Introduction to Integers
Until now in mathematics, you have worked mostly with whole numbers — 0, 1, 2, 3, 4 and so on. But what happens when the temperature drops below zero? What happens when you spend more money than you have? What happens when a submarine goes below sea level? In all these situations, we need numbers that are less than zero — we need negative numbers. When we combine negative numbers, zero and positive numbers together, we get a wonderful number system called integers. Integers are one of the most important number systems in mathematics, and learning about them opens up a whole new world of problem-solving. In Class 6, you will learn what integers are, how to represent them on a number line, how to compare them, and how they appear in everyday life. Think of integers as a number line that stretches infinitely in both directions — to the right for positive numbers and to the left for negative numbers, with zero sitting right in the middle as the dividing point. Understanding integers will help you in science (temperatures below zero), geography (altitudes below sea level), banking (debts), and many other areas. Let us explore this fascinating number system.
What is Introduction to Integers?
Integers are the set of whole numbers and their negatives. The set of integers is written as:
Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}
The three dots (...) mean the numbers continue forever in both directions.
Integers are of three types:
1. Positive Integers: Numbers greater than zero: 1, 2, 3, 4, 5, ... These are the same as natural numbers. They are written with or without a plus sign: +3 or simply 3.
2. Zero (0): Zero is an integer that is neither positive nor negative. It is the boundary between positive and negative numbers. Zero is the starting point on the number line.
3. Negative Integers: Numbers less than zero: -1, -2, -3, -4, -5, ... They are always written with a minus sign in front: -3, -7, -15. Negative integers represent quantities that are "below", "less than", "opposite to" or "lost" compared to a reference point.
Number Line Representation: Integers are beautifully represented on a number line. Draw a horizontal line with arrows on both ends (showing it extends forever). Mark a point in the centre and label it 0. To the right of 0, mark points at equal intervals and label them 1, 2, 3, 4, 5, ... To the left of 0, mark points at equal intervals and label them -1, -2, -3, -4, -5, ...
Key facts about the number line:
- Numbers increase as you move to the right.
- Numbers decrease as you move to the left.
- Every positive number has a corresponding negative number on the opposite side of zero (for example, +5 and -5).
- The distance of a number from zero (ignoring the sign) is called its absolute value. For example, |-7| = 7 and |+7| = 7.
Predecessor and Successor:
The successor of an integer is the integer that comes right after it (one step to the right on the number line). Successor of n is n + 1.
The predecessor of an integer is the integer that comes right before it (one step to the left on the number line). Predecessor of n is n - 1.
Example: Successor of -3 is -2. Predecessor of -3 is -4. Successor of 0 is 1. Predecessor of 0 is -1.
Types and Properties
Let us explore the different aspects of integers and how they are used:
1. Positive and Negative Integers in Daily Life
Integers are not just abstract numbers — they represent real situations:
- Temperature: On a winter day in Shimla, the temperature might be -4°C (4 degrees below zero). In summer in Delhi, it might be +45°C. Zero degrees Celsius is the freezing point of water.
- Altitude: Mount Everest is +8,849 metres (above sea level). The Dead Sea shore is -430 metres (below sea level). Sea level itself is 0 metres.
- Money: If you have Rs. 500 in your bank, that is +500. If you owe someone Rs. 200, that is -200. Zero balance means you have neither money nor debt.
- Floors of a building: The ground floor is 0. The 3rd floor is +3. The 2nd basement is -2.
- Time zones: India is UTC +5:30 (ahead of Greenwich). New York is UTC -5 (behind Greenwich).
2. Comparing Integers
On the number line, the number to the right is always greater:
- Any positive integer is greater than zero: 1 > 0, 100 > 0.
- Any negative integer is less than zero: -1 < 0, -100 < 0.
- Any positive integer is greater than any negative integer: 3 > -5, 1 > -1000.
- Among positive integers: the larger the number, the greater it is: 15 > 7.
- Among negative integers: the one closer to zero is greater. So -2 > -5 (because -2 is to the right of -5 on the number line). This is the trickiest part — the "bigger looking" negative number is actually smaller!
Think of temperature: -2°C is warmer (greater) than -5°C. Minus 5 is colder (lesser) than minus 2.
3. Ordering Integers
Ascending order means from smallest to largest (left to right on the number line).
Example: -8, -3, -1, 0, 2, 6 is in ascending order.
Descending order means from largest to smallest (right to left on the number line).
