Civil Service Exam (Professional) Numerical Ability — Integers, PEMDAS & DivisibilityStudy Notes

Integers are whole numbers that can be positive, negative, or zero. The set of integers is represented as Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}. Unlike fractions or decimals, integers have no fractional parts. Zero is special—it's neither positive nor negative, neither prime nor composite, but it is an even number. Understanding integers is crucial for all mathematical operations and forms the foundation for more complex numerical concepts.

Working with integers follows specific rules, especially when dealing with positive and negative numbers. The Law of Signs governs how we add, subtract, multiply, and divide integers. For addition: same signs add and keep the sign, different signs subtract and take the sign of the larger absolute value. For subtraction: change to addition of the opposite. For multiplication and division: same signs give positive results, different signs give negative results. These rules are essential for solving complex problems accurately.

PEMDAS is the acronym for the correct order of mathematical operations: Parentheses, Exponents, Multiplication & Division (left to right), Addition & Subtraction (left to right). This order ensures everyone gets the same answer when solving complex expressions. Start with innermost parentheses and work outward, then handle exponents, followed by multiplication and division from left to right, and finally addition and subtraction from left to right. Remember: multiplication and division have equal priority, as do addition and subtraction.

The absolute value of a number is its distance from zero on the number line, always expressed as a positive value. Denoted by |n|, absolute value represents the magnitude of a number regardless of its sign. For positive numbers and zero, the absolute value is the number itself. For negative numbers, the absolute value is the positive version of that number. This concept is crucial in understanding distances, differences, and in solving equations involving both positive and negative values.

Factors are numbers that divide evenly into another number without leaving a remainder. Every number has at least two factors: 1 and itself. Multiples are the results of multiplying a number by whole numbers (1, 2, 3, ...). Prime numbers have exactly two factors (1 and themselves), while composite numbers have more than two factors. Understanding factors and multiples is essential for finding Greatest Common Factor (GCF) and Least Common Multiple (LCM), which are frequently tested concepts.

Divisibility rules are shortcuts that help determine if one number divides evenly into another without performing long division. These rules save time and are particularly useful in exams. For example, a number is divisible by 2 if it's even, by 3 if the sum of its digits is divisible by 3, by 5 if it ends in 0 or 5, and by 9 if the sum of its digits is divisible by 9. Mastering these rules allows for quick factorization and problem-solving in various mathematical contexts.

Scientific notation is a way to express very large or very small numbers in the form m × 10ⁿ, where m is between 1 and 10 (but not equal to 10), and n is an integer. This notation is particularly useful in science and engineering where extreme values are common. To convert to scientific notation, move the decimal point to create a number between 1 and 10, then count how many places you moved—this becomes your exponent. Moving left gives positive exponents, moving right gives negative exponents.