Chapter 1 — Integers

What are integers? Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. The set of integers is denoted by Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Basic ideas

• Positive integers (> 0): 1, 2, 3, ...
• Negative integers (< 0): -1, -2, -3, ...
• Zero (0): neither positive nor negative

Representation on a number line: Integers are placed at equal intervals on a horizontal line. 0 is at the centre; positive integers lie to the right of 0, negatives to the left. The further right a number is, the greater it is.

Absolute value: The absolute value of an integer a, written |a|, is its distance from 0 on the number line. Example: |−4| = 4, |3| = 3.

Comparing integers: On the number line a number to the right is greater. For example, 2 > −1 because 2 is to the right of −1.

Basic operations and rules

• Addition (using signs):
  • Same sign: add magnitudes, keep the sign (e.g., 5 + 3 = 8, −5 + (−3) = −8).
  • Different signs: subtract smaller magnitude from larger, take sign of larger magnitude (e.g., 7 + (−4) = 3; −7 + 4 = −3).
• Subtraction: a − b = a + (−b). Convert subtraction into addition of the additive inverse and apply addition rules.
• Multiplication and division (sign rules):
  • Product/quotient of numbers with same sign is positive (e.g., (−2)×(−3)=6, 6÷2=3).
  • Product/quotient of numbers with different signs is negative (e.g., (−2)×3=−6, −6÷2=−3).

Key properties of integers

• Closed under addition, subtraction and multiplication (result stays an integer).
• Additive identity: 0 (a + 0 = a).
• Multiplicative identity: 1 (a × 1 = a).
• Additive inverse: for each a there is −a such that a + (−a) = 0.
• Commutative and associative laws hold for addition and multiplication; distributive law links them.

How to visualise operations: Use the number line to perform addition and subtraction by moving right for positive steps and left for negative steps. For multiplication think of repeated addition or groups.

This introduction builds the foundation for solving integer problems involving temperature changes, bank balances, elevations, and more.

What is a number line? A number line is a straight horizontal line on which numbers are placed at equal intervals. It helps visualise integers, their order, distance from zero and basic operations like addition and subtraction.

How to represent integers on a number line

• Choose a horizontal line and mark a point as 0 (the origin).
• From 0, mark equal spaced points to the right and left. Points to the right of 0 are positive integers (1, 2, 3, ...). Points to the left of 0 are negative integers (−1, −2, −3, ...).
• Label the marks with integers. Draw arrowheads at both ends to show the line continues infinitely.
• An integer a is placed at a distance |a| units from 0: to the right if a > 0, to the left if a < 0. The integer 0 is at the origin.

Key ideas explained

• Opposite integers: a and −a are on opposite sides of 0 and are equidistant from 0.
• Order: On the number line, any integer to the right is greater than any integer to its left. Example: 4 is to the right of −2, so 4 > −2.
• Absolute value: The absolute value |a| is the distance of a from 0 on the number line (always non‑negative).
• Distance between two integers: The distance between a and b on the number line is |a − b|.
• Counting integers between two integers: Number of integers from a to b inclusive is |a − b| + 1. If exclusive (not including endpoints) it is |a − b| − 1 (if positive).

Using the number line for operations

• Addition: From the first integer, move right (for positive addend) or left (for negative addend) by the magnitude of the second integer.
• Subtraction: Subtracting b from a (a − b) is same as adding the opposite: a + (−b). Move right if −b is positive, left if −b is negative.

This visual tool makes it easy to compare integers, find distances, and perform simple integer arithmetic.

What are we comparing? Comparison of integers means deciding which of two integers is greater, smaller or if they are equal. Ordering means arranging a set of integers from the smallest to the greatest (ascending) or greatest to smallest (descending).

Visual idea — the number line: Draw a horizontal line with 0 in the middle, positive integers to the right (1, 2, 3, ...), and negative integers to the left (−1, −2, −3, ...). A point that is further to the right is always greater.

Rules for comparing two integers

• If one integer is positive and the other is negative, the positive integer is always greater. Example: 5 > −2.
• If both integers are positive, the one with the larger absolute value is greater. Example: 8 > 3 because 8 has larger value than 3.
• If both integers are negative, the one with the smaller absolute value is greater. Example: −3 > −7 because |−3| = 3 < |−7| = 7, so −3 is to the right of −7 on the number line.
• Zero is greater than any negative integer and less than any positive integer: for any positive n, n > 0 > −n.

