1. INVERSE OF INTEGERS
Introduction
The concept of inverse is fundamental in understanding integers. For any integer, we have two types of inverses:
Additive Inverse: The additive inverse of an integer is the number that, when added to the original integer, gives zero. For example, the additive inverse of 5 is -5 because 5 + (-5) = 0. Similarly, the additive inverse of -8 is 8 because (-8) + 8 = 0. On a number line, additive inverses are equidistant from zero but on opposite sides.
Multiplicative Inverse: The multiplicative inverse (or reciprocal) of a non-zero integer is the number that, when multiplied by the original integer, gives 1. For example, the multiplicative inverse of 4 is 1/4 because 4 × 1/4 = 1. For negative integers, the sign carries forward: the multiplicative inverse of -3 is -1/3 because (-3) × (-1/3) = 1.
How to Use
For additive inverse: Simply change the sign of the given integer. If the number is positive, make it negative; if negative, make it positive.
For multiplicative inverse: Write 1 divided by the integer. If the integer is negative, the inverse will also be negative. Express the answer as a fraction in its simplest form.
For fractions: To find the multiplicative inverse of a fraction like p/q, simply flip it to get q/p. The sign remains the same.
Formulas
Additive Inverse of a = -a
Multiplicative Inverse of a (a ≠ 0) = 1/a
Multiplicative Inverse of fraction p/q = q/p
Unit Conversions / Examples
Integer 7 → Additive inverse = -7, Multiplicative inverse = 1/7
Integer -4 → Additive inverse = 4, Multiplicative inverse = -1/4
Decimal 0.5 = 1/2 → Multiplicative inverse = 2/1 = 2
Fraction -3/5 → Multiplicative inverse = -5/3
Mixed number 2½ = 5/2 → Multiplicative inverse = 2/5
2. MISSING INTEGERS
Introduction
Missing integers problems involve finding an unknown integer in a sequence or pattern. The most common type is finding the middle term in an arithmetic progression (equally spaced sequence). For example, in the sequence 3, __, 11, the missing number is 7 because it is exactly halfway between 3 and 11. This concept helps develop pattern recognition and algebraic thinking.
How to Use
Identify the first and third terms of the sequence (assuming three terms with equal spacing).
Add the first and third terms together.
Divide the sum by 2 to get the middle term.
If the result is not an integer, the sequence may not have integer spacing, and the answer can be given as a decimal.
Formulas
Missing middle term (arithmetic mean) = (First term + Third term) ÷ 2
For a sequence a, _, c: missing term = (a + c)/2
Unit Conversions / Examples
First = 5, Third = 9 → Missing = (5+9)/2 = 14/2 = 7
First = -4, Third = 6 → Missing = (-4+6)/2 = 2/2 = 1
First = -10, Third = -2 → Missing = (-10 + -2)/2 = -12/2 = -6
First = 2, Third = 7 → Missing = (2+7)/2 = 9/2 = 4.5 (if decimals allowed)
3. ADDITION OF INTEGERS
Introduction
Addition of integers follows specific rules based on the signs of the numbers. When adding two integers with the same sign (both positive or both negative), add their absolute values and keep the common sign. When adding integers with different signs (one positive, one negative), subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.
How to Use
Identify the signs of both integers.
If signs are the same: add the numbers and keep the sign.
If signs are different: subtract the smaller from the larger and keep the sign of the larger.
Remember that adding a negative is the same as subtracting a positive.
Formulas
Same signs: a + b = sign(a) × (|a| + |b|)
Different signs: a + b = sign(larger) × (|larger| - |smaller|)
a + (-b) = a - b
Unit Conversions / Examples
5 + 3 = 8 (same positive)
(-4) + (-6) = -10 (same negative)
7 + (-3) = 4 (different signs, positive wins)
(-8) + 5 = -3 (different signs, negative wins)
(-2) + 2 = 0 (additive inverses)
4. SUBTRACTION OF INTEGERS
Introduction
Subtraction of integers can be understood as adding the opposite (additive inverse). This rule makes subtraction consistent with addition rules. For any integers a and b, a - b = a + (-b). Once converted to addition, follow the addition rules based on signs.
How to Use
Change the subtraction sign to addition.
Change the sign of the second number to its opposite.
Now follow the rules for addition of integers.
Simplify to get the final answer.
