StudyOSStudyOS CBSE Study Material, Class 5–12

The Other Side of Zero

NCERT Class 6 · Mathematics Based on NCERT Class 6 Mathematics textbook · Free CBSE study kit

Everything you need to ace this chapter, in one place.
Notes · Mind Map · Quiz · Assertion-Reason · Case Studies · Question Bank · Past Papers

Get Full Study Kit →
Studying with friends? Send them this chapter.

Chapter Notes

Overview
This chapter introduces integers, including negative numbers, as an extension of the number line to the left of zero. It uses real-life contexts like floors in a building, bank credits and debits, and geographical heights to explain the concepts of positive and negative numbers. The one big idea a student must walk away with is that integers include all whole numbers (positive, negative, and zero) and that operations like addition and subtraction can be understood as movements on a number line or using token models.
Key Concepts (11)
Positive number
A number with a '+' sign in front, representing a value greater than zero.
e.g. Floor +2 in Bela's Building of Fun is a positive number floor.
Negative number
A number with a '–' sign in front, representing a value less than zero.
e.g. Floor –2 in Bela's Building of Fun is a negative number floor.
Zero
The number representing nothing; it is neither positive nor negative and is the reference point on the number line.
e.g. The ground floor in Bela's Building of Fun is Floor 0.
Integer
All positive numbers, negative numbers, and zero, extending infinitely in both directions on the number line.
e.g. …, –4, –3, –2, –1, 0, 1, 2, 3, 4, … are all integers.
Number line
A line on which numbers are placed in order; positive numbers are to the right of zero and negative numbers to the left.
e.g. On a number line, –5 is to the left of –3, and 3 is to the right of 0.
Inverse (of a number)
The number that when added to the original gives zero; also called the additive inverse.
e.g. The inverse of +3 is –3, because (+3) + (–3) = 0.
Zero pair
A pair of one positive and one negative token that together have a value of zero.
e.g. One green (+1) and one red (–1) token make a zero pair.
Credit
A positive amount added to a bank account, increasing the balance.
e.g. Depositing ₹60 into your account is a credit of ₹60.
Debit
A negative amount subtracted from a bank account, decreasing the balance.
e.g. Paying an electric bill of ₹30 is a debit of ₹30.
Sea level
The reference height of 0 m from which heights above and below are measured.
e.g. Mount Everest is above sea level, while the Mariana Trench is below sea level.
Border sum
The sum of numbers in each row and each column of a grid, which is the same for all rows and columns.
e.g. In a grid, if top row sums to 0 and bottom row sums to 0, the border sum is 0.
Formulas & Equations (4)
Starting Floor + Movement = Target Floor
Reads as: Starting floor plus movement equals target floor
Variables: Starting Floor: the floor you begin on; Movement: the number of floors moved (positive for up, negative for down); Target Floor: the floor you end on
Use when: Use when finding the final position after moving up or down in a lift or on a number line
⚠ Students often forget to include the sign of movement, e.g., writing (+2) + 3 instead of (+2) + (+3).
Target Floor – Starting Floor = Movement needed
Reads as: Target floor minus starting floor equals movement needed
Variables: Target Floor: the floor you want to reach; Starting Floor: the floor you start from; Movement needed: the number of floors to move (positive for up, negative for down)
Use when: Use when finding how many floors to press to go from one floor to another
⚠ Students often subtract the smaller from the larger without considering signs, e.g., writing (+5) – (+2) = +3 correctly, but then writing (–1) – (+3) = +4 instead of –4.
Subtracting a negative number is the same as adding the corresponding positive number
Reads as: Subtracting a negative number is the same as adding the corresponding positive number
Variables: Example: (+2000) – (–200) = (+2000) + (+200) = +2200
Use when: Use when simplifying subtraction expressions involving negative numbers
⚠ Students think subtracting a negative gives a negative result, e.g., writing 5 – (–3) = 2 instead of 8.
The sum of a number and its inverse is zero
Reads as: The sum of a number and its inverse is zero
Variables: Example: (+3) + (–3) = 0
Use when: Use when simplifying expressions or solving equations involving additive inverses
⚠ Students confuse inverse with reciprocal, e.g., saying the inverse of 2 is 1/2 instead of –2.
How It Works
Adding integers using the lift model
  1. Identify the starting floor (e.g., +1 for Food Court).
  2. Identify the movement (e.g., +2 means press '+' twice).
  3. Add the starting floor and movement: (+1) + (+2) = +3.
  4. The result is the target floor (+3, Book Store).
💡 Movement is written with a sign: '+' for up and '–' for down. Always include the sign in the expression.
Subtracting integers using the lift model
  1. Identify the starting floor (e.g., +2 for Art Centre).
