Numerical Ability and Principles of Real Numbers Study Guide

Classification of Real Numbers

The hierarchy of real numbers is categorized into several distinct sets based on their properties and origin. The most fundamental set is the Natural Numbers ($N$), which consists of the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...\}$. These numbers serve two primary functions: counting elements within a set (cardinal numbers) and defining the position or order of elements (ordinal numbers). Natural numbers are ordered, allowing for the determination of which values are greater or lesser, and the set is infinite, meaning there is no upper limit. Moving beyond naturals, the set of Integers ($Z$) encompasses all natural numbers (positives), their opposites (negatives), and zero, expressed as $\{..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...\}$. Integers are applied practically to represent temperatures above and below zero, altitudes relative to sea level, and movement between floors in an elevator. Between these integers exist Rational Numbers ($Q$), also known as fractional numbers. These include any value that can be expressed as a quotient of two integers ($\frac{a}{b}$) where the denominator is not zero ($b \neq 0$). Since every integer can be divided by one (e.g., $\frac{5}{1}$, $\frac{21}{1}$, $\frac{144}{1}$), all integers belong to the set of rational numbers. Conversely, Irrational Numbers ($I$) are those that cannot be expressed as a rational fraction; famous examples include $\pi$ and $\sqrt{2}$. The union of all these sets—natural, integer, rational, and irrational—forms the set of Real Numbers ($R$).

Properties of Real Numbers

If we let $a$, $b$, and $c$ represent real numbers, several fundamental properties define their operations. The first is Cerradura (Closure): performing addition, subtraction, multiplication, or division on two real numbers always results in another real number. The second is Asociatividad (Associative Property) of addition and multiplication, which allows for the grouping of terms using parenthesis without altering the result: $(a + b) + c = a + (b + c)$ for addition and $(ab)c = a(bc)$ for multiplication. For example, $(2 + 3) + 4 = 2 + (3 + 4)$ and $(2 \times 3) \times 4 = 2 \times (3 \times 4)$. Third is Conmutatividad (Commutative Property), which states that the order of terms does not affect the outcome: $a + b = b + a$ and $ab = ba$. Examples include $2 + 3 = 3 + 2$ and $2 \times 3 = 3 \times 2$. The fourth and fifth properties involve neutral elements: the Neutro Aditivo (Additive Identity) is $0$ because adding it does not change the quantity ($a + 0 = a$), and the Neutro Multiplicativo (Multiplicative Identity) is $1$ because multiplying by it leaves the quantity unchanged ($a \times 1 = a$). Note that multiplying by zero results in zero, so the identities must not be confused. The sixth property is the Inverso Aditivo (Additive Inverse): for every real number $a$, there exists a unique number $-a$ such that $a + (-a) = 0$. Finally, the Inverso Multiplicativo (Multiplicative Inverse or Reciprocal) exists for every non-zero real number such that $a \times \frac{1}{a} = 1$. Additionally, the Distributividad (Distributive Property) allows a product to be distributed across a sum: $a(b + c) = ab + ac$, as seen in $2 \times (3 + 4) = (2)(3) + (2)(4)$.

Laws of Signs for Real Numbers

Adding and subtracting integers requires specific rules based on the signs. For addition, if numbers share the same sign, they are summed and the common sign is preserved (e.g., $-25 + (-12) = -37$ and $20 + 15 = 35$). If they have different signs, they are subtracted, and the result takes the sign of the number with the larger absolute value (e.g., $-9 + 13 = 4$ and $12 - 45 = -33$). In subtraction, if the signs are the same, they are summed and the sign is kept (e.g., $-7 - 8 = -15$). If the signs are different, and particularly if the sustraendo (subtrahend) is negative, the negative signs multiply to become positive (e.g., $10 - (-6) = 10 + 6 = 16$). If the minuendo (minuend) is negative, it reverts to the rule for same signs (e.g., $-8 - 5 = -13$). Multiplication and division follow consistent rules: $(+) \times (+) = (+)$, $(-) \times (-) = (+)$, $(+) \times (-) = (-)$, and $(-) \times (+) = (-)$. The same logic applies to division. Examples include $-2(-15) = 30$, $10(-5) = -50$, $-30 \div -2 = 15$, and $-50 \div 5 = -10$. Note that these multiplication/division sign rules cannot be blindly applied to addition and subtraction.

Laws of Exponents

There are eight primary laws governing exponential operations. First, multiplying two quantities with the same base involves adding exponents: $a^m \times a^n = a^{m+n}$ (e.g., $5^2 \times 5^6 = 5^8$). Second, dividing two quantities with the same base involves subtracting exponents: $\frac{a^m}{a^n} = a^{m-n}$ (e.g., $\frac{3^{10}}{3^3} = 3^7$). Third, the product of multiple factors raised to a power applies the power to all factors: $(ab)^n = a^n b^n$ (e.g., $(3 \times 2)^2 = 3^2 \times 2^2$). Fourth, the division of quantities raised to a power applies the power to both numerator and denominator: $(\frac{a}{b})^n = \frac{a^n}{b^n}$ (e.g., $(\frac{6}{9})^2 = \frac{6^2}{9^2}$). Fifth, raising an exponential expression to another power requires multiplying the exponents: $(a^m)^n = a^{mn}$ (e.g., $(8^3)^2 = 8^{3 \times 2} = 8^6$). Sixth, any expression with a negative exponent is equal to its reciprocal: $a^{-n} = \frac{1}{a^n}$ (e.g., $5^{-2} = \frac{1}{5^2}$). Seventh, any quantity (except zero) raised to the power of zero equals one: $a^0 = 1$ (e.g., $120^0 = 1$). Eighth, a number raised to a fractional power results in a root of that number: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ (e.g., $x^{\frac{1}{2}} = \sqrt{x}$).