Step-by-step method to compare two integers a and b

1. If a and b have different signs, the positive one is greater.
2. If both are positive, compare their usual sizes (larger absolute value is larger).
3. If both are negative, compare their absolute values; the one with the smaller absolute value is larger.
4. Alternatively, place both on a number line — the one to the right is greater.

Ordering a set of integers

To arrange integers in ascending order (smallest to largest): place them on the number line and read from left to right. For descending order, read from right to left.

Important properties (useful in solving problems)

• Transitive property: If a < b and b < c, then a < c.
• Addition property: If a < b, then a + c < b + c (for any integer c).
• Multiplication property: If a < b and c > 0, then ac < bc. If c < 0, then ac > bc (inequality reverses when multiplying by a negative).

Real-life contexts

• Temperature: −5°C is colder than 2°C, so 2 > −5.
• Bank balance: a balance of −500 (overdraft) is less than a balance of 1000.
• Floors of a building: basement −2 is below ground and is less than ground floor 0 and 3rd floor.
• Sea level: depth −10 m is lower than −2 m; higher altitude (+50 m) is greater than sea level 0 m.

What is absolute value? The absolute value of an integer is its distance from 0 on the number line, always taken as a non-negative number. It is written using vertical bars: |a|. For example, |5| = 5 and |−5| = 5, because both 5 and −5 are 5 units away from 0.

Formal (piecewise) definition:

• |x| = x, if x ≥ 0
• |x| = −x, if x < 0 (so that the result is positive)

Key ideas and properties:

• Absolute value is always non-negative: |x| ≥ 0 for every x.
• Distance interpretation: |a − b| is the distance between a and b on the number line.
• Symmetry: |x| = |−x|.

How to read equations with absolute value: If |x| = a and a > 0, then x = a or x = −a. If |x| = 0, then x = 0. If |x| = b with b < 0, no solution (absolute value cannot be negative).

Definition: The opposite (or additive inverse) of an integer a is the integer that when added to a gives 0. It is denoted by −a. So the opposite of a is −a and a + (−a) = 0.

How to find the opposite: If a is positive, its opposite is the corresponding negative number; if a is negative, its opposite is the corresponding positive number. The opposite of 0 is 0 itself.

Key properties:

• Uniqueness: Every integer has exactly one additive inverse.
• Sign rule: Opposite of a positive number is negative and vice versa. Example: opposite of 8 is −8; opposite of −3 is 3.
• Double opposite: −(−a) = a.
• Addition: a + (−a) = 0 for every integer a.
• Distributive with addition: −(a + b) = (−a) + (−b).
• Multiplication by −1: (−1)×a = −a.

Using a number line: Plot a on the number line. The opposite −a is the point at the same distance from 0 but on the other side. Points a and −a are symmetric about 0. This visual makes it clear why their sum is 0.

Relation to subtraction: Subtraction a − b can be seen as adding the opposite: a − b = a + (−b). Thus understanding opposites turns subtraction into addition.

What are integers? Integers are whole numbers that can be positive, negative or zero: {..., -3, -2, -1, 0, 1, 2, 3, ...}.

What is addition of integers? Addition of integers means combining two or more integers to get a sum. We use the number line idea: move to the right for a positive integer and to the left for a negative integer.

Rules (short)

• Same sign: Add their absolute values; the sum has the same sign. Example: (-3) + (-7) = -(3+7) = -10.
• Different signs: Subtract the smaller absolute value from the larger absolute value; the result takes the sign of the integer with the larger absolute value. Example: 8 + (-5) = +(8-5) = 3; (-8) + 5 = -(8-5) = -3.
• Adding zero: Adding 0 leaves the integer unchanged: a + 0 = a.

Using a number line (visual method)

• Start at 0 or the first integer’s point on the number line.
• For a positive number, move right by that many units; for a negative number, move left by that many units.
• When adding several integers, perform the moves sequentially.

Properties

• Commutative: a + b = b + a
• Associative: (a + b) + c = a + (b + c)

Worked steps — examples inside explanation

• Example 1: (-4) + (-6). Same sign (both negative). Add absolutes: 4+6=10 → answer = -10.
• Example 2: 7 + (-3). Different signs. Subtract: 7-3=4. Larger absolute is 7 (positive) → answer = +4.
• Example 3 using number line: Start at -2, add +5: move right 5 steps from -2 to +3 → result = 3.