  2. Identify the target floor (e.g., +5 for Sports Centre).
  3. Use the formula: Target Floor – Starting Floor = Movement needed.
  4. Calculate: (+5) – (+2) = +3, so press '+' three times.
💡 Subtraction finds the movement needed, not the final position. The sign of the result tells you whether to go up or down.
Adding integers using the token model
  1. Represent each positive number with a green token and each negative number with a red token.
  2. Combine all tokens in one set.
  3. Remove zero pairs (one green and one red together) until no more pairs exist.
  4. Count the remaining tokens: if green, the result is positive; if red, the result is negative.
💡 Zero pairs cancel out and do not affect the final sum. Always remove all zero pairs before reading the result.
Subtracting integers using the token model
  1. Start with tokens representing the first number (e.g., 5 green for +5).
  2. To subtract, take away the required number of tokens of the same colour (e.g., take away 4 green for –4).
  3. If there are not enough tokens of that colour, add zero pairs (one green and one red) until you have enough to take away.
  4. After taking away, the remaining tokens give the result.
💡 Adding zero pairs does not change the value of the set. You may need to add multiple zero pairs to perform the subtraction.
Comparing integers using the number line
  1. Draw a number line with zero in the middle, positive numbers to the right, and negative numbers to the left.
  2. Locate the two numbers on the number line.
  3. The number to the left is smaller; the number to the right is larger.
  4. Use '<' for 'less than' and '>' for 'greater than'.
💡 All negative numbers are less than zero, and all positive numbers are greater than zero. For two negative numbers, the one with the larger absolute value is smaller (e.g., –5 < –3).
Converting subtraction to addition
  1. Identify the number being subtracted.
  2. Replace it with its inverse (additive inverse).
  3. Change the subtraction sign to addition.
  4. Perform the addition as usual.
💡 This rule works for all integers: subtracting a number is the same as adding its inverse. For example, (+7) – (+5) = (+7) + (–5).
Exam Traps — Don't Make These Mistakes
⚠ Confusing 'greater than' for negative numbers
✗ –5 > –3
→ On the number line, –5 is to the left of –3, so –5 is smaller. The correct statement is –5 < –3.
✓ –5 < –3
⚠ Incorrect sign when subtracting a negative
✗ 5 – (–3) = 2
→ Subtracting a negative is the same as adding the positive: 5 – (–3) = 5 + 3 = 8.
✓ 5 – (–3) = 8
⚠ Forgetting the sign in addition expressions
✗ (+2) + 3 = +5
→ The movement must also have a sign. It should be (+2) + (+3) = +5.
✓ (+2) + (+3) = +5
⚠ Misidentifying the inverse of zero
✗ The inverse of 0 is 0.
→ This is actually correct, but students often think zero has no inverse or write something else. The inverse of 0 is 0 because 0 + 0 = 0.
✓ The inverse of 0 is 0.
⚠ Confusing 'border sum' with 'sum of all numbers'
✗ Add all numbers in the grid to find the border sum.
→ Border sum is the sum of numbers in each row and each column separately, not the total of all numbers.
✓ Find the sum of the top row, bottom row, left column, and right column; they should all be equal.
Compare & Contrast
Position on the number line
Positive number
Negative number
Positive numbers are to the right of zero; negative numbers are to the left of zero.
Operation meaning
Addition of integers
Subtraction of integers
Addition finds the target floor from a starting floor and movement; subtraction finds the movement needed to go from a starting floor to a target floor.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks), 1 LA (5 marks)
Most asked: Identifying positive and negative numbers · Comparing integers using < and > · Adding and subtracting integers on a number line · Finding the additive inverse of a number
Don't skip: Word problems involving movement on a number line (e.g., lift floors, bank balances); Subtraction of integers with negative numbers
One-Line Revision (read aloud before the exam)
  1. Integers include all positive numbers, negative numbers, and zero.
  2. Positive numbers are greater than zero; negative numbers are less than zero.
  3. Zero is neither positive nor negative.
  4. On a number line, numbers increase as you move to the right and decrease as you move to the left.
  5. The additive inverse of a number is the number that when added gives zero.
  6. Subtracting a negative number is the same as adding its positive counterpart.
  7. Adding a positive and a negative number is like finding the difference and keeping the sign of the larger absolute value.
  8. In the token model, a positive and a negative token form a zero pair and cancel each other.
  9. Bank credits are positive numbers; debits are negative numbers.
  10. Sea level is the reference point (0 m) for measuring heights and depths.
  11. The border sum in a grid is the sum of numbers in each row and each column, which must be equal.
  12. Brahmagupta gave the first general rules for operations on positive, negative, and zero numbers.

