Absolute Value of Integers
Think about a number line. The number 5 is 5 steps to the right of zero. The number −5 is 5 steps to the left of zero. Both are the same distance from zero — just in opposite directions.
The absolute value of a number tells us how far that number is from zero on the number line, without caring about the direction (left or right). It is always a positive number or zero.
Absolute value is also called the modulus of a number. It is written using two vertical bars: |−5| = 5 and |5| = 5. This topic helps us compare integers and understand distance on the number line.
What is Absolute Value of Integers - Grade 6 Maths (Integers)?
Definition: The absolute value of an integer is its distance from zero on the number line.
Symbol: The absolute value of a number n is written as |n|.
Key facts:
- The absolute value is always zero or positive. It is never negative.
- |5| = 5 (5 is 5 units from zero)
- |−5| = 5 (−5 is also 5 units from zero)
- |0| = 0 (zero is 0 units from itself)
- Two numbers that are the same distance from zero but on opposite sides are called opposites. For example, 7 and −7 are opposites.
In simple words:
- If the number is positive, its absolute value is the number itself.
- If the number is negative, remove the minus sign to get the absolute value.
- If the number is zero, the absolute value is zero.
Absolute Value of Integers Formula
Formula:
|n| = n, if n ≥ 0
|n| = −n, if n < 0
Where:
- |n| = absolute value of n
- If n is positive or zero, the absolute value is just n.
- If n is negative, we multiply by −1 to make it positive. For example, |−8| = −(−8) = 8.
Distance between two integers:
Distance between a and b = |a − b|
This gives the number of steps between two points on the number line.
Derivation and Proof
Understanding absolute value on the number line:
- Draw a number line with 0 in the middle.
- Mark +3 on the right side (3 steps from 0).
- Mark −3 on the left side (3 steps from 0).
- Count the steps from 0 to +3: it is 3 steps. So |3| = 3.
- Count the steps from 0 to −3: it is also 3 steps. So |−3| = 3.
- Both numbers are the same distance from zero, just in different directions.
Finding distance between two integers:
- To find the distance between −4 and 3, calculate |−4 − 3| = |−7| = 7.
- This means −4 and 3 are 7 steps apart on the number line.
- You can check by counting: from −4 to 0 is 4 steps, and from 0 to 3 is 3 steps. Total = 4 + 3 = 7 steps.
Types and Properties
Types of problems involving absolute value:
- Type 1: Finding the absolute value — Given a number, find its absolute value. Example: |−12| = 12.
- Type 2: Comparing using absolute value — Which is farther from zero, −8 or 5? Compare |−8| = 8 and |5| = 5. Since 8 > 5, −8 is farther from zero.
- Type 3: Finding distance between two integers — Use |a − b|. Distance between −3 and 4 = |−3 − 4| = |−7| = 7.
- Type 4: Solving absolute value statements — If |x| = 6, then x = 6 or x = −6 (two possible values).
- Type 5: Ordering by absolute value — Arrange numbers by their distance from zero. Example: −2, 5, −7, 1 → By absolute value: 1, −2, 5, −7 (distances: 1, 2, 5, 7).
Solved Examples
Example 1: Example 1: Finding Absolute Value of Positive Numbers
Problem: Find |9| and |25|.
Solution:
- 9 is a positive number. It is 9 units from zero.
- |9| = 9
- 25 is a positive number. It is 25 units from zero.
- |25| = 25
Answer: |9| = 9 and |25| = 25.
Example 2: Example 2: Finding Absolute Value of Negative Numbers
Problem: Find |−7| and |−15|.
Solution:
- −7 is 7 units from zero (on the left side of the number line).
- |−7| = 7
- −15 is 15 units from zero.
- |−15| = 15
Answer: |−7| = 7 and |−15| = 15.
Example 3: Example 3: Absolute Value of Zero
Problem: Find |0|.
Solution:
- Zero is at the centre of the number line.
- Its distance from itself is 0.
Answer: |0| = 0.
Example 4: Example 4: Comparing Distances from Zero
Problem: Which is farther from zero: −11 or 8?
Solution:
- |−11| = 11
- |8| = 8
- 11 > 8
Answer: −11 is farther from zero than 8.
Example 5: Example 5: Finding Distance Between Two Integers
Problem: Find the distance between −6 and 4 on the number line.
Solution:
Using the formula:
- Distance = |−6 − 4|
- = |−10|
- = 10
Check: From −6 to 0 is 6 steps. From 0 to 4 is 4 steps. Total = 6 + 4 = 10.
Answer: The distance is 10 units.
