Official Syllabus Item: 25, 35 Arithmetic & Number Theory

Integers & Fundamental Operations

Q: Is subtraction of integers commutative and associative?

No. Subtraction is neither commutative (a - b != b - a) nor associative ((a - b) - c != a - (b - c)).

Q: What is the additive inverse of an integer?

The additive inverse of an integer a is -a such that a + (-a) = 0. For example, the additive inverse of 7 is -7, and of -15 is +15. Zero is its own additive inverse.

Q: Why does the product of two negative integers equal a positive integer?

By the distributive law, a × [b + (-b)] = 0. Expanding (-a) × [b + (-b)] = 0 gives -(ab) + [(-a) × (-b)] = 0, which implies (-a) × (-b) = +ab.

Q: What is the difference between -a² and (-a)²?

In -a², the exponent applies only to a: -a² = -(a × a) (negative). In (-a)², the negative sign is squared: (-a)² = (-a) × (-a) = +a² (positive).

Q: How should sign conventions be declared in RIMC descriptive word problems?

Explicitly state: 'Let elevation above ground/sea level be (+) and descent below ground/sea level be (-)'. Stating this frame earns 1 mark in the RIMC marking scheme.

Q: What is the product rule when multiplying multiple negative integers?

An even number of negative factors yields a positive product: (-1)^(2k) = +1. An odd number of negative factors yields a negative product: (-1)^(2k+1) = -1.

Q: How does RIMC grade questions involving integer brackets like -[-a - (-b)]?

Examiners award step marks for resolving brackets from inside out: (1) Inner round brackets (), (2) Braces {}, (3) Square brackets [], and (4) Outer signs. Avoid doing all steps in one line.

Q: What is the difference between whole numbers and integers regarding predecessors?

In whole numbers W, 0 has no predecessor. In integers Z, every element has a predecessor (a - 1); the predecessor of 0 in Z is -1.

Understand properties of integers, addition and subtraction rules, number line representation, and additive identity for RIMC Entrance Exam.

Overview & Fundamental Concept

The system of whole numbers is inadequate when solving equations of the form x + a = b where a > b. To give meaning to subtraction in all circumstances, the number system is extended by adjoining the negative counterparts of natural numbers, creating the set of Integers (Z). In RIMC descriptive mathematics, examiners assess candidates on directional number line representations, absolute values, axiomatic properties (closure, commutativity, associativity, additive identity, and inverse), sign rules for products and quotients, and real-world modeling of sea level depths, temperatures, elevator descents, and financial balances.

Key Theoretical Principles

Zero is neither positive nor negative; it is the neutral origin on the integer number line.
The absolute value of an integer |a| represents its non-negative distance from zero on the number line.
Subtracting a negative number is equivalent to adding its positive counterpart: a - (-b) = a + b.
Integers are closed under addition, subtraction, and multiplication, but NOT under division.
Subtraction of integers is NEITHER commutative (a - b != b - a) NOR associative ((a - b) - c != a - (b - c)).
Multiplication is distributive over both addition and subtraction: a(b ± c) = ab ± ac.

Illustrated Conceptual Diagrams & Visual Schematics

Directed Integer Number Line & Vector Sign Rules

Directional Vector Arithmetic: On the integer number line, adding a positive number moves right; subtracting a negative number inverts direction and moves right ($a - (-b) = a + b$). Adding a negative number moves left ($a + (-b) = a - b$). Multiplications: $(+) imes (+) = +$, $(-) imes (-) = +$, $(+) imes (-) = -$.

Topper Strategy & Descriptive Step Breakdown

Mastering RIMC Mathematics Standard

Begin every real-world integer problem by explicitly declaring your directional Cartesian sign convention (e.g., 'Let elevation above sea level be (+) and depth below sea level be (-)'). Show bracket-clearing steps individually: do not combine sign resolution and multiplication in a single step.

Step 1: Given Information & Variable Declaration (1-2 Marks)

Clearly write down the data provided in the question. State any assumption (e.g., 'Let the cost price be ₹ x').

