Comprehensive Guide to University Preparatory Mathematics: Numbers, Powers, Roots, and Proportionality

Introduction to Number Systems and the Integers

The fundamental need for humans to enumerate and count led to the development of numerical sets. The first set created for this purpose is the set of Natural Numbers, denoted by $\mathbb{N}$, which includes elements starting from one such as $\mathbb{N} = \{1, 2, 3, 4, 5, 6, 7, \dots\}$. To represent debts or negative balances, as well as a null value, the set was expanded to include zero and negative numbers. This formed the set of Integers, denoted by $\mathbb{Z}$, containing $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, \dots\}$.

In the set of integers, the addition of numbers with the same sign requires adding their absolute values and preserving the sign. For numbers with different signs, the difference between absolute values is calculated, and the result retains the sign of the number with the greater absolute value. Multiplication and division follow specific sign rules where multiplying or dividing two numbers of the same sign always results in a positive value, while different signs result in a negative value. Every integer $a$ has an additive inverse or opposite denoted as $-a$, where the opposite of $7$ is $-7$ and the opposite of $-12$ is $12$.

Historical examples highlight the application of integers in timelines. Pythagoras died at the age of $74$ in the year $497\,\text{BC}$, implying he was born in $571\,\text{BC}$. Archimedes was born in $287\,\text{BC}$ and lived for $75$ years, meaning he died in $212\,\text{BC}$. Eratosthenes was born in $276\,\text{BC}$ and died in $194\,\text{BC}$, living for $82$ years. Thales of Mileto, being born in $624\,\text{BC}$, was the earliest born among these figures. In another scenario, the result of calculating $90.606 - 19.878$ yields $70.728$.

Successors, Antecessors, and Parity in Integers

For any integer $n$, the successor is defined as $(n + 1)$ and the antecessor is $(n - 1)$. Parity is a core property where the expression $2n$ always represents an even integer, while $(2n - 1)$ or $(2n + 1)$ represents odd integers. Consecutive integers follow the pattern $(n - 1), n, (n + 1)$. Even consecutive numbers are expressed as $2n$ and $2n + 2$, while consecutive odd numbers follow the form $2n + 1$ and $2n + 3$. Notably, the number $0$ is explicitly considered an even number because it can be written in the form $2n$.

Operations between even and odd numbers follow predictable patterns. The sum or difference of two even numbers is even, two odd numbers is even, and an even and an odd number results in an odd number. In multiplication, the product is only odd if both factors are odd; if at least one factor is even, the product is even. Perfect squares are defined as $n^2$ with $n \neq 0$, producing the sequence $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, \dots$

Practical logic machines illustrate these concepts. In Machine A, an input $n$ produces the even successor of the double, modeled as $2n + 2$. In Machine B, an input results in the antecessor of the square, modeled as $n^2 - 1$. If the smallest prime number, which is $2$, is entered into Machine A, the result is $(2 \times 2) + 2 = 6$. If this $6$ is then entered into Machine B, the final output is $6^2 - 1 = 35$. This resulting number $35$ is a composite number greater than $34$, a multiple of $7$, the antecessor of the perfect square $36$, and has a digit sum of $8$.

Multiples, Divisors, and Primes

Relationships between integers are established through the equation $a = b \times c$, where $a$ is a multiple of both $b$ and $c$, while $b$ and $c$ are divisors of $a$ (with $b, c \neq 0$). Divisibility rules provide shortcuts: a number is divisible by $2$ if its last digit is even; by $3$ if the sum of its digits is a multiple of $3$; by $5$ if it ends in $0$ or $5$; by $6$ if it is divisible by both $2$ and $3$; and by $10$ if it ends in $0$. Mathematical observations confirm that $0$ is a multiple of every number, $1$ is a divisor of all numbers, and every number is a multiple and divisor of itself. However, $0$ is not a divisor of any number.

Numbers are categorized as Primes if they have exactly two distinct positive divisors, namely $1$ and the number itself. The sequence begins with $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, \dots$, with $2$ being the only even prime. Composite numbers have more than two divisors and include $4, 6, 8, 9, 10, 12, 14, 15, 16, \dots$. The number $1$ is unique as it is neither prime nor composite. The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of prime factors. For example, the prime factorization of $1.080$ is $2^3 \times 3^3 \times 5$. To find the total number of positive divisors for a number $P$ with prime decomposition $a^p \times b^q \times c^r$, the formula used is $(p + 1) \times (q + 1) \times (r + 1)$.