Example: 6, 2, 0, -1, -3, -8 is in descending order.
4. Opposite of an Integer
Every integer has an opposite. The opposite of +5 is -5, and the opposite of -5 is +5. The opposite of 0 is 0 itself. Opposites are the same distance from zero but on different sides. On the number line, they are mirror images of each other across zero. When you add an integer and its opposite, you always get zero: 5 + (-5) = 0.
5. Absolute Value
The absolute value of an integer is its distance from zero on the number line, regardless of direction. It is always positive (or zero).
|+8| = 8, |-8| = 8, |0| = 0.
Think of it as removing the sign and keeping just the number. Absolute value tells you "how far from zero" without caring about which direction.
Solved Examples
Example 1: Example 1: Representing real-life situations with integers
Problem: Represent each situation with an integer: (a) A temperature of 5 degrees below zero. (b) A profit of Rs. 1,200. (c) 300 metres below sea level. (d) Going up 8 floors from the ground floor. (e) Losing 3 wickets in cricket.
Solution:
(a) 5 degrees below zero: -5°C (negative because it is below the reference point of 0°).
(b) Profit of Rs. 1,200: +1,200 (positive because profit is a gain).
(c) 300 metres below sea level: -300 m (negative because it is below the reference point of sea level = 0).
(d) Going up 8 floors: +8 (positive because going up is above the reference ground floor = 0).
(e) Losing 3 wickets: -3 (negative because losing wickets is a decrease).
Example 2: Example 2: Plotting integers on a number line
Problem: Plot the following integers on a number line: -4, -1, 0, 2, 5.
Solution:
Step 1: Draw a horizontal line with arrows on both ends.
Step 2: Mark a point near the centre and label it 0.
Step 3: Mark equally spaced points to the right of 0 and label them 1, 2, 3, 4, 5.
Step 4: Mark equally spaced points to the left of 0 and label them -1, -2, -3, -4, -5.
Step 5: Now identify and highlight the required numbers: -4, -1, 0, 2, 5.
The order from left to right on the number line is: -4, -1, 0, 2, 5 (this is also the ascending order).
Example 3: Example 3: Comparing integers
Problem: Insert < or > between each pair: (a) -3 ___ 2, (b) -7 ___ -4, (c) 0 ___ -5, (d) -1 ___ -1, (e) 4 ___ -100.
Solution:
(a) -3 < 2: Any negative number is less than any positive number.
(b) -7 < -4: Among negative numbers, the one closer to zero is greater. -4 is closer to zero than -7, so -4 is greater. Think of temperature: -4°C is warmer than -7°C.
(c) 0 > -5: Zero is greater than any negative number.
(d) -1 = -1: They are equal (trick question — we use = not < or >).
(e) 4 > -100: Any positive number is greater than any negative number, no matter how "big" the negative number looks. +4 is to the right of -100 on the number line.
Example 4: Example 4: Arranging integers in ascending and descending order
Problem: Arrange the following in (a) ascending order and (b) descending order: 3, -7, 0, -2, 8, -5, 1.
Solution:
First, plot all numbers mentally on the number line. From left to right: -7, -5, -2, 0, 1, 3, 8.
(a) Ascending order (smallest to largest):
-7, -5, -2, 0, 1, 3, 8
(b) Descending order (largest to smallest):
8, 3, 1, 0, -2, -5, -7
Tip: For ascending order, think of walking along the number line from left to right. For descending order, walk from right to left.
Example 5: Example 5: Finding successors and predecessors
Problem: Find the successor and predecessor of: (a) 7, (b) -3, (c) 0, (d) -1, (e) -100.
Solution:
Successor = number + 1 (one step right). Predecessor = number - 1 (one step left).
(a) 7: Successor = 7 + 1 = 8. Predecessor = 7 - 1 = 6.
(b) -3: Successor = -3 + 1 = -2. Predecessor = -3 - 1 = -4.
(c) 0: Successor = 0 + 1 = 1. Predecessor = 0 - 1 = -1.
(d) -1: Successor = -1 + 1 = 0. Predecessor = -1 - 1 = -2.
(e) -100: Successor = -100 + 1 = -99. Predecessor = -100 - 1 = -101.
Example 6: Example 6: Absolute value of integers
Problem: Find the absolute value of: (a) -12, (b) +8, (c) 0, (d) -256, (e) 1.
Solution:
The absolute value is the distance from zero — always positive or zero.
