How Numbers Below Zero Actually Work
Learn where negative numbers come from, how to place and compare integers on a number line, and how to add and subtract them without losing the sign.
What are negative numbers used for?
Negative numbers describe quantities that point the opposite way to positive ones. A loss is the opposite of a profit, a fall in temperature the opposite of a rise, and a depth below sea level the opposite of a height above it. Together with 0, the positive and negative whole numbers form the integers.
This page covers everything in the ICSE Class 6 Mathematics chapter on negative numbers, integers and the number line: what integers represent, how to order them, how to add and subtract by moving along a line, and how to keep the sign correct in word problems.
This page covers everything in the ICSE Class 6 Mathematics chapter on negative numbers, integers and the number line: what integers represent, how to order them, how to add and subtract by moving along a line, and how to keep the sign correct in word problems.
How do you write quantities of opposite sense as integers?
You choose one direction as positive, and then the opposite direction must be negative. Which one you choose is up to you, but you must say so and stay consistent.
A profit of Rs 500 is , so a loss of Rs 500 is . A temperature rise of 4 degrees is , so a fall of 4 degrees is . A hill 200 m above sea level is , so a valley floor 200 m below is . Ten steps forward is , and ten steps back is .
Notice that 0 is neither positive nor negative — it is the reference point from which both directions are measured. Sea level itself is 0.
For example, if a shop makes Rs 1,200 profit on Monday and Rs 800 loss on Tuesday, the two days are and , and the week so far stands at .
The boundary case worth remembering: a bigger loss means a more negative number, so is a worse result than even though 900 is the bigger digit.
A profit of Rs 500 is , so a loss of Rs 500 is . A temperature rise of 4 degrees is , so a fall of 4 degrees is . A hill 200 m above sea level is , so a valley floor 200 m below is . Ten steps forward is , and ten steps back is .
Notice that 0 is neither positive nor negative — it is the reference point from which both directions are measured. Sea level itself is 0.
For example, if a shop makes Rs 1,200 profit on Monday and Rs 800 loss on Tuesday, the two days are and , and the week so far stands at .
The boundary case worth remembering: a bigger loss means a more negative number, so is a worse result than even though 900 is the bigger digit.
How do you compare and order integers on a number line?
Draw a line, mark 0 in the middle, put the positive integers to the right and the negative ones to the left, spaced equally. The rule is then simple: the number further to the right is always greater.
So , and also , because sits to the right of .
This produces the result students find strangest at first: among negative numbers, the one with the smaller digits is larger. Compare and . Since is closer to 0, . Written in ascending order, a mixed set like becomes .
Every negative integer is less than 0, and every positive integer is greater than 0, so any negative number is less than any positive number without needing to compare digits at all.
In real life, a winter temperature of degrees is colder than degrees, which matches the maths exactly.
So , and also , because sits to the right of .
This produces the result students find strangest at first: among negative numbers, the one with the smaller digits is larger. Compare and . Since is closer to 0, . Written in ascending order, a mixed set like becomes .
Every negative integer is less than 0, and every positive integer is greater than 0, so any negative number is less than any positive number without needing to compare digits at all.
In real life, a winter temperature of degrees is colder than degrees, which matches the maths exactly.
How do you add and subtract integers on the number line?
Treat every calculation as movement. Start at the first number. Adding a positive moves you right; adding a negative moves you left. Subtracting reverses the direction of the movement.
Work out . Start at and move 5 steps right: you land on . So .
Work out . Start at 2 and move 6 steps left: you land on . So .
Work out . Start at and move 3 more steps left, reaching .
The additive inverse (or opposite) of an integer is the number the same distance from 0 on the other side. The additive inverse of is , and of is . Adding a number to its own inverse always gives 0:
That is what makes subtraction and addition two views of the same move: subtracting 6 is identical to adding .
Work out . Start at and move 5 steps right: you land on . So .
Work out . Start at 2 and move 6 steps left: you land on . So .
Work out . Start at and move 3 more steps left, reaching .
The additive inverse (or opposite) of an integer is the number the same distance from 0 on the other side. The additive inverse of is , and of is . Adding a number to its own inverse always gives 0:
That is what makes subtraction and addition two views of the same move: subtracting 6 is identical to adding .
How do you solve word problems with integers?
Work in three steps: choose which direction is positive, convert every quantity in the question into a signed number, then add them in order.
A lift starts on the ground floor, goes up 7 floors, comes down 3, then comes down 5 more. Taking up as positive: . The lift finishes at , one floor below the ground floor — in a basement.
A second example: the temperature in a hill town is 5 degrees at noon and falls 9 degrees by night. Then , so the night temperature is 4 degrees below zero.
Always finish by writing the sign and the unit, and then translating it back into words. An answer of "" is incomplete; "4 degrees below zero" shows you understood it.
A common misconception is that a negative answer means a mistake. In these questions a negative answer is often the correct and meaningful result — it simply points the other way.
A lift starts on the ground floor, goes up 7 floors, comes down 3, then comes down 5 more. Taking up as positive: . The lift finishes at , one floor below the ground floor — in a basement.
A second example: the temperature in a hill town is 5 degrees at noon and falls 9 degrees by night. Then , so the night temperature is 4 degrees below zero.
Always finish by writing the sign and the unit, and then translating it back into words. An answer of "" is incomplete; "4 degrees below zero" shows you understood it.
A common misconception is that a negative answer means a mistake. In these questions a negative answer is often the correct and meaningful result — it simply points the other way.
Exam tip
The mistake most students make with two signs together
The trap is a calculation like . Students read the two minus signs as "more negative" and answer 2.
Subtracting a negative reverses a leftward move, so it sends you right: .
The reliable way to handle it is to rewrite every subtraction as an addition of the opposite before you move: becomes , and becomes . Convert first, then travel along the line once. Doing the sign work and the arithmetic in one step is where marks are lost.
Subtracting a negative reverses a leftward move, so it sends you right: .
The reliable way to handle it is to rewrite every subtraction as an addition of the opposite before you move: becomes , and becomes . Convert first, then travel along the line once. Doing the sign work and the arithmetic in one step is where marks are lost.
Did you know
Why do lifts in tall buildings show a minus floor?
A lift panel treats the ground floor as 0, so basement levels are genuinely negative positions — which is why they appear as and rather than as ordinary floor numbers.
It is one of the few places in daily life where you can watch a number line being used vertically, with 0 sitting exactly at the reference level you walk in on.
It is one of the few places in daily life where you can watch a number line being used vertically, with 0 sitting exactly at the reference level you walk in on.
Key takeaways
Integers in 30 seconds
- Integers are the positive whole numbers, the negative whole numbers and 0; negative numbers describe the opposite sense of a quantity such as loss, fall or depth.
- 0 is neither positive nor negative — it is the reference point for both directions.
- On a number line the number further right is greater, so and every negative number is less than every positive number.
- Adding a positive moves right, adding a negative moves left, and subtracting reverses the movement; the additive inverse of a number added to it gives 0.
- In word problems, fix the positive direction, convert every quantity to a signed number, and state the answer with its sign, unit and meaning.
You will remember all of this far better after answering five questions on it than after reading it twice.
- 0 is neither positive nor negative — it is the reference point for both directions.
- On a number line the number further right is greater, so and every negative number is less than every positive number.
- Adding a positive moves right, adding a negative moves left, and subtracting reverses the movement; the additive inverse of a number added