{"page_number":22,"title":"Page 022","overview":"This page provides fundamental definitions and properties of different number systems (rational, real, and complex numbers) and introduces the concept of a mathematical function, illustrating it with common examples and explaining that functions are defined by rules, not necessarily single formulas.","text_summary":"The page begins by defining and describing properties of various number systems:\n*   **Rational Numbers:** These are numbers that can be expressed as a ratio of two integers. The page states that if two rational numbers are added, subtracted, multiplied, or divided (except by zero), the result is always another rational number.\n*   **Real Numbers (R):** These encompass positive and negative numbers that can be represented by infinite decimals, including those with a terminating sequence of zeros. Similar to rational numbers, performing addition, subtraction, multiplication, or division (except by zero) on two real numbers yields another real number.\n*   **Complex Numbers (C):** These numbers are expressed in the form *x + iy*, where *x* and *y* are real numbers, and *i* represents the imaginary unit, defined as the square root of -1. The page notes that operations of addition, subtraction, multiplication, or division (except by zero) between two complex numbers will always result in another complex number.\n\nThe second major section, titled \"FUNCTIONS,\" introduces this core mathematical concept:\n*   **Definition of a Function:** A function *f* is described as a mathematical rule that assigns a unique output *f(x)* to each input number *x* (within specified limitations on its value). The rule itself, rather than the specific values it produces, defines the function.\n*   **Example: The Square Function:** The function \"square\" is given as an example, where it assigns to any number *x* its square, *x²*.\n*   **Common Functions in Analysis:** The text lists several types of functions frequently encountered in mathematical analysis, including those defined by formulas like *f(x) = x²*, trigonometric functions such as *sin(x), cos(x), tan(x)*, the logarithmic function *log(x)* (specifically mentioning the natural logarithm with base *e* ≈ 2.71828...), the exponential function *exp(x)* or *eˣ*, and the square root function *√x*.\n*   **Functions Without Single Formulas:** It is emphasized that functions do not necessarily need to be defined by a single formula. The absolute value function *|x|* is provided as an example, defined as *x* when *x* is greater than or equal to zero (*x ≥ 0*), and as *-x* when *x* is less than zero (*x < 0*). The page clarifies that *≥* means \"greater than or equal to\" and *<* means \"less than.\"","content_markdown":"# Page 022\n\n### Page Overview\nThis page provides fundamental definitions and properties of different number systems (rational, real, and complex numbers) and introduces the concept of a mathematical function, illustrating it with common examples and explaining that functions are defined by rules, not necessarily single formulas.\n\n### Text Content Summary\nThe page begins by defining and describing properties of various number systems:\n*   **Rational Numbers:** These are numbers that can be expressed as a ratio of two integers. The page states that if two rational numbers are added, subtracted, multiplied, or divided (except by zero), the result is always another rational number.\n*   **Real Numbers (R):** These encompass positive and negative numbers that can be represented by infinite decimals, including those with a terminating sequence of zeros. Similar to rational numbers, performing addition, subtraction, multiplication, or division (except by zero) on two real numbers yields another real number.\n*   **Complex Numbers (C):** These numbers are expressed in the form *x + iy*, where *x* and *y* are real numbers, and *i* represents the imaginary unit, defined as the square root of -1. The page notes that operations of addition, subtraction, multiplication, or division (except by zero) between two complex numbers will always result in another complex number.\n\nThe second major section, titled \"FUNCTIONS,\" introduces this core mathematical concept:\n*   **Definition of a Function:** A function *f* is described as a mathematical rule that assigns a unique output *f(x)* to each input number *x* (within specified limitations on its value). The rule itself, rather than the specific values it produces, defines the function.\n*   **Example: The Square Function:** The function \"square\" is given as an example, where it assigns to any number *x* its square, *x²*.\n*   **Common Functions in Analysis:** The text lists several types of functions frequently encountered in mathematical analysis, including those defined by formulas like *f(x) = x²*, trigonometric functions such as *sin(x), cos(x), tan(x)*, the logarithmic function *log(x)* (specifically mentioning the natural logarithm with base *e* ≈ 2.71828...), the exponential function *exp(x)* or *eˣ*, and the square root function *√x*.\n*   **Functions Without Single Formulas:** It is emphasized that functions do not necessarily need to be defined by a single formula. The absolute value function *|x|* is provided as an example, defined as *x* when *x* is greater than or equal to zero (*x ≥ 0*), and as *-x* when *x* is less than zero (*x < 0*). The page clarifies that *≥* means \"greater than or equal to\" and *<* means \"less than.\"\n\n### Visual Elements (Diagrams, Figures, Graphs, Portraits, Illustrations)\n*No visual elements on this page.*","has_visuals":0,"visual_count":0,"visuals":[]}