(a) |-12| = 12 (12 steps from zero on the number line)
(b) |+8| = 8 (8 steps from zero)
(c) |0| = 0 (zero steps from zero — zero is at zero!)
(d) |-256| = 256 (256 steps from zero)
(e) |1| = 1 (1 step from zero)
Key insight: |-12| = |+12| = 12. A number and its opposite always have the same absolute value.
Example 7: Example 7: Temperature word problem
Problem: The temperature in Srinagar at night was -6°C. During the day it rose by 13 degrees. What was the daytime temperature?
Solution:
Night temperature = -6°C.
Temperature rise = 13 degrees.
Daytime temperature = -6 + 13 = +7°C.
On the number line: start at -6, move 13 steps to the right. You pass through -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7. You reach +7.
Answer: The daytime temperature was 7°C.
Example 8: Example 8: Altitude word problem
Problem: A diver is at a depth of 40 metres below sea level. He swims up 15 metres. Then he dives down another 10 metres. What is his final position?
Solution:
Starting position = -40 m (40 metres below sea level).
After swimming up 15 m: -40 + 15 = -25 m.
After diving down 10 m: -25 - 10 = -35 m.
Answer: The diver is at -35 m, which means 35 metres below sea level.
Example 9: Example 9: Integers between two numbers
Problem: List all integers between -4 and 3.
Solution:
"Between" means we do not include the endpoints themselves.
The integers between -4 and 3 are: -3, -2, -1, 0, 1, 2.
There are 6 integers between -4 and 3.
Quick formula: The number of integers between two integers a and b (where a < b) is (b - a - 1).
Here: 3 - (-4) - 1 = 3 + 4 - 1 = 6. ✓
Example 10: Example 10: Building floors problem
Problem: A building has 5 floors above the ground and 3 basement levels. The ground floor is Floor 0. (a) Represent all floors using integers. (b) How many floors must a person travel from Basement 3 to Floor 4?
Solution:
(a) Floor 5 = +5, Floor 4 = +4, Floor 3 = +3, Floor 2 = +2, Floor 1 = +1, Ground Floor = 0, Basement 1 = -1, Basement 2 = -2, Basement 3 = -3.
(b) From Basement 3 (-3) to Floor 4 (+4):
Number of floors = 4 - (-3) = 4 + 3 = 7 floors.
Or count: from -3 to -2 (1 floor), -2 to -1 (1), -1 to 0 (1), 0 to 1 (1), 1 to 2 (1), 2 to 3 (1), 3 to 4 (1) = 7 floors total.
Answer: The person must travel 7 floors.
Real-World Applications
Integers are used extensively in everyday life and various fields:
Weather and Temperature: Temperatures below freezing are represented by negative integers. Weather forecasts use integers to communicate temperatures. In India, places like Dras (in Ladakh) experience temperatures as low as -45°C in winter. Understanding negative temperatures is essential for weather scientists and ordinary people planning their clothing and travel.
Finance and Banking: In banking, credits (deposits) are positive and debits (withdrawals or expenses) are negative. A bank balance of -Rs. 500 means you owe the bank Rs. 500 (overdraft). Profit is positive, loss is negative. Understanding integers helps in managing personal finance and business accounts.
Geography and Altitude: Elevations above sea level are positive, and depths below sea level are negative. The Mariana Trench in the Pacific Ocean is about -11,034 metres (the deepest point on Earth). The Kuttanad region in Kerala is one of the few places in India below sea level (-2.2 m). Maps use integers to show height and depth.
Sports: In golf, scores are relative to par — under par is negative (good) and over par is positive. In cricket, run rate differences can be positive or negative. In football, goal difference is calculated as goals scored minus goals conceded, which can be negative.
Science: Electrons have negative charge (-1) and protons have positive charge (+1). In chemistry, oxidation states use positive and negative integers. Temperatures in science are measured on the Kelvin, Celsius and Fahrenheit scales, all of which can involve negative numbers.
Lifts and Elevators: Modern buildings number their floors with positive integers above ground and negative integers (or B1, B2, B3) for basement levels. This is a direct application of the integer number line with ground floor as zero.
Key Points to Remember
- Integers are the set of positive numbers, negative numbers and zero: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}.
- Positive integers are to the right of zero on the number line. Negative integers are to the left of zero.
- Zero is neither positive nor negative — it is the dividing point.
- On the number line, numbers increase to the right and decrease to the left.
- Any positive integer is greater than any negative integer: 1 > -1000.