Tips

• Turn addition of integers into simple add/subtract of absolute values using the sign rules above.
• When unsure, draw a number line and perform the moves.

What is subtraction of integers? Subtraction of integers means finding how much one integer differs from another. Subtraction can be done by converting it into addition: subtracting an integer is the same as adding its additive inverse. In symbols, a − b = a + (−b).

Key idea (additive inverse): Every integer x has an additive inverse −x such that x + (−x) = 0. So to subtract b from a, add −b to a.

Rules (number-line view):

• If you subtract a positive integer (a − positive), move left on the number line.
• If you subtract a negative integer (a − (−b)), you move right because a − (−b) = a + b.
• Start at the point a on the number line, then move |b| units left if b is positive, or |b| units right if b is negative.

Sign rules (quick forms):

• a − b = a + (−b).
• a − (−b) = a + b.
• (−a) − b = −(a + b).
• (−a) − (−b) = −a + b = b − a.

Properties to remember:

• Subtraction is not commutative: a − b ≠ b − a (in fact a − b = −(b − a)).
• Subtracting zero: a − 0 = a and 0 − a = −a.
• The magnitude of the difference is |a − b|, which is the distance between a and b on the number line.

How to do it step by step (two common methods):

1. Algebraic method: rewrite subtraction as addition of the inverse: compute a + (−b) using integer addition rules.
2. Number-line method: locate a; if subtracting a positive b, move left b units; if subtracting a negative (−b), move right b units; result is the landing point.

Using these approaches makes subtraction of integers systematic and easy to visualize.

What is multiplication of integers?

Multiplication of integers is repeated addition of an integer a specified number of times, extended to include negative numbers. It follows simple sign rules and the basic properties of multiplication (commutative, associative, distributive).

Sign rules (short summary):

• Positive × Positive = Positive (e.g., 3 × 4 = 12)
• Positive × Negative = Negative (e.g., 3 × −4 = −12)
• Negative × Positive = Negative (e.g., −3 × 4 = −12)
• Negative × Negative = Positive (e.g., −3 × −4 = 12)
• If one factor is 0, the product is 0 (e.g., 0 × 5 = 0)

Why negative × negative = positive? Count the number of negative factors: if the number of negative factors is even, the product is positive; if odd, the product is negative. Equivalently, multiplying by a negative reverses direction; doing it twice reverses direction twice, returning to the original (positive) direction.

Properties used with integers

• Commutative: a × b = b × a
• Associative: (a × b) × c = a × (b × c)
• Distributive over addition: a × (b + c) = a × b + a × c
• Zero property: a × 0 = 0
• Identity: a × 1 = a

Multiplying more than two integers

Apply sign rule by counting negative factors (even → product positive, odd → product negative) and multiply absolute values as usual, then attach the sign determined.

What is Division of Integers?

Division of integers means splitting one integer (the dividend) into equal groups each of size equal to another integer (the divisor). If a and b are integers and b ≠ 0, then a ÷ b = c means that when a is divided into groups of size b we get c groups. Equivalently, a = b × c.

Key ideas and rules

• Sign rule: The quotient of two integers is positive if the dividend and divisor have the same sign, and negative if they have opposite signs. (Example: (+12) ÷ (+3) = +4; (−12) ÷ (−3) = +4; (+12) ÷ (−3) = −4.)
• Division by zero: Division by 0 is not defined. You cannot divide any integer by 0.
• Zero as dividend: 0 divided by any nonzero integer is 0. (0 ÷ b = 0 for b ≠ 0.)
• Inverse relation to multiplication: a ÷ b = c if and only if a = b × c (for b ≠ 0).
• Division properties: Division is not commutative (a ÷ b ≠ b ÷ a in general) and not associative ((a ÷ b) ÷ c ≠ a ÷ (b ÷ c) in general).

Division as repeated subtraction and on a number line

Division can be seen as repeated subtraction: a ÷ b asks how many times we can subtract b from a until nothing (or less than b) is left. On a number line, for positive divisor b, move to the right in steps of size b when the divisor is positive and to the left if the divisor is negative; count the number of steps to reach the dividend from zero to get the quotient (sign included).

Quotient and remainder (for positive integers)

When dividing a positive integer by a positive integer that does not divide it exactly, we get a quotient and a remainder: a = b × q + r where 0 ≤ r < b. This is the usual division algorithm used in arithmetic.

What are integers and integer operations? Integers are whole numbers and their negatives: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Integer operations are addition, subtraction, multiplication and division performed on integers. Each operation has certain properties that help in simplifying calculations and solving problems.