Calculations for the Greatest Common Divisor (M.C.D.) involve multiplying common prime factors with their lowest exponent. The Least Common Multiple (m.c.m.) involves multiplying all distinct prime factors with their highest exponent. For instance, the M.C.D. and m.c.m. of $A = 2^3 \times 3^2$ and $B = 2^2 \times 3^3 \times 5$ are $M.C.D. = 2^2 \times 3^2 = 36$ and $m.c.m. = 2^3 \times 3^3 \times 5 = 1.080$. In a circular track scenario where three cyclists take $120$, $140$, and $180$ seconds respectively per lap, they will meet at the starting line again in $2.520\,\text{seconds}$, which is the m.c.m. of their lap times.

Operational Priority and Algorithm of Division

Complexity in arithmetic is managed through the order of operations: Parentheses (from the innermost outward), Powers, Multiplication and Division (from left to right), and finally Addition and Subtraction. An example of this is the expression $3 - \{2 - [1 - (12 : 4 \times 3)] - 9\}$. Following the hierarchy: the innermost division and multiplication $(12 : 4 \times 3 = 9)$ result in $[1 - 9] = -8$. Then, the bracketed terms yield $2 - (-8) - 9 = 1$. The final result is $3 - 1 = 2$. Similarly, $4^2 - 3^2 : 2 \times 5$ is calculated as $16 - 9 : 2 \times 5 = 16 - 4.5 \times 5 = 16 - 22.5 = -6.5$.

The division algorithm states that for any two numbers $a$ (dividend) and $b$ (divisor), if the quotient is $c$ and the remainder is $r$, then $a = b \times c + r$. If Paulina divides a number by $6$, $8$, and $9$ and always gets a remainder of $2$, the number is a multiple of the m.c.m. of $(6, 8, 9)$ plus $2$. Given $m.c.m.(6, 8, 9) = 72$, the number would be $72 + 2 = 74$. In another scenario, three barrels containing $875$, $625$, and $750\,\text{liters}$ of oil are adjusted to the M.C.D. of these amounts, which is $125\,\text{liters}$. To reach this, $500\,\text{liters}$ must be removed from the second barrel.

Rational Numbers and Algebraic Fractions

Rational Numbers, denoted by $\mathbb{Q}$, are defined as all numbers that can be expressed as a fraction $\frac{a}{b}$ where $a, b \in \mathbb{Z}$ and $b \neq 0$. This set includes integers, as they can be written with a denominator of $1$. Fractions are classified as Proper if the absolute value of the numerator is less than the denominator ($|a| < |b|$), resulting in a value less than $1$. They are Improper if $|a| > |b|$, resulting in a value greater than $1$. Improper fractions can be converted into mixed numbers represented by $A \frac{b}{c}$, where the fractional value is calculated as $\frac{A \times c + b}{c}$.

In the set $\mathbb{Q}$, addition and subtraction with the same denominator involves combining numerators while keeping the denominator constant. With different denominators, the m.c.m. of the denominators must be determined. For multiplication, numerators are multiplied together, and denominators are multiplied together. Division is performed by multiplying by the reciprocal of the divisor. Every rational number $\frac{a}{b}$ has an additive inverse $-\frac{a}{b}$ and, if non-zero, a multiplicative inverse or reciprocal $\frac{b}{a}$. Comparison between rational numbers can be achieved by cross-multiplying, finding common denominators, or converting to decimals. For example, to compare $a = \frac{7}{8}$, $b = \frac{11}{12}$, and $c = \frac{9}{10}$, evaluating the common denominator or decimal equivalents ($0.875, 0.916, 0.9$) shows the order $a < c < b$.

Experimental examples include a dosage machine using a fill fraction $F = \frac{k}{8}$. If $k$ is a positive integer such that $F$ is proper but $\frac{k + 3}{8}$ is improper, then $k < 8$ and $k + 3 \geq 8$. This implies $k \geq 5$. The possible values for $k$ are $5, 6, 7$, and their sum is $18$. In social demographics, if a town has a population with $\frac{1}{10}$ affected by an epidemic, and subsequently $\frac{1}{10}$ of the sick recover while $\frac{1}{10}$ of the healthy get sick, the healthy population becomes original healthy minus the newly sick plus the recovered: $\frac{9}{10} - (\frac{1}{10} \times \frac{9}{10}) + (\frac{1}{10} \times \frac{1}{10}) = \frac{9}{10} - \frac{9}{100} + \frac{1}{100} = \frac{82}{100} = \frac{41}{50}$.

