Positive & Negative Numbers
Positive and negative numbers are used to represent quantities
lying on opposite sides of a reference value such as zero.
They are important in questions involving number lines,
temperature, profit and loss, elevation, direction, increase
and decrease, and many other competitive-exam situations.
1. Positive Numbers
A real number greater than zero is called a
positive number.
Examples:
A positive number may be written with a plus (+) sign, although the
plus sign is normally omitted.
+8 = 8
If x is positive, then x > 0.
2. Negative Numbers
A real number less than zero is called a
negative number.
Examples:
A negative number is written with a minus (−) sign before it.
If x is negative, then x < 0.
3. Is Zero Positive or Negative?
Zero is neither positive nor negative.
0 is neither > 0 nor < 0
Common Exam Trap:
Never classify 0 as a positive or a negative number.
4. Positive and Negative Numbers are Not Limited to Integers
The terms positive and negative can be used for many kinds of
real numbers, not only integers.
| Number |
Classification by Sign |
| 8 |
Positive |
| −8 |
Negative |
| 3/7 |
Positive |
| −3/7 |
Negative |
| √5 |
Positive |
| −√5 |
Negative |
| 0 |
Neither positive nor negative |
5. Positive and Negative Numbers on the Number Line
On a standard horizontal number line:
- Zero is the reference point.
- Positive numbers lie to the right of zero.
- Negative numbers lie to the left of zero.
- Values increase as we move to the right.
- Values decrease as we move to the left.
... −4, −3, −2, −1, 0, 1, 2, 3, 4 ...
Number-Line Rule:
A number lying farther to the right is greater than a number lying
to its left.
6. Comparing Positive and Negative Numbers
Positive Number vs Negative Number
Every positive number is greater than every negative number.
5 > −100
Even though 100 has a larger magnitude than 5, −100 lies to the
left of 5 on the number line.
Comparing Two Positive Numbers
Among positive numbers, the number with the greater value is greater.
15 > 8
Comparing Two Negative Numbers
Among two negative numbers, the number closer to zero is greater.
−3 > −8
Important:
For negative numbers, a larger absolute value generally means a
smaller number. Thus −20 < −5.
7. Opposite Numbers
Two numbers having the same distance from zero but lying on opposite
sides of zero are called opposite numbers.
Examples:
- 5 and −5
- 12 and −12
- 3/4 and −3/4
- √2 and −√2
The opposite of a number a is −a.
The sum of a number and its opposite is zero:
a + (−a) = 0
8. A Very Important Sign Concept: −a Need Not Be Negative
The expression −a means the opposite of a.
It does not automatically mean that the final value is negative.
If a = 7:
−a = −7
But if a = −7:
−a = −(−7) = 7
Important Exam Trap:
The minus sign before a variable means “take its opposite.”
Therefore, −x is not necessarily a negative number.
9. Absolute Value
The absolute value of a real number is its distance
from zero on the number line, without considering direction.
The absolute value of x is written as:
|x|
Examples:
- |7| = 7
- |−7| = 7
- |0| = 0
- |−3/5| = 3/5
For every real number x:
|x| ≥ 0
A number and its opposite have the same absolute value:
|a| = |−a|.
10. Basic Absolute-Value Rules
For a real number x:
- If x ≥ 0, then |x| = x.
- If x < 0, then |x| = −x.
For example, if x = −9:
|−9| = −(−9) = 9
Remember: Absolute value can never be negative.
11. Distance Between Two Numbers
The distance between two real numbers a and b on the number line is:
|a − b|
Example: Find the distance between −6 and 8.
Distance = |−6 − 8|
= |−14|
= 14
12. Addition of Positive and Negative Numbers
Same Signs
When two numbers have the same sign, add their absolute values and
retain the common sign.
(+7) + (+5) = +12
(−7) + (−5) = −12
Different Signs
When the signs are different:
- Subtract the smaller absolute value from the larger absolute value.
- Give the result the sign of the number having the larger absolute value.
Example:
(−12) + 7 = −5
Since |−12| > |7|, the answer has a negative sign.
Similarly:
12 + (−7) = 5
13. Subtraction of Signed Numbers
Subtraction can be converted into addition by adding the opposite
of the number being subtracted.
a − b = a + (−b)
Examples:
8 − (−5)
= 8 + 5
= 13
−8 − 5
= −8 + (−5)
= −13
−8 − (−5)
= −8 + 5
= −3
Shortcut:
Subtracting a negative number is equivalent to adding the
corresponding positive number.