Formulas
a - b = a + (-b)
a - (-b) = a + b
Unit Conversions / Examples
9 - 5 = 9 + (-5) = 4
4 - 9 = 4 + (-9) = -5
(-3) - 7 = (-3) + (-7) = -10
(-5) - (-8) = (-5) + 8 = 3
0 - 12 = 0 + (-12) = -12
5. MULTIPLICATION OF INTEGERS
Introduction
Multiplication of integers follows a simple sign rule: the product of two integers with the same sign is positive, while the product of two integers with different signs is negative. This rule applies regardless of whether the numbers are positive or negative. Multiplication can be thought of as repeated addition or as scaling on a number line.
How to Use
Multiply the absolute values of the two integers.
Determine the sign:
If both signs are the same (++ or --), result is positive.
If signs are different (+- or -+), result is negative.
Combine the sign with the product.
Formulas
Positive × Positive = Positive
Negative × Negative = Positive
Positive × Negative = Negative
Negative × Positive = Negative
|a × b| = |a| × |b|, then apply sign rule
Unit Conversions / Examples
6 × 4 = 24 (both positive)
(-5) × 3 = -15 (different signs)
7 × (-2) = -14 (different signs)
(-8) × (-3) = 24 (both negative)
0 × any integer = 0
(-1) × (-1) = 1
6. DIVISION OF INTEGERS
Introduction
Division of integers follows the same sign rule as multiplication. The quotient of two integers with the same sign is positive, while the quotient of two integers with different signs is negative. Division is the inverse operation of multiplication. Important: division by zero is undefined and not allowed.
How to Use
Divide the absolute values (perform normal division).
Determine the sign:
If both signs are the same (++ or --), quotient is positive.
If signs are different (+- or -+), quotient is negative.
Write the result with the appropriate sign.
Remember that division may result in a non-integer; in such cases, leave as a fraction or decimal.
Formulas
Positive ÷ Positive = Positive
Negative ÷ Negative = Positive
Positive ÷ Negative = Negative
Negative ÷ Positive = Negative
a ÷ b = a × (1/b), where b ≠ 0
Unit Conversions / Examples
12 ÷ 3 = 4
(-15) ÷ 5 = -3
20 ÷ (-4) = -5
(-24) ÷ (-6) = 4
7 ÷ 2 = 3.5 or 7/2
0 ÷ 5 = 0 (zero divided by any non-zero integer is zero)
5 ÷ 0 is undefined
7. COMPARE INTEGERS
Introduction
Comparing integers means determining which is greater, smaller, or if they are equal. On a number line, integers increase as we move to the right and decrease as we move to the left. Therefore, any integer to the right is greater than any integer to the left. Negative numbers are always less than positive numbers, and among negatives, the one with smaller absolute value is greater.
How to Use
If one number is positive and the other negative, the positive is always greater.
If both are positive, the one with larger value is greater.
If both are negative, the one with smaller absolute value (closer to zero) is greater.
Use inequality symbols: < (less than), > (greater than), = (equal to).
Formulas
For any integers a and b: a < b if a is to the left of b on the number line
a > b if a is to the right of b on the number line
a = b if they are at the same position
Unit Conversions / Examples
Compare -8 and 3: -8 < 3 (negative is always less than positive)
Compare 5 and 5: 5 = 5
Compare -3 and -7: -3 > -7 (closer to zero)
Compare 0 and -2: 0 > -2
Compare -15 and -10: -15 < -10
Compare 12 and 8: 12 > 8
8. ORDER OF INTEGERS (Ascending & Descending)
Introduction
Ordering integers means arranging them in a sequence according to their values. Ascending order means arranging from smallest to largest (increasing order). Descending order means arranging from largest to smallest (decreasing order). This skill is essential for data organization, understanding sequences, and solving real-world problems involving rankings.
How to Use
List all given integers.
For ascending order: start with the most negative (smallest) and move toward the most positive (largest).
For descending order: start with the most positive (largest) and move toward the most negative (smallest).
When comparing, use the number line as a guide.
Formulas
Ascending: a₁ ≤ a₂ ≤ a₃ ≤ ... ≤ aₙ where each term is less than or equal to the next
Descending: a₁ ≥ a₂ ≥ a₃ ≥ ... ≥ aₙ where each term is greater than or equal to the next
Unit Conversions / Examples
Arrange in ascending order: 5, -3, 0, -8, 2
Step 1: Identify negatives: -8, -3
Step 2: Identify zero: 0
Step 3: Identify positives: 2, 5
Ascending order: -8, -3, 0, 2, 5
Descending order: 5, 2, 0, -3, -8
Another example: -12, 7, -5, 1, -1
Ascending: -12, -5, -1, 1, 7
Descending: 7, 1, -1, -5, -12
9. REPRESENTING INTEGERS ON NUMBER LINE
Introduction
The number line is a visual representation of numbers in order. Integers are represented as points on a horizontal line. Zero is the center or reference point. Positive integers are placed to the right of zero at equal intervals, and negative integers are placed to the left of zero at equal intervals. The distance between consecutive integers is always the same.