Step 2: Formula & Conceptual Foundation (1-2 Marks)

State the exact mathematical theorem or formula in standard algebraic notation before entering numeric values.

Step 3: Systematic Calculation & Reduction (2-4 Marks)

Carry out each step on a fresh line. Avoid combining three operations in one step so examiners can award partial credit.

Step 4: Final Statement with Proper Units (1 Mark)

End with a concluding statement highlighting the answer in a neat box or underline with explicit units (e.g., '₹ 1,500', '45 km/h').

Key Formulas & Conversion Cheat Sheet

Additive Inverse & Subtraction Definition

$$a - b = a + (-b); \quad a - (-b) = a + b; \quad a + (-a) = 0$$

Sign Rules for Multiplication

$$(+) \times (+) = (+), \quad (-) \times (-) = (+), \quad (+) \times (-) = (-)$$

Distributive Laws of Integers

$$a \times (b + c) = a \times b + a \times c; \quad a \times (b - c) = a \times b - a \times c$$

Absolute Value Triangle Inequality

$$|a + b| \le |a| + |b| \quad \text{and} \quad |a - b| \ge ||a| - |b||$$

Product of Negative Signs Rule

$$(-1)^n = +1 \text{ if } n \text{ is even}; \quad (-1)^n = -1 \text{ if } n \text{ is odd}$$

Solved Model Examples (Descriptive Format with Step Mark Distribution)

Solved Model 1 Model Solution

Evaluate: (-15) + [28 - {(-12) - (18 - (7 - 4))}].

Step-by-Step Model Solution:

Step 1: Evaluate under vinculum/inner brackets: 7 - 4 = 3.\nExpression = (-15) + [28 - {(-12) - (18 - 3)}].\nStep 2: Evaluate round brackets: 18 - 3 = 15.\nExpression = (-15) + [28 - {(-12) - 15}].\nStep 3: Evaluate braces: (-12) - 15 = -27.\nExpression = (-15) + [28 - {-27}].\nStep 4: Remove braces: 28 - (-27) = 28 + 27 = 55.\nExpression = (-15) + 55.\nStep 5: Final addition: 55 - 15 = 40.\nFinal Answer: 40.

Solved Model 2 Model Solution

Verify whether a - (b - c) = (a - b) - c for a = -8, b = -5, and c = 3. What conclusion do you draw regarding the associative property of subtraction of integers?

Step-by-Step Model Solution:

Step 1: Evaluate LHS: a - (b - c) = (-8) - [(-5) - 3] = (-8) - [-8] = -8 + 8 = 0.\nStep 2: Evaluate RHS: (a - b) - c = [(-8) - (-5)] - 3 = [-8 + 5] - 3 = -3 - 3 = -6.\nStep 3: Compare LHS and RHS: LHS (0) != RHS (-6).\nConclusion: Subtraction of integers is NOT associative.

Solved Model 3 Model Solution

An elevator descends into a mine shaft at the rate of 6 metres per minute. If the descent starts from 10 metres above the ground level, how long will it take to reach a depth of 350 metres below the ground level?

Step-by-Step Model Solution:

Step 1: Sign convention: Ground = 0 m; Above ground = (+); Below ground = (-); Descent rate = -6 m/min.\nStep 2: Initial position h1 = +10 m; Final position h2 = -350 m.\nStep 3: Total displacement = h2 - h1 = (-350) - (+10) = -360 m.\nTotal downward distance = |-360| = 360 m.\nStep 4: Time taken = Total Distance / Speed = 360 / 6 = 60 minutes = 1 hour.\nFinal Answer: It will take 1 hour (60 minutes) to reach 350 m below ground level.

RIMC Practice & Drill Exercises

Attempt these questions on paper following proper step-wise conventions, then click Reveal Solution to verify your answers.

Question 1 5 Marks

Write down a pair of integers whose: (a) sum is -7; (b) difference is -10; (c) sum is 0.