Chapter 6 — Integers

Integers are the set of whole numbers that include positive numbers, negative numbers and zero. We write the positive integers as 1, 2, 3, ... and the negative integers as −1, −2, −3, ... Zero (0) is a part of this set and acts as the neutral element for addition. Integers do not include fractions or decimals. Learning integers means learning how to read, write and use numbers that can be below or above a reference point.

Integer examples occur in everyday life. Temperatures may be below zero (for example −5°C), bank accounts may show negative balances (debt) like −200 rupees, and floors in a building may be below ground level (B1 as −1). Using integers lets us record direction and magnitude: +10 could mean 10 steps forward, while −10 means 10 steps backward. Knowing integers helps to model such situations and to perform calculations that involve both increases and decreases.

We represent integers on paper and in problems using a minus sign (−) before the number for negatives. The same digits without a sign are positive by default. It is important to notice the sign because it changes the meaning: +4 and −4 are different integers. Zero is unique: it is neither positive nor negative and adding zero to any integer leaves it unchanged. Practice by listing integers around zero and by naming simple real situations where integers appear. This builds the foundation for operations, ordering and algebraic thinking that follow in later lessons.

The number line is a simple but powerful tool to represent integers visually. Draw a straight horizontal line and mark a central point as 0. To the right at equal intervals mark 1, 2, 3, ... representing positive integers that increase as you go right. To the left at the same equal intervals mark −1, −2, −3, ... representing negative integers that decrease as you go left. Each mark is one unit away from its neighbour, so the arrangement reflects how integers change by plus or minus one.

Using the number line helps with many tasks. To compare two integers, look at their positions: the one to the right is greater. For example, 4 is to the right of −1, so 4 > −1. To find the distance or difference between two integers, count the units between their marks or compute the absolute value of their difference: distance between a and b = |a − b|. The number line also visualises addition and subtraction: start at the first number, then move right for adding a positive number or left for adding a negative number. For subtraction, subtracting a positive moves left, while subtracting a negative moves right because subtracting −b equals adding +b.

Number-line models make rules easier to remember because students can see movement and symmetry. They also illustrate opposites: a and −a lie symmetrically at equal distance from 0. Encourage drawing number lines when solving problems with integers, especially for small numbers and word problems, as it strengthens understanding and reduces sign errors.

Comparing integers means deciding which is larger or smaller. Use the number line as the main tool: integers to the right are greater than those to the left. If you are given two integers a and b, check their positions; if a lies to the right of b then a > b, otherwise a < b. For example, 1 is to the right of −4 so 1 > −4. This visual rule works in all cases and helps avoid confusion when signs are involved. Positive integers are always greater than zero and therefore greater than every negative integer, while negative integers are always less than zero.

Among negative integers we use absolute values to compare: the number with the smaller absolute value is greater because it is closer to zero. For example, −2 is greater than −7 because 2 < 7 and −2 is nearer to zero. When two integers have the same absolute value but opposite signs, the positive one is always larger: for instance 5 > −5. These ideas let us compare any pair of integers without mistakes.

Ordering a set of integers means arranging them from smallest to largest (ascending) or vice versa (descending). To order them, either place each number on a drawn number line and read from left to right for ascending order, or sort by sign first then by absolute value: list all negatives (more negative means smaller), then zero, then positives (larger absolute value means larger). Use symbols >, < and = to write comparisons and practise with mixed lists to become quick. Regular exercises where students place numbers on number lines, explain their reasoning, and check by counting steps build strong comparison skills useful for inequalities and algebra later.

Absolute value measures how far an integer is from zero, ignoring its sign. We write absolute value with vertical bars: |a| denotes the absolute value of a. By definition, |a| is the non-negative distance from 0 to a on the number line. Thus |5| = 5 because 5 is five units to the right of zero, and |−5| = 5 because −5 is five units to the left of zero. Absolute value is always zero or positive.

Absolute value is useful in many situations. It gives distance between two integers: the distance between a and b equals |a − b|. For example, the distance between −3 and 2 is |−3 − 2| = |−5| = 5 units. Absolute value helps when we only care about size, not direction. In real-life problems we often want the magnitude of a change (for example a rise or fall in temperature), and absolute value gives that magnitude.

Properties to remember: |0| = 0; |a| = |−a|, because both a and −a are equally distant from zero. When solving problems use the number line to check absolute values visually. Also, absolute value turns negative differences to positive distances: even if a − b is negative, |a − b| is positive. Practice computing absolute values in various contexts so students become confident in interpreting magnitude and distance correctly.

Every integer has an opposite, also called its additive inverse. The opposite of a number a is written as −a. When you add a number to its opposite, the sum is zero: a + (−a) = 0. This is central to solving equations and to understanding subtraction as the addition of an opposite. Zero is its own opposite because 0 = −0.