14. Multiplication Sign Rules
The sign of a product depends on the signs of its factors.
| Signs |
Result |
| (+) × (+) |
+ |
| (−) × (−) |
+ |
| (+) × (−) |
− |
| (−) × (+) |
− |
Examples:
- 6 × 4 = 24
- (−6) × (−4) = 24
- 6 × (−4) = −24
- (−6) × 4 = −24
Same signs → Positive
Different signs → Negative
15. Division Sign Rules
The same sign rule applies to division:
| Signs |
Result |
| (+) ÷ (+) |
+ |
| (−) ÷ (−) |
+ |
| (+) ÷ (−) |
− |
| (−) ÷ (+) |
− |
Examples:
- (−24) ÷ (−6) = 4
- (−24) ÷ 6 = −4
- 24 ÷ (−6) = −4
The sign rules do not change the fundamental rule that
division by zero is undefined.
16. Sign of a Product Containing Several Negative Factors
When several non-zero factors are multiplied, count the number of
negative factors.
-
An even number of negative factors gives a
positive product.
-
An odd number of negative factors gives a
negative product.
Example 1:
(−2)(−3)(−4)(−5)
There are four negative factors. Since 4 is even, the product is
positive.
Product = 120
Example 2:
(−2)(−3)(−4)(5)
There are three negative factors. Since 3 is odd, the product is
negative.
Product = −120
If even one factor is zero, the entire product is
zero, irrespective of the other signs.
17. Multiple Signs and Brackets
Questions involving positive and negative numbers often contain two
or more signs together. Simplifying the signs correctly before
performing the arithmetic operation helps avoid mistakes.
| Sign Combination |
Equivalent Sign |
| +(+a) |
+a |
| +(−a) |
−a |
| −(+a) |
−a |
| −(−a) |
+a |
Examples:
- 8 + (+3) = 11
- 8 + (−3) = 5
- 8 − (+3) = 5
- 8 − (−3) = 11
Quick Rule:
Two like signs together effectively give a positive sign, while
two unlike signs effectively give a negative sign in these
bracket/sign situations.
18. Powers of Negative Numbers
The sign of a power of a negative number depends on whether the
exponent is even or odd.
(−a)even = positive
(−a)odd = negative
Examples:
- (−2)2 = 4
- (−2)3 = −8
- (−3)4 = 81
- (−3)5 = −243
Important:
The negative number must be inside brackets when the negative sign
is intended to be part of the base.
19. Difference Between −a2 and (−a)2
This is an important source of mistakes in competitive examinations.
Consider a = 5.
−52 = −(52) = −25
But:
(−5)2 = (−5)(−5) = 25
Exam Trap:
−52 = −25
but
(−5)2 = 25
20. Sign of a Number from Its Power
Powers can sometimes provide information about the sign of a real
number.
-
If x2 > 0, then x may be positive or negative,
but x ≠ 0.
-
If x2 = 0, then x = 0.
-
For real x, x2 can never be negative.
-
If x3 > 0, then x > 0.
-
If x3 < 0, then x < 0.
An even power does not preserve the distinction between positive
and negative non-zero bases, whereas an odd power does preserve
the sign.
21. Positive and Negative Numbers in Temperature
Temperature is one of the most common real-life applications of
signed numbers.
- Temperature above 0°C is represented by a positive value.
- Temperature below 0°C is represented by a negative value.
Example:
The temperature rises from −6°C to 4°C. Find the increase.
Increase = Final temperature − Initial temperature
= 4 − (−6)
= 4 + 6
= 10°C
Moving from −6°C to +4°C is an increase of 10°C, not 2°C.
22. Elevation: Above and Below a Reference Level
Positive and negative numbers can represent positions above and
below a chosen reference level.
For example, if sea level is taken as zero:
- 250 m above sea level may be represented as +250 m.
- 80 m below sea level may be represented as −80 m.
The vertical distance between these two positions is:
|250 − (−80)| = |330| = 330 m
23. Profit, Loss, Credit and Debt
Signed numbers are useful when opposite financial situations need
to be represented relative to zero.
For example, under a chosen sign convention:
- Profit may be represented by a positive number.
- Loss may be represented by a negative number.
- Credit may be represented by a positive number.
- Debt may be represented by a negative number.
Example:
A person has a balance of ₹500 and then incurs a debit of ₹750.
Represent the net position using signed numbers.
Net position = 500 + (−750)
= −₹250
The negative sign indicates that the final position is ₹250 below
the chosen zero balance.
In word problems, first identify what the question has chosen as
positive and what it has chosen as negative. Then perform the
calculation.
24. Increase and Decrease
A positive change generally represents an increase, while a
negative change represents a decrease.
Example 1:
A quantity changes from 20 to 32.