Key properties

• Closure: The result of an operation on two integers is an integer for addition, subtraction and multiplication. That is, if a and b are integers then a + b, a - b and a × b are integers. Division is not closed in integers because a ÷ b need not be an integer (and b must not be 0).
• Commutative property: Changing the order does not change the result for addition and multiplication.
  • a + b = b + a
  • a × b = b × a
  Subtraction and division are not commutative.
• Associative property: Changing grouping (parentheses) does not change the result for addition and multiplication.
  • (a + b) + c = a + (b + c)
  • (a × b) × c = a × (b × c)
  Subtraction and division are not associative.
• Identity elements:
  • Additive identity: a + 0 = a
  • Multiplicative identity: a × 1 = a
• Additive inverse: Every integer a has an additive inverse −a such that a + (−a) = 0. (Multiplicative inverse 1/a is generally not an integer unless a = ±1.)
• Distributive property: Multiplication distributes over addition and subtraction:
  • a × (b + c) = a×b + a×c
  • a × (b − c) = a×b − a×c
• Zero rules and division:
  • a × 0 = 0
  • 0 ÷ a = 0 for a ≠ 0
  • Division by zero (a ÷ 0) is undefined.
• Sign rules for multiplication and division:
  • Product or quotient of two integers with the same sign is positive: (+)×(+) = +, (−)×(−) = +.
  • Product or quotient of two integers with opposite signs is negative: (+)×(−) = −, (−)×(+) = −.
• Addition of signed integers (short rules):
  • If signs are the same, add magnitudes and keep the sign.
  • If signs differ, subtract smaller magnitude from larger magnitude and take the sign of the larger magnitude.
  • Subtraction can be treated as addition of the additive inverse: a − b = a + (−b).

Understanding these properties helps to rearrange and simplify expressions involving integers, check work quickly, and apply methods like grouping or distributing to compute accurately.

What this topic means: Word problems with integers require translating everyday situations into integer expressions or equations, applying the rules of integer operations, and interpreting the answer in the given context.

Common contexts: temperatures (above/below zero), elevations (above/below sea level), bank transactions (deposits/withdrawals), gains/losses, score changes, position changes (up/down), and profit or debt situations.

Steps to solve integer word problems:

• Read the problem carefully and identify quantities and directions (increase/decrease, above/below, deposit/withdrawal).
• Assign signs: positive for rise/above/deposit/gain, negative for fall/below/withdrawal/loss.
• Write the corresponding integer expression or equation.
• Use integer operation rules (addition, subtraction, multiplication, division) to simplify or solve for unknowns.
• Interpret the result in the context of the problem and check the answer.

Key idea about subtraction: Convert subtraction to addition of the opposite: a - b = a + (−b). This helps apply addition rules consistently.

Number line model: Visualize operations on a number line: moving right for positive numbers and left for negative numbers. This helps to understand addition and subtraction intuitively.

What this topic covers
Practice Exercises and Patterns in the chapter on Integers help students master operations on integers, recognise regular sequences, and apply sign rules quickly. Emphasis is on using the number line, converting subtraction into addition of opposites, spotting arithmetic patterns (constant difference), and recognising sign behaviour in multiplication and powers.

Key ideas and strategies

• Use a number line to visualise addition and subtraction: move right for positive, left for negative.
• Convert subtraction to addition: a − b = a + (−b). This simplifies many problems.
• Learn the sign rules for multiplication and division and practise until they are automatic.
• Look for arithmetic patterns: if consecutive terms change by the same integer, they form an arithmetic sequence.
• Observe special patterns: repeated multiplication by −1 alternates signs; sum of opposites is zero; square of any integer is non‑negative.
• Use parity (even/odd) and divisibility patterns when asked about multiples or every nth term.

How to approach practice exercises

1. Read the problem and identify whether it is arithmetic (find next term), operational (add/subtract/multiply/divide integers) or word‑based (real life context).
2. If sequence/pattern: compute differences (or ratios) between successive terms to find the rule.
3. If operation: simplify using sign rules and check using the number line or simple examples.
4. For word problems, translate deposits/withdrawals, temperature rises/falls, floors above/below ground into positive/negative integers first, then compute.

Common pitfalls
Not converting subtraction into addition of the opposite; forgetting that two negatives multiply to a positive; misplacing direction on the number line; mixing up the sign when factoring out a negative.