Prime Factors, Multiples, and Divisors

Prime numbers are positive integers greater than $1$ that are divisible only by themselves and $1$. The sequence begins: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...$. Composite numbers can be broken down into prime factorizations (e.g., $20 = 2^2 \times 5$). The Minimum Common Multiple (mcm) is the smallest common multiple of two or more numbers. For $2$ and $3$, multiples of $2$ include $2, 4, 6, 8, ...$ and multiples of $3$ include $3, 6, 9, ...$, making the mcm $6$. A quick calculation method involves a table where numbers are divided by prime factors until all quotients reach $1$. For $60$ and $190$, the prime factors are $2, 2, 3, 5, 19$, totaling $1140$ ($2 \times 2 \times 3 \times 5 \times 19$). The Maximum Common Divisor (mcd) is the largest factor common to the numbers. For $10$ and $20$, the divisors of $10$ are $\{1, 2, 5, 10\}$ and for $20$ they are $\{1, 2, 4, 5, 10, 20\}$, making the mcd $10$. To calculate mcd quickly, divide all numbers simultaneously by common prime factors until the quotients are relative primes. For $24$ and $36$, the common factors are $2, 2, 3$, resulting in $2 \times 2 \times 3 = 12$. The mcd is used for dividing or partitioning values.

Rational Number Operations (Fractions)

When adding or subtracting fractions with the same denominator, one simply adds or subtracts the numerators and maintains the denominator (e.g., $\frac{11}{15} + \frac{14}{15} = \frac{25}{15} = 1 \frac{10}{15} = 1 \frac{2}{3}$). If denominators differ, find the mcm of the denominators as a common denominator. Divide this common denominator by the original denominator and multiply by the numerator for each fraction to find new summands (e.g., $\frac{2}{15} + \frac{2}{3} + \frac{9}{10}$ with mcm $30$ becomes $\frac{4 + 20 + 27}{30} = \frac{57}{30} = 1 \frac{27}{30} = 1 \frac{9}{10}$). For operations with mixed numbers, convert them to improper fractions first by multiplying the denominator by the integer and adding the numerator (e.g., $5 \frac{2}{3} + 6 \frac{1}{2} + 3 \frac{1}{6} = \frac{17}{3} + \frac{13}{2} + \frac{19}{6} = \frac{34 + 39 + 19}{6} = \frac{92}{6} = 15 \frac{1}{3}$). Multiplication of fractions involves multiplying numerators by numerators and denominators by denominators ($\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$). Division is performed by multiplying the dividend by the multiplicative inverse of the divisor ($\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$).

Decimal Operations and Square Roots

To add or subtract decimals, numbers must be aligned by their decimal points. If digits are missing in the minuend, placeholders (zeros) should be added (e.g., $45.2 + 26 + 3.872 + 1.3 = 76.372$). For multiplication, multiply as if they were natural numbers, then count the total decimal digits in the factors and place the point that many places from the right in the product (e.g., $4.321 \times 0.23 = 0.99383$). In division, there are three scenarios: decimal divided by natural, natural divided by decimal, and decimal divided by decimal. In cases involving decimal divisors, multiply both dividend and divisor by powers of $10$ (like $10, 100$) to convert the divisor into a whole number. Square roots involves finding a number that, when multiplied by itself, equals the radicand. The process includes dividing the radicand into pairs of digits from right to left, searching for the largest whole number square for the first period, and systematically calculating digits. For example, if the root is $12$, the radicand is $12 \times 12 = 144$. If the root is $10$, the radicand is $100$. For $625$, the root is $25$.

Numerical Sequences and Patterns

A numerical sequence is an ordered set of numbers based on a rule ($n$ represents the position). To find the pattern, calculate the difference between terms; this becomes the constant factor. Then, adjust by adding or subtracting to match the first position. For the sequence $3, 8, 13, 18, 23, 28$, the increment is $+5$. Testing $n=1$ gives $5(1) = 5$. To get to $3$, we subtract $2$, establishing the rule $5n - 2$. For position $25$, the value is $5(25) - 2 = 123$. In another sequence $5, 7, 9, 11, 13$, the increment is $2$. Testing $n=1$ gives $2(1) = 2$. To get to $5$, we add $3$, resulting in the generalization $2n + 3$. Checking the third position: $2(3) + 3 = 9$, confirming the rule.

Proportionality and Rule of Three

A numerical proportion is an equality between two ratios ($\frac{a}{b} = \frac{c}{d}$) where the product of extremes equals the product of means ($ad = bc$). For example, if a car does $60$ laps in $105$ minutes, the time for $40$ laps is found by $\frac{105}{60} = \frac{x}{40}$, resulting in $x = \frac{105 \times 40}{60} = 70$ minutes. Proportionality can be direct (both magnitudes increase) or inverse (one increases, the other decreases). Multiplicative inverse is applied for inverse relations. Inverse rule of three involves flipping the fraction for evaluation. For example, if $5$ workers take $8$ hours, how many workers are needed for $10$ hours? Since more hours require fewer workers (inverse), we calculate $\frac{x}{5} = \frac{8}{10}$, so $x = \frac{5 \times 8}{10} = 4$ workers. Multiple proportionality involves three or more magnitudes. If $25$ workers in $8$ hours paint $4\,km$, how many workers are needed for $10$ hours and $15\,km$? The relationship is worker/hours (inverse) and worker/km (direct): $\frac{25}{x} = \frac{10}{8} \times \frac{4}{15}$. This simplifies to $\frac{25}{x} = \frac{40}{120}$, so $40x = 3000$, resulting in $x = 75$ workers.