: $a^m \times a^n = a^{m+n}$
Quotient Property: $\frac{a^m}{a^n} = a^{m-n}$
Power of a Power Property: $(a^m)^n = a^{m \times n}$
Power of a Product Property: $(ab)^n = a^n b^n$ (Does not apply to addition/subtraction).
Power of a Quotient Property: $(\frac{a}{b})^n = \frac{a^n}{b^n}$
Negative Exponent Rules: * Rule 1: $b^{-n} = \frac{1}{b^n}$ (Yields the reciprocal of the base; does not make the expression negative). * Rule 2: $\frac{1}{b^{-n}} = b^n$ * Rule 3: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n = \frac{b^n}{a^n}$

Fractional Exponents and Radicals

Relationship: Radicals undo exponent operations. The symbol $\sqrt[n]{x}$ includes the index ( $n$ ) and the radicand ( $x$ ). * Square Root ( $n=2$ ): If no index is written, it is assumed to be $2$ . $\sqrt{25} = 5$ because $5^{2} = 25$ . * Cube Root ( $n=3$ ): $\sqrt[3]{8} = 2$ because $2^{3} = 8$ . * Even Roots of Negatives: Undefined for real numbers (e.g., $\sqrt{-4}$ is non-real). * Odd Roots of Negatives: Result in real numbers (e.g., $\sqrt[3]{-8} = -2$ ).
Converting Between Forms: * Rule 1: $x^{\frac{1}{n}} = \sqrt[n]{x}$ * Rule 2: $x^{\frac{m}{n}} = \sqrt[n]{x^m}$ . The denominator is the index; the numerator is the power.
Simplifying Radicals: * A radical is in simplest form if the radicand contains no perfect power factors of the index (other than $1$ ). * Example: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ . * Example: $\sqrt[3]{54} = \sqrt[3]{27 \times 2} = 3\sqrt[3]{2}$ .

Scientific Notation

Definition: A way to express very large or very small numbers in the form $a \times 10^b$ , where 1 \le |a| < 10 and $b$ is an integer.
Conversion: * Positive Exponent: Represents a large number (move decimal right for standard notation). * Negative Exponent: Represents a small number (move decimal left for standard notation).
Operations: * Multiplication: Multiply decimal numbers, add exponents of $10$ . * Division: Divide decimal numbers, subtract exponents of $10$ . * Calculation Note: Many calculators use "E" or "e" to represent the power of $10$ (e.g., $4.3 \times 10^{12}$ appears as $4.3E12$ ).

Geometry: Pythagorean Theorem, Area, and Volume

Pythagorean Theorem: Used for right triangles to find side lengths or diagonals. Formula: $a^2 + b^2 = c^2$ , where $a$ and $b$ are legs and $c$ is the hypotenuse. * Example: Homeowner ladder calculation for a $12\,\text{ft}$ roof placed $3\,\text{ft}$ out: $3^2 + 12^2 = c^2 \rightarrow 9 + 144 = 153 \rightarrow c = \sqrt{153} \approx 12.37\,\text{ft}$ .
Area Formulas (2D Space): * Rectangle: $A = l \times w$ * Square: $A = s^2$ * Triangle: $A = \frac{1}{2}bh$ * Circle: $A = \pi r^2$ ( $\pi \approx 3.14$ ).
Volume Formulas (3D Space): * Rectangular Prism: $V = l \times w \times h$ * Cylinder: $V = \pi r^2 h$ * Sphere: $V = \frac{4}{3} \pi r^3$
Units: * Area is measured in squared units ( $\text{ft}^2$ , $\text{cm}^2$ ). * Volume is measured in cubic units ( $\text{ft}^3$ , $\text{cm}^3$ ). Note: $1\,\text{liter (L)} = 1000\,\text{cm}^3$ .

Unit Conversion

Conversion Factor: A fraction equal to $1$ relating two different units. * Step 1: List the given value. * Step 2: Multiply by conversion factors so units cancel diagonally.
Multi-step conversions: Linking known factors (e.g., Hours to Minutes to Seconds).
Area and Volume Conversion: * Squared Units: Must square the linear conversion factor. Example: $1\,\text{ft} = 12\,\text{in}$ , therefore $1\,\text{ft}^2 = (12\,\text{in})^2 = 144\,\text{in}^2$ . * Cubic Units: Must cube the linear conversion factor. Example: $1\,\text{ft} = 30.48\,\text{cm}$ , therefore $1\,\text{ft}^3 = (30.48\,\text{cm})^3 \approx 28316.85\,\text{cm}^3$ .