Knowing opposites helps when changing subtraction into addition. The rule a − b = a + (−b) converts subtraction into adding the opposite, so you can use addition rules. For example, 6 − 9 becomes 6 + (−9), and using addition rules gives −3. Subtracting a negative reverses the sign: a − (−b) = a + b, because the opposite of −b is +b. This explains why subtracting a debt increases your balance: removing a negative amount adds a positive amount.

On the number line opposites are symmetric: the point at a and the point at −a are at equal distance from 0 but on opposite sides. This symmetry is useful when solving problems involving cancellation: if a student owes Rs.50 (−50) and receives Rs.50 (+50), the net is zero. Understanding opposites also supports algebraic thinking when manipulating expressions and solving equations—moving a term to the other side changes its sign because we add the opposite. Practice finding opposites for many integers and check by adding the pair to confirm they sum to zero. Use real examples to solidify the idea: temperature changes, walking directions and money transactions often involve opposites and help students internalise the concept.

Addition of integers follows simple sign rules that students can learn with number-line models. When adding two integers with the same sign, add their absolute values and retain the common sign. For example, (−6) + (−3) = −(6 + 3) = −9; similarly 7 + 5 = 12. When adding integers with opposite signs, subtract the smaller absolute value from the larger absolute value and give the sign of the number with the larger absolute value. For example, 8 + (−5) = 3 because 8 − 5 = 3 and 8 has the larger absolute value, so the sum is +3.

Using the number line: to add a positive number, move to the right; to add a negative number, move to the left. Start at the first integer, then move the required units. This visual method shows why the sign rules work. Also remember the role of zero: adding zero leaves the number unchanged, a + 0 = a, and adding opposites gives zero, a + (−a) = 0.

Practice examples with different combinations of signs and sizes to build confidence. Teach students to convert subtraction into addition of opposites when needed: a − b = a + (−b). Encourage checking sums by reversing the operation: if a + b = c, then c − b should equal a. Repeated practice, both on number line sketches and numeric computations, reduces careless sign errors and improves speed.

Subtraction of integers is most easily handled by converting it into addition of the opposite. The key rule is a − b = a + (−b). This method avoids special subtraction rules and makes every subtraction an addition problem, so the addition rules for signs apply. For example, 5 − (−2) becomes 5 + 2 = 7, since subtracting a negative adds a positive. Similarly, −3 − 4 = −3 + (−4) = −7 by adding two negatives.

On the number line, subtraction means starting at the first number and moving left to subtract a positive number, or moving right when subtracting a negative number (because subtracting a negative equals adding a positive). Visual movement reduces mistakes: to compute 1 − (−3) start at 1 and move right 3 units to reach 4. Always be careful with parentheses and signs: an expression like 6 − (−2) differs from 6 − −2 only in writing but both mean 6 + 2. Practice many kinds of examples including double negatives until the conversion becomes automatic.

Remember special cases: a − 0 = a and 0 − a = −a. After solving, check by adding the subtracted number to your result to see if you recover the original: (a − b) + b should equal a. This inverse check helps catch sign errors quickly. Teach students to write each step and to use number-line sketches for tricky cases.

Multiplication with integers uses the same magnitude calculation as with natural numbers but follows sign rules for the result. Multiply the absolute values to get the magnitude. For the sign, if both factors have the same sign (both positive or both negative) then the product is positive; if the factors have opposite signs, the product is negative. Thus (−4) × (−3) = +12 while (−4) × 3 = −12. This sign rule follows because multiplying by a negative can be thought of as a direction reversal.

For products with more than two factors, count the number of negative factors. If the count of negative factors is even, the product is positive; if odd, the product is negative. For example, (−2) × 3 × (−1) has two negatives so the product is positive: (−2 × 3) × (−1) = (−6) × (−1) = 6. Use repeated addition to see the magnitude when factors are positive. For negative factors, think of multiplication as repeated movement in the opposite direction to gain intuition.

Properties like distributivity still hold: a(b + c) = ab + ac even for negative numbers. Use multiplication tables for magnitudes and apply sign rules carefully. Remind students that multiplication by 1 leaves the number unchanged and multiplication by 0 gives 0. Encourage checking by dividing the product by one of the factors to confirm the other factor returns.

Division of integers follows similar sign rules to multiplication and uses absolute values for magnitude. When dividing two integers, divide the absolute values to find the size of the quotient. For the sign, if the dividend and divisor have the same sign, the quotient is positive; if their signs differ, the quotient is negative. For example, 18 ÷ 3 = 6, (−18) ÷ (−3) = 6, (−18) ÷ 3 = −6 and 18 ÷ (−3) = −6. Always remember that division by zero is not allowed: a ÷ 0 is undefined for any integer a.