Decimal Operations and Transformations

Decimals are classified based on their digits. Finite Decimals are converted to fractions by placing the digits over a power of $10$ matching the number of decimal places. Infinite Periodic Decimals are converted by taking the difference between the full number and the non-periodic part, divided by a sequence of nines corresponding to the length of the period. Infinite Semi-periodic Decimals use a numerator formed by subtracting the non-periodic part from the full number and a denominator of nines (for the period) followed by zeros (for the ante-period).

Performing operations with decimals requires alignment of decimal points for addition and subtraction. Multiplication treats the factors as integers first, then places the decimal point so the result has a total number of decimal places equal to the sum of the decimal places in the factors. For division, it is often useful to multiply both the dividend and divisor by a power of $10$ to eliminate decimals. In an engineering series circuit with resistances $1.5\,\Omega$, $2.6\,\Omega$, and $0.83\,\Omega$, the total resistance is $1.5 + 2.6 + 0.83 = 4.93\,\Omega$. Comparisons can also be precise; for time sensors recording $1.321$, $1.3\overline{21}$, and $1.32\overline{1}$, the decimal expansion shows the smallest value (and thus the fastest pilot) is $\text{Pilot A} (1.32100 \dots)$, followed by $\text{Pilot C} (1.32111 \dots)$, and then $\text{Pilot B} (1.32121 \dots)$.

The Theory of Powers and Exponential Algebra

A power $a^n$ represents a base $a$ multiplied by itself $n$ times for positive integer exponents. Definitions include $a^0 = 1$ (for $a \neq 0$), $0^n = 0$, and $1^n = 1$. Negative exponents are defined as $a^{-n} = \frac{1}{a^n}$. The sign of a power depends on the base and the parity of the exponent: a positive base remains positive, a negative base becomes positive if the exponent is even, and it remains negative if the exponent is odd. Properties of powers include product of equal bases ($a^n \times a^m = a^{n+m}$), quotient of equal bases ($a^n : a^m = a^{n-m}$), product of equal exponents ($a^n \times b^n = (a \times b)^n$), quotient of equal exponents ($\frac{a^n}{b^n} = (\frac{a}{b})^n$), and power of a power ($(a^n)^m = a^{n \times m}$).

Practical modeling often utilizes powers. In biology, a bacterial population starting at $3^4$ that triples every hour results in $3^4 \times 3^5 = 3^9$ bacteria after $5$ hours. Computer science uses binary storage where a Gigabythe is technically $2^{30}\,\text{bytes}$, while commercial manufacturers define it as $10^9\,\text{bytes}$, leading to a shortfall of $2^{30} - 10^9\,\text{bytes}$. Acoustic intensity is measured logarithmically; an intensity of $10^8$ compared to $10^6$ is $10^{8-6} = 10^2 = 100$ times more powerful. In geometry, the volume of a cubic container with an edge of $10^2\,\text{cm}$ is calculated as $(10^2)^3 = 10^6\,\text{cm}^3$.

Exponential equations require reducing both sides to the same base to solve for an unknown in the exponent. If $a^n = a^m$, then $n = m$ for $a \in \mathbb{R} - \{-1, 0, 1\}$. For instance, solving $(0.01)^{-x+5} = 100$ involves converting to base $10$: $(10^{-2})^{-x+5} = 10^2$, leading to $2x - 10 = 2$, or $x = 6$.

Roots and Irrational Numbers

Irrational Numbers, denoted by $\mathbb{I}$ or $\mathbb{Q}'$, are infinite non-periodic decimals that cannot be expressed as fractions. Examples include $\pi \approx 3.14159$, $e \approx 2.71828$, and non-exact roots like $\sqrt{2}$ or $\sqrt{3}$. A root $\sqrt[n]{a} = b$ is defined such that $b^n = a$. If $n$ is even, $a$ must be non-negative for the root to be real. If $n$ is odd, $a$ can be any real number. Roots can be expressed as powers with fractional exponents: $\sqrt[n]{a^k} = a^{k/n}$. An important observation is that $\sqrt{a^2} = |a|$, ensuring the result is non-negative.

Properties of roots facilitate manipulation. They include multiplication and division of roots with the same index: $\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \times b}$ and $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$. Factors can be moved into the radical: $b\sqrt[n]{a} = \sqrt[n]{b^n \times a}$. For roots of different indices, a common index must be found via the m.c.m. of the indices. Nested roots follow the property $\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \times m]{a}$. In engineering, resistance and voltage determine current via $I = \frac{V}{R}$; if $V = 10$ and $R = 5\,\Omega$, then $I = \sqrt{\frac{10}{5}} = \sqrt{2}\,\text{A}$.