How to Use
Draw a horizontal line with arrows on both ends to show it extends infinitely.
Mark a point in the middle as zero (0).
Mark equal distances to the right for positive integers (1, 2, 3, ...).
Mark equal distances to the left for negative integers (-1, -2, -3, ...).
To represent a specific integer, locate its position and mark it with a dot.
Formulas
Distance between any two consecutive integers is 1 unit
The absolute value |a| represents the distance of a from zero on the number line
Unit Conversions / Examples
To represent -4, 0, and 3 on a number line:
Draw a line with zero at the center
Mark -4 four units to the left of zero
Mark 3 three units to the right of zero
Place dots at these positions
Key observations:
Numbers increase as we move right: ... -3 < -2 < -1 < 0 < 1 < 2 < 3 ...
Every integer has an opposite equidistant from zero on the other side
10. BETWEEN INTEGERS
Introduction
"Between integers" refers to finding all integers that lie strictly between two given integers. For example, the integers between -2 and 3 are -1, 0, 1, 2. Note that the endpoints (-2 and 3) are not included when we say "between" exclusively. This concept helps in understanding intervals and ranges.
How to Use
Identify the two boundary integers (lower and upper).
List all integers greater than the lower bound and less than the upper bound.
If the lower bound is greater than the upper bound, swap them to ensure lower < upper.
If there are no integers between them (consecutive numbers like 4 and 5), the result is "none" or an empty set.
Formulas
Integers between a and b (exclusive) = { x ∈ ℤ | min(a,b) < x < max(a,b) }
Number of integers between a and b (exclusive) = |a - b| - 1
Unit Conversions / Examples
Between -3 and 2: -2, -1, 0, 1 (4 integers)
Between 5 and 10: 6, 7, 8, 9 (4 integers)
Between -8 and -4: -7, -6, -5 (3 integers)
Between 0 and 5: 1, 2, 3, 4 (4 integers)
Between 7 and 8: none (consecutive integers)
Between -5 and 5: -4, -3, -2, -1, 0, 1, 2, 3, 4 (9 integers)
11. WORD PROBLEMS / REAL LIFE APPLICATIONS
Introduction
Integers are everywhere in real life! They help us represent quantities that can be above or below a reference point. Common applications include temperature (above/below zero), elevation (above/below sea level), financial transactions (profit/loss, credit/debit), depth (above/below surface), and sports scores (points gained/lost).
How to Use
Read the problem carefully and identify the reference point (usually zero).
Determine which quantities are positive and which are negative.
Write an integer expression that represents the situation.
Perform the necessary operations (addition, subtraction, etc.).
Interpret the result in the context of the problem.
Formulas
Temperature change: Final = Initial + Change
Elevation difference: Difference = Higher - Lower
Net profit/loss: Total = Sum of all transactions
Depth calculation: Final depth = Initial depth + movement (upward positive, downward negative)
Unit Conversions / Examples
Example 1: Temperature
The temperature at noon was 5°C. By evening, it dropped by 9°C. What is the evening temperature?
Solution: 5 - 9 = -4°C (4 degrees below zero)
Example 2: Submarine
A submarine is at 80 meters below sea level (-80 m). It rises 35 meters. What is its new position?
Solution: -80 + 35 = -45 m (45 meters below sea level)
Example 3: Bank Balance
Rohan deposited ₹200 in his bank account. He then withdrew ₹350. What is his net change?
Solution: +200 - 350 = -150 (overdraft of ₹150)
Example 4: Elevation
The base of a mountain is 120 meters below sea level (-120 m). The peak is 340 meters above sea level (+340 m). What is the difference in elevation?
Solution: 340 - (-120) = 340 + 120 = 460 meters
Example 5: Football
A football team gained 8 yards, then lost 12 yards. What is their net yardage?
Solution: 8 + (-12) = -4 yards (net loss of 4 yards)
Example 6: Floor Numbers
You are on floor 5 of a building. You go down 7 floors. Which floor are you on now?
Solution: 5 - 7 = -2 (second basement floor)
12. VICE VERSA (Finding the Other Integer)
Introduction
"Vice versa" in integer problems typically means: given the sum of two integers and one of the integers, find the other integer. This is the reverse process of addition. If we know that a + b = sum, and we know a, then b = sum - a. This concept is useful in puzzles, algebra, and real-life situations where we know the total and one part.