Think of division as the reverse of multiplication. After finding a quotient c such that a ÷ b = c, check by multiplying c × b to see if you recover a. This inverse relation helps find mistakes in sign or magnitude. When dividing several integers in sequence, perform the division step by step and keep track of signs at each stage. For example, 48 ÷ (−4) ÷ (−2) is evaluated as (48 ÷ (−4)) ÷ (−2) = (−12) ÷ (−2) = +6 because two negatives make a positive.

Notice that integer division does not always yield an integer; if the absolute values do not divide evenly, the exact quotient will be a fraction or decimal and not an integer. In class exercises focus on cases where division is exact to remain within integers. In word problems, carefully record whether a quantity represents a debt (negative) or a positive amount and apply the sign rule. Use number-line thinking for simple examples: dividing by −1 changes the sign of a number, so 7 ÷ (−1) = −7. Regular practice with checks and contextual examples builds confidence when working with signed divisors and dividends.

Integers obey many arithmetic properties that make calculations predictable. For addition: closure means the sum of two integers is an integer; commutativity means a + b = b + a; associativity means (a + b) + c = a + (b + c). There is an additive identity 0 such that a + 0 = a, and every integer a has an additive inverse −a with a + (−a) = 0. These properties hold for positive and negative numbers alike and help rearrange sums for easier calculation.

For multiplication: closure means the product of two integers is an integer; commutativity means a × b = b × a; associativity means (ab)c = a(bc). Multiplicative identity is 1 since a × 1 = a. Distributivity links multiplication and addition: a(b + c) = ab + ac, and this applies even when some numbers are negative. Multiplication by zero always gives zero: a × 0 = 0. These properties allow simplification and mental calculation strategies, for example to factor expressions or expand brackets.

However, subtraction and division are not generally commutative or associative, so apply caution. Use the properties to rearrange computations into easier steps, check work by inverse operations, and to prove simple equalities. Practising these properties with both positive and negative integers shows their wide applicability and is useful preparation for algebra where they are used routinely.

Word problems show how integers model real-life situations. Typical contexts include temperature changes, bank balances (credits and debts), elevation above or below sea level, and gains or losses in games. To solve a word problem: first read carefully and identify quantities and whether each is positive or negative. Use clues in words: "below", "loss", "debt", "decrease" often mean negative, while "above", "gain", "profit", "increase" usually mean positive. Write each change as an integer and then combine them using the correct operations.

For multi-step problems, process one change at a time and keep units (°C, metres, rupees) visible to avoid mistakes. Representing the situation on a number line helps visualise movements: starting at one value, apply each change by moving right for positive additions and left for negative additions. For example, if the temperature is −2°C, it rises by 9°C and then falls by 4°C, compute: start −2, add +9 to get 7, then add −4 to get 3°C. Always check signs: subtracting a negative equals adding a positive, which can be counterintuitive at first.

Use inverse checks after solving: if you calculated a net change by adding, you can subtract one of the changes to see if you return to the previous value. In distribution problems ensure divisibility to remain within integers or state when a fractional result is needed. Practise a variety of contexts so students learn to translate words to integer operations quickly and accurately. Encourage neat working, labelled steps and a final sentence that answers the question with correct units, for clarity and marks in exam-style questions.

Factors and multiples extend to negative integers. If a × b = c, then a and b are factors of c, and the signs of factors determine the sign of the product. For example, −3 × 4 = −12, so −3 and 4 are factors of −12. Every positive divisor d of a number has a corresponding negative divisor −d that also divides the number with opposite sign. Multiples of an integer n include both positive and negative values: ..., −2n, −n, 0, n, 2n, 3n, ... . When finding LCM and GCD for signed integers, we usually use absolute values and then attach a sign if needed, so LCM(6, −8) = LCM(6, 8) = 24.

Integer sequences show clear patterns. Arithmetic sequences have a fixed common difference which may be positive or negative; for example 5, 2, −1, −4 has common difference −3 and each term decreases equally. Alternating sequences like 1, −1, 1, −1 repeat sign pattern and are periodic. Recognising these patterns helps in predicting the next terms and in writing rules for nth terms, where the nth term of an arithmetic sequence is a_n = a_1 + (n − 1)d, valid whether d is positive or negative.

Checking and estimation are important skills to avoid mistakes. Use inverse operations: if a + b = c then c − b should be a; if a ÷ b = c then c × b should return a. For sign checks, predict the sign before computing: the sum's sign usually matches the term with larger absolute value; the product or quotient's sign follows whether the number of negative factors is odd or even. Use rounding or simple estimation to test if the magnitude is reasonable. These habits build accuracy and confidence in working with integers in calculations, problem solving and higher mathematics.