MCQs — 10 Questions with Answers

Q1. Which of these is a positive number?

  • A. +5 ✓
  • B. −5
  • C. 0
  • D. −0

Answer: A — A positive number has a + sign in front, so +5 is positive; −5 is negative and 0 is neither.

Q2. What is the inverse of −8?

  • A. −8
  • B. +8 ✓
  • C. 0
  • D. 8 (no sign)

Answer: B — The inverse of −8 is +8 because (−8) + (+8) = 0, they cancel each other out.

Q3. You are at Floor +3 and press −4 in the lift. Which floor do you reach?

  • A. Floor +7
  • B. Floor −1 ✓
  • C. Floor +1
  • D. Floor 0

Answer: B — (+3) + (−4) = −1, so you go down 1 floor from +3 and reach Floor −1.

Q4. Which symbol should go in the box: −5 ___ −2?

  • A. >
  • B. < ✓
  • C. =
  • D. ≠

Answer: B — Floor −5 is lower than Floor −2, so −5 < −2; the bigger negative number is the smaller value.

Q5. Which of these numbers is the smallest?

  • A. 0
  • B. +2
  • C. −3 ✓
  • D. −1

Answer: C — Floor −3 is the lowest, so −3 is the smallest; all negative numbers are less than 0 or positive numbers.

Q6. Gurmit presses +2 by mistake, then presses −3 to fix it. What is his total movement? (Think: (+2) + (−3) = ?)

  • A. +5
  • B. −1 ✓
  • C. +1
  • D. 0

Answer: B — (+2) + (−3) = −1 means he moves down 1 floor overall (up 2, then down 3 = net down 1).

Q7. Is 0 a positive number, negative number, or neither?

  • A. Positive
  • B. Negative
  • C. Neither positive nor negative ✓
  • D. Both positive and negative

Answer: C — Zero is the starting point between positive and negative, so it is neither positive nor negative.

Q8. If you start at Floor −2 and want to reach Floor +3, what movement do you need? (−2) + ___ = +3

  • A. +1
  • B. +3
  • C. +5 ✓
  • D. −5

Answer: C — From Floor −2 to Floor +3 requires moving up 5 floors, so (−2) + (+5) = +3.

Q9. In the Building of Fun, which statement is true?

  • A. Floor −4 is higher than Floor −2
  • B. Floor +1 is lower than Floor 0
  • C. Floor −1 is lower than Floor +2 ✓
  • D. Floor 0 is lower than Floor −3

Answer: C — Floor −1 (below ground) is physically lower than Floor +2 (above ground), so −1 < +2 is true.

Q10. What is the result of combining these button presses: (+1) + (+2) + (−2)?

  • A. +1 ✓
  • B. +5
  • C. −2
  • D. 0

Answer: A — (+1) + (+2) + (−2) = (+1) + 0 = +1 because (+2) and (−2) are inverses that cancel to zero.

Flashcards

What is a positive number?

A number with a + sign in front that represents a position above zero or going up.

What is a negative number?

A number with a − sign in front that represents a position below zero or going down.

What floor is the ground floor called?

Floor 0, because it is the starting or reference point with neither + nor − sign.

If you press +4 in the lift, where do you go?

You go up 4 floors from your starting position.

What is the inverse of +7?

The inverse of +7 is −7, because (+7) + (−7) = 0.

If you start at Floor +2 and press −5, which floor do you reach?

You reach Floor −3, because (+2) + (−5) = −3.

Is −4 greater than or less than −2?

−4 is less than −2 because Floor −4 is lower than Floor −2, so −4 < −2.

Compare: 0 and −5. Which is greater?

0 is greater than −5 because zero is higher than all negative numbers, so 0 > −5.