Change = Final value − Initial value
= 32 − 20 = +12
Therefore, the quantity increased by 12.
Example 2:
A quantity changes from 20 to 13.
Change = 13 − 20 = −7
Therefore, the quantity decreased by 7.
Signed Change = Final Value − Initial Value
25. Direction and Signed Numbers
Opposite directions can be represented using opposite signs.
The choice of which direction is positive is a convention and
should be stated or inferred from the question.
For example, suppose:
- East = positive
- West = negative
Then moving 8 km east may be represented by +8 km, while moving
5 km west may be represented by −5 km.
Net displacement:
+8 + (−5) = +3 km
Therefore, the final displacement is 3 km east of the starting point.
Positive does not inherently mean east, north, upward or profit.
The sign depends on the convention defined in the problem.
26. Successive Changes
When several increases and decreases occur successively, represent
each change with its appropriate sign and add them.
Example:
A value changes successively by +12, −7, −5 and +9.
Find the net change.
Net change = 12 − 7 − 5 + 9
= +9
Therefore, the overall effect is an increase of 9 units.
27. Solved Example: Comparison
Arrange −7, 4, −2, 0 and 9 in ascending order.
On the number line, values increase from left to right.
−7 < −2 < 0 < 4 < 9
Therefore, the ascending order is:
−7, −2, 0, 4, 9.
28. Solved Example: Absolute Value
Evaluate:
|−18| − |7 − 12|
|−18| = 18
|7 − 12| = |−5| = 5
Therefore:
18 − 5 = 13
29. Solved Example: Multiple Signs
Simplify:
−15 − (−8) + (−6)
= −15 + 8 − 6
= −7 − 6
= −13
30. Solved Example: Product Sign
Find the value of:
(−2)(−3)(4)(−5)
There are three negative factors.
Since the number of negative factors is odd, the result will be
negative.
Magnitude = 2 × 3 × 4 × 5 = 120
Therefore, the answer is −120.
31. Solved Example: Temperature Change
At midnight the temperature was −4°C. By afternoon it became
9°C. By how many degrees did the temperature rise?
Rise = 9 − (−4)
= 9 + 4
= 13°C
32. Solved Example: Position Relative to Zero
A lift is initially at floor −3 relative to the ground floor
represented by 0. It moves upward by 8 floors. At what numbered
level does it arrive?
Final level = −3 + 8
= 5
Therefore, it arrives at level +5 relative to the chosen reference.
33. Common Exam Traps
Trap 1:
Zero is neither positive nor negative.
Trap 2:
Among negative numbers, the number closer to zero is greater.
Thus −3 > −10.
Trap 3:
−x is not necessarily negative. If x itself is negative,
then −x is positive.
Trap 4:
Absolute value represents distance from zero and therefore cannot
be negative.
Trap 5:
Subtracting a negative number means adding its positive opposite:
a − (−b) = a + b.
Trap 6:
(−5)2 = 25, but −52 = −25.
Brackets matter.
Trap 7:
An even number of negative factors gives a positive product,
whereas an odd number gives a negative product, provided none of
the factors is zero.
Trap 8:
In application problems, positive and negative signs depend on the
chosen reference and sign convention. Do not assume a direction
without reading the question.
34. Quick Revision
- A positive real number is greater than zero.
- A negative real number is less than zero.
- Zero is neither positive nor negative.
- Positive numbers lie to the right of zero on the standard number line.
- Negative numbers lie to the left of zero.
- Every positive number is greater than every negative number.
- Among negative numbers, the number closer to zero is greater.
- a and −a are opposite numbers.
- a + (−a) = 0.
- −a does not necessarily represent a negative value.
- |a| represents the distance of a from zero.
- |a| ≥ 0 for every real number a.
- The distance between a and b is |a − b|.
- For addition with the same signs, add magnitudes and retain the sign.
- For addition with different signs, subtract magnitudes and use the sign of the larger magnitude.
- a − b = a + (−b).
- Same signs in multiplication or division give a positive result.
- Different signs in multiplication or division give a negative result.
- An even number of negative non-zero factors gives a positive product.
- An odd number of negative factors gives a negative product.
- (−a)even is non-negative and is positive when a ≠ 0.
- (−a)odd has the opposite sign to positive a.
- −a2 and (−a)2 are generally different.
- Signed change = Final Value − Initial Value.
- Always identify the reference point and sign convention in application problems.
Practice these previous-year questions based on positive and negative
numbers, signs and signed-number operations. Try each question before
opening the answer and explanation.
Practice these exam-oriented questions on positive and negative numbers.
Try to solve each question before viewing the answer and explanation.