Estimation of roots is necessary for non-exact values. For example, $\sqrt{40}$ falls between $6$ and $7$ because $6^2 = 36$ and $7^2 = 49$. Algebraic simplification often involves perfect squares, such as $5\sqrt{12} - 2\sqrt{27} = 5\sqrt{4 \times 3} - 2\sqrt{9 \times 3} = 10\sqrt{3} - 6\sqrt{3} = 4\sqrt{3}$. In geometry, the diagonal of a square plaza with sides of $100\,\text{m}$ is given by $\sqrt{100^2 + 100^2} = 100\sqrt{2}\,\text{m}$.

Ratios, Proportions, and Proportionality

A Ratio is a comparison between two quantities via division, written as $a : b$ or $\frac{a}{b}$, where $a$ is the antecedent and $b$ the consequent. The value of the ratio is the quotient $c = \frac{a}{b}$. A Proportion is an equality of two ratios: $\frac{a}{b} = \frac{c}{d}$. The Fundamental Theorem of Proportions states that the product of the extremes equals the product of the means ($a \times d = b \times c$). For example, if a mixture requires blue and yellow paint in a ratio of $5 : 2$ and $400\,\text{ml}$ of blue is used, the yellow needed is $x$. Thus $\frac{5}{2} = \frac{400}{x}$, resulting in $x = 160\,\text{ml}$, and a total mixture of $560\,\text{ml}$.

Direct Proportionality occurs when two variables $x$ and $y$ maintain a constant quotient ($\frac{y}{x} = k$), implying both increase or decrease together. A graph of this relationship is a straight line passing through the origin. Inverse Proportionality occurs when two variables maintain a constant product ($x \times y = k$). If one increases, the other decreases proportionally. For example, if $6$ hoses fill a pool in $10\,\text{hours}$, then $15$ hoses ($k = 60$) would fill it in $\frac{60}{15} = 4\,\text{hours}$.

Series of ratios involve multiple quantities, such as ages of three sisters in a ratio of $2 : 5 : 3$ summing to $30\,\text{years}$. This can be solved by setting $2k + 5k + 3k = 30$, yielding $k = 3$. Their ages are therefore $6$, $15$, and $9$. In financial allocations, a start-up with an utility of $\$4.500.000$ distributed to partners in a ratio of $3 : 5 : 7$ means the partner with the highest share gets $\frac{7}{15}$ of the total ($\$2.100.000$) and the lowest gets $\frac{3}{15}$ ($\$900.000$), a difference of $\$1.200.000$.

Percentages and Percent Change

Percentage is an application of direct proportionality where the whole is considered as $100$. The percentage $Q$ of a total $C$ is given by $Q = \frac{P}{100} \times C$. Common percentages have simple fractional equivalents, such as $12.5\% = \frac{1}{8}$, $20\% = \frac{1}{5}$, $25\% = \frac{1}{4}$, $33\frac{1}{3}\% = \frac{1}{3}$, $50\% = \frac{1}{2}$, and $75\% = \frac{3}{4}$. Successive percentages follow the rule that $a\%$ of $b\%$ of $C$ is equal to $\frac{a}{100} \times \frac{b}{100} \times C$. For example, the amount allocated to security cameras from a total fund of $\$15.000.000$ where security gets $40\%$ and cameras get $20\%$ of that $\$40\%$ is computed as $0.20 \times (0.40 \times 15.000.000) = \$1.200.000$.

Percent Variation includes Increase and Decrease. An increase of $P\%$ results in $C + \frac{P}{100} \times C$, while a decrease results in $C - \frac{P}{100} \times C$. If a television costing $\$187.520$ is sold with a $20\%$ profit, the price is $1.2 \times 187.520 = \$225.024$. In market variations, if a stock rises $20\%$ and then falls $25\%$, its relative value is $1.20 \times 0.75 = 0.9$, denoting a net loss of $10\%$.

Specialized cases include area changes where a field's length increases by $20\%$ and width by $50\%$. The new area is $1.2L \times 1.5W = 1.8LW$, which is an $80\%$ increase over the original area. In console storage, a system update of $45\,\text{GB}$ moving the usage bar from $25\%$ to $40\%$ ($15\%$ difference) means the total disk capacity $C$ is derived from $0.15C = 45$, resulting in $C = 300\,\text{GB}$.