- Among negative integers, the one closer to zero is greater: -2 > -5 (think temperature: -2°C is warmer than -5°C).
- The absolute value of an integer is its distance from zero: |-7| = 7, |+7| = 7.
- Every integer has an opposite: the opposite of +5 is -5 and vice versa. An integer plus its opposite equals zero.
- Successor of n is n+1. Predecessor of n is n-1.
- Integers are used in temperature, altitude, banking, sports scoring and many other real-life situations.
Practice Problems
- Represent each situation with an integer: (a) 15°C below zero, (b) a gain of Rs. 800, (c) 50 metres above sea level, (d) a loss of 5 runs, (e) going down 4 floors.
- Plot these integers on a number line and arrange in ascending order: 4, -6, 0, -2, 7, -4, 1.
- Fill in with < or >: (a) -9 ___ -3, (b) 0 ___ -7, (c) -15 ___ 2, (d) -1 ___ -100.
- Find the successor and predecessor of: (a) -5, (b) 0, (c) 99, (d) -1.
- Find the absolute value of: -18, +25, -1, 0, -999.
- How many integers are there between -5 and 5? List them all.
- The temperature at midnight was -8°C. By noon it rose 20 degrees. What was the noon temperature?
- Write four integers that are greater than -3 but less than 4.
Frequently Asked Questions
Q1. What are integers in simple words?
Integers are all whole numbers, including positive numbers (1, 2, 3, ...), negative numbers (-1, -2, -3, ...) and zero. They do not include fractions or decimals. Think of the number line stretching infinitely in both directions — every point at a whole number position is an integer.
Q2. Is zero a positive or negative integer?
Zero is neither positive nor negative. It is a special integer that sits right in the middle of the number line, separating positive integers from negative integers. Zero acts as the reference point — positive numbers are above or to the right of zero, and negative numbers are below or to the left of zero.
Q3. Why is -2 greater than -5?
On the number line, -2 is to the right of -5, which means -2 is greater. Think of it as temperature: -2°C is warmer than -5°C. Or think of debt: owing Rs. 2 is better than owing Rs. 5. The negative number closer to zero is always greater because it represents a smaller loss or deficit.
Q4. Are fractions and decimals integers?
No. Integers are only whole numbers (both positive and negative) and zero. Numbers like 1/2, 3.5, -2.7 and 0.1 are NOT integers because they fall between whole numbers on the number line. However, numbers like -4.0 or 7.0 are integers because they are equal to whole numbers (-4 and 7).
Q5. What is the smallest integer?
There is no smallest integer! Integers extend infinitely in the negative direction: -1, -2, -3, -4, ... and it never stops. No matter how small a negative integer you think of, there is always a smaller one. Similarly, there is no largest integer because positive integers also extend infinitely.
Q6. What is absolute value?
Absolute value is the distance of a number from zero on the number line, without considering which direction. It is always positive or zero. The absolute value of -7 is 7 (written as |-7| = 7) because -7 is 7 steps away from zero. The absolute value of +7 is also 7. Think of absolute value as answering the question: How far from zero? without caring about left or right.
Q7. Where do we use negative numbers in real life?
Negative numbers are used everywhere: temperatures below zero (winter in the mountains), depths below sea level (ocean depth, underground), bank overdrafts (when you owe money), scoring below par in golf, losses in business, and floors below ground level in buildings. Even in video games, losing health points is represented by negative numbers.
Q8. What is the opposite of an integer?
The opposite of an integer is the number that is the same distance from zero but on the other side. The opposite of +5 is -5. The opposite of -12 is +12. The opposite of 0 is 0 (it is its own opposite). When you add any integer and its opposite, the result is always zero: 8 + (-8) = 0.
Q9. How are integers different from whole numbers?
Whole numbers are 0, 1, 2, 3, 4, ... They include zero and all positive numbers but do NOT include negative numbers. Integers include all whole numbers PLUS all negative numbers: ..., -3, -2, -1, 0, 1, 2, 3, ... So every whole number is an integer, but not every integer is a whole number (because negative integers are not whole numbers).
Q10. Why do we need negative numbers?
Without negative numbers, we cannot represent many real situations. We cannot express temperatures below zero, depths below sea level, debts, or losses. Before negative numbers were accepted (which took centuries in the history of mathematics), these situations had to be described in words. Negative numbers gave us a clean, mathematical way to represent quantities below a reference point.