Example 6: Example 6: Distance Between Two Negative Integers
Problem: Find the distance between −3 and −9.
Solution:
- Distance = |−3 − (−9)|
- = |−3 + 9|
- = |6|
- = 6
Answer: The distance between −3 and −9 is 6 units.
Example 7: Example 7: Solving |x| = 4
Problem: Find all values of x if |x| = 4.
Solution:
- |x| = 4 means x is 4 units from zero.
- There are two points that are 4 units from zero: 4 (on the right) and −4 (on the left).
Answer: x = 4 or x = −4.
Example 8: Example 8: Ordering by Absolute Value
Problem: Arrange these numbers in order of their distance from zero (smallest to largest): 3, −8, −1, 6, −4.
Solution:
- |3| = 3
- |−8| = 8
- |−1| = 1
- |6| = 6
- |−4| = 4
Order by absolute value: −1 (1), 3 (3), −4 (4), 6 (6), −8 (8).
Answer: −1, 3, −4, 6, −8.
Example 9: Example 9: True or False
Problem: True or false: |−10| > |7|?
Solution:
- |−10| = 10
- |7| = 7
- 10 > 7 is true.
Answer: True. The absolute value of −10 is greater than the absolute value of 7.
Example 10: Example 10: Real-Life Application
Problem: The temperature in the morning was −5°C and in the afternoon it was 8°C. What was the change in temperature?
Solution:
- Change = |8 − (−5)|
- = |8 + 5|
- = |13|
- = 13
Answer: The temperature changed by 13°C.
Real-World Applications
Absolute value in real life:
- Temperature — When we say it is 10 degrees below zero (−10°C), the absolute value tells us how cold it is compared to zero (10 degrees).
- Distance — On a number line, absolute value gives the distance between two points. In real life, distance is always positive.
- Elevation — Sea level is 0. A place 200 m above sea level (+200) and a place 200 m below sea level (−200) are both 200 m from sea level.
- Money — If you owe Rs 500 (−500) or have Rs 500 (+500), the amount involved is Rs 500 in both cases.
- Sports — In golf, scores are shown as above or below par. A score of −3 (3 under par) and +3 (3 over par) both differ from par by 3 strokes.
Key Points to Remember
- The absolute value of a number is its distance from zero on the number line.
- Absolute value is always zero or positive — never negative.
- |n| = n if n is positive or zero. |n| = −n if n is negative.
- |0| = 0.
- Two opposite numbers like 5 and −5 have the same absolute value.
- If |x| = k (where k > 0), then x = k or x = −k.
- Distance between integers a and b = |a − b|.
- Absolute value is also called modulus.
- Absolute value helps in comparing which number is farther from zero.
- In real life, absolute value is used for temperature differences, distances, and elevations.
Practice Problems
- Find the absolute value of: −14, 22, 0, −1, 100.
- Which is farther from zero: −20 or 18? Use absolute value.
- Find the distance between 7 and −5 on the number line.
- If |x| = 9, what are the possible values of x?
- Arrange in order of absolute value (smallest first): −6, 2, −3, 10, −1.
- The temperature was −8°C in the morning and 5°C in the afternoon. Find the change.
- True or false: |−4| = |4|. Give a reason.
- Find two integers whose absolute value is 12.
Frequently Asked Questions
Q1. What is absolute value?
Absolute value is the distance of a number from zero on the number line. It is always positive or zero. For example, |−7| = 7 and |7| = 7.
Q2. Can the absolute value of a number be negative?
No. Absolute value measures distance, and distance is always zero or positive. Even the absolute value of a negative number is positive. For example, |−100| = 100.
Q3. What is the absolute value of zero?
|0| = 0. Zero is the only number whose absolute value is 0, because it is at zero distance from itself.
Q4. What is the difference between absolute value and the number itself?
For positive numbers, there is no difference: |5| = 5. For negative numbers, the absolute value removes the minus sign: |−5| = 5. The absolute value ignores the direction and only tells the size (distance from zero).
Q5. How do you write absolute value?
Use two vertical bars around the number. For example, the absolute value of −3 is written as |−3| = 3. The vertical bars are called the modulus symbol.
Q6. If |a| = |b|, does that mean a = b?
Not always. It means a and b are the same distance from zero. They could be equal (both 5), or they could be opposites (5 and −5). So |a| = |b| means a = b or a = −b.
Q7. What is the absolute value of −1?
|−1| = 1. The number −1 is 1 unit away from zero on the number line.
Q8. How is absolute value used in real life?
Absolute value is used to find differences regardless of direction. Temperature change, distance between locations, elevation above or below sea level, and comparing scores all use absolute value.