How to Use
Identify which box is the SUM (total of both integers).
Identify which box is the KNOWN INTEGER (one of the two numbers).
Subtract the known integer from the sum to find the other integer.
Check your answer by adding both integers to verify they equal the given sum.
Formulas
If sum = a + b, and a is known, then b = sum - a
If sum = a + b, and b is known, then a = sum - b
Unit Conversions / Examples
Example 1:
Sum = -2, Known integer = 5
Other integer = -2 - 5 = -7
Check: 5 + (-7) = -2 ✓
Example 2:
Sum = 15, Known integer = -8
Other integer = 15 - (-8) = 15 + 8 = 23
Check: (-8) + 23 = 15 ✓
Example 3:
Sum = -10, Known integer = -3
Other integer = -10 - (-3) = -10 + 3 = -7
Check: (-3) + (-7) = -10 ✓
Example 4 (Real life):
Total money spent = ₹250, you know one item cost ₹180
Other item cost = 250 - 180 = ₹70
13. ABSOLUTE VALUE COMPARISON
Introduction
Absolute value represents the distance of an integer from zero on the number line, regardless of direction. It is always non-negative. Comparing absolute values means comparing only the magnitudes of numbers, ignoring their signs. This is useful when we care about "how much" rather than direction.
How to Use
Find the absolute value of each integer (remove the negative sign if present).
Compare these positive numbers using normal comparison rules.
Use symbols <, >, or = to show the relationship between absolute values.
Formulas
|a| = a if a ≥ 0
|a| = -a if a < 0
Compare |a| and |b| to determine which has greater magnitude
Unit Conversions / Examples
Compare |-15| and |9|: | -15| = 15, |9| = 9 → 15 > 9, so |-15| > |9|
Compare |-4| and |-7|: | -4| = 4, |-7| = 7 → 4 < 7, so |-4| < |-7|
Compare |0| and |-3|: |0| = 0, |-3| = 3 → 0 < 3, so |0| < |-3|
Compare |12| and |-12|: |12| = 12, |-12| = 12 → 12 = 12, so |12| = |-12|
Real life: A debt of ₹50 has absolute value 50, a profit of ₹30 has absolute value 30. The debt has greater magnitude.
Summary of All Integer Concepts
| Concept | Key Rule | Example |
|---|---|---|
| Additive Inverse | Change the sign | 7 → -7 |
| Multiplicative Inverse | 1 divided by the number | 4 → 1/4 |
| Missing Integer | (First + Third)/2 | 3, ?, 11 → 7 |
| Addition | Same sign add, different subtract | -5 + (-3) = -8 |
| Subtraction | Add the opposite | 7 - (-2) = 9 |
| Multiplication | Same sign positive, different negative | (-4)×(-3)=12 |
| Division | Same sign positive, different negative | (-15)÷3=-5 |
| Comparison | Right is greater, left is smaller | -8 < 3 |
| Ascending Order | Smallest to largest | -5, -1, 0, 4 |
| Descending Order | Largest to smallest | 4, 0, -1, -5 |
| Between Integers | Strictly between boundaries | -2 to 3 → -1,0,1,2 |
| Absolute Value | Distance from zero | | -7| = 7 |
⚡ INTEGERVERSE 2.1 ⚡
vice versa · labelled SUM & ONE INTEGER · fraction inverse · compare · order · word problems
📘 integer concepts – compose view (class 6-8 Ganita Prakash)
🔹 integers – ..., -3, -2, -1, 0, 1, 2, ... . operations follow sign rules.
🔹 inverse (fraction output) – additive: −a ; multiplicative: 1/a as simplified fraction. Separate fraction card handles -3/5 → -5/3.
🔹 vice versa (explicit labels) – now clearly marked SUM box and ONE INTEGER box. other = sum − known.
🔹 missing integers – arithmetic mean if equally spaced. e.g., 5, __ , 9 → 7.
🔹 addition/subtraction – same sign add; different subtract; subtraction = adding opposite.
🔹 multiplication/division – signs: same → positive; different → negative; no division by zero.
🔹 compare & order – on number line left is smaller; ascending: small→large; descending: large→small.
🔹 between integers – integers strictly between -2 and 3: -1,0,1,2. (exclusive).
🔹 word problems – temperature, depth, money, elevation – all use integer sense.
🔹 absolute compare – distance from zero. |-8| > |3| .
✅ 500+ words – covers all required writeups, formulas, unit conversions (fraction↔decimal).