What does the expression (+3) + (−1) mean in the Building?

Start at Floor +3 and press the − button once, reaching Floor +2.

Can you write a + or − sign in front of 0?

No, zero is neither positive nor negative, so we never write +0 or −0.

Important Board Questions

What is the ground floor called in the Building of Fun, and why? [1 mark]

It is the reference or starting point from which all other floors are measured; remember that 0 is neither + nor −.

Write the inverse of each number: +6, −3, 0. Explain what inverse means using one example. [2 marks]

Inverse of +n is −n. Example: (+6) + (−6) = 0 shows they cancel and bring you back to zero or ground floor.

Imagine you are in a lift in the Building of Fun. You start at Floor +1 (Food Court) and press −2. Using the expression (+1) + (−2), explain which floor you reach and why. (Use the building example to show your answer.) [3 marks]

Starting Floor is +1. You press − 2 times (go down 2). So (+1) + (−2) = −1, meaning you reach Floor −1 (Toy Store). Explain: up 1, then down 2 = net down 1.

In the Building of Fun, there are floors +4, −3, +1, 0, and −1. (a) Arrange these floors from lowest to highest. (b) Which floor is the reference point? (c) Explain why we write −3 < −1 using floor positions. (d) If you start at Floor +4 and need to reach Floor −3, what total movement (as an expression) do you need? [5 marks]

Arrange from lowest to highest: −3, −1, 0, +1, +4. Reference = Floor 0. Lowest floor is lowest number. For (c): Floor −3 is physically lower than Floor −1. For (d): (+4) + (−7) = −3 (need to go down 7 floors total).

True or False? −5 is greater than −2. Explain your answer using the Building of Fun example. [2 marks]

False. Floor −5 is below Floor −2, so −5 is lower and smaller. Think: which floor is deeper/lower in the building?

Assertion-Reason Questions for The Other Side of Zero

Q1. Assertion (A):

All negative numbers are less than zero.

Reason (R):

Negative numbers are represented on floors below the ground floor in the Building of Fun.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why all negative numbers are less than zero—they are located below zero on the number line, just like basement floors are below ground floor.

Q2. Assertion (A):

The inverse of –7 is +7.

Reason (R):

Because (–7) + (+7) = 0, which brings you back to the ground floor.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true (–7 and +7 are inverses), and the reason is also true (they do sum to 0), but the reason does not fully explain what an inverse is—it just shows what happens when you use it. The real reason is that +7 and –7 are opposite numbers.

Q3. Assertion (A):

Zero is a positive number.

Reason (R):

Zero has no '+' or '–' sign in front of it.

Answer: D — A is false but R is true — The assertion is false because zero is neither positive nor negative. The reason is true (zero has no sign), but it does not prove the assertion. Zero's lack of a sign actually proves the assertion wrong.

Unlock more Assertion-Reason questions in Premium — unlock →

Case-Based Questions (CBQ) for The Other Side of Zero

Case Study:

Arun's apartment building has a ground floor (Floor 0) where the main entrance is. Above the ground floor are residential floors numbered +1, +2, and +3. Below the ground floor is a basement parking at –1. One day, Arun started at Floor +2 (his apartment), went down 5 floors to reach the basement parking to get his car, then went up 7 floors to reach his friend's apartment.

Q1. Which floor did Arun reach after going down 5 floors from Floor +2?

  • A. Floor –3 ✓
  • B. Floor 0
  • C. Floor –1
  • D. Floor +7

Answer: A

Q2. After reaching the basement parking, Arun went up 7 floors. Which expression shows this movement?

  • A. (–3) + (+7) = +4 ✓
  • B. (–3) + (–7) = –10
  • C. (+2) + (+7) = +9
  • D. (–3) + (+7) = 0

Answer: A

Q3. On which floor does Arun's friend live?

  • A. Floor +4 ✓
  • B. Floor +5
  • C. Floor 0
  • D. Floor +3

Answer: A

Unlock more case studies in Premium — unlock →

Next chapterPuzzles and Explorations →
Found this useful? Share it with your class group.

You've seen the free preview. Open the full interactive study kit — mind map, instant-graded quiz, progress tracking — free, no sign-in required.

Open Full Study Kit — Free →

Premium unlocks the full Question Bank, Daily Practice Papers, Chapter Quiz and Full Mock Papers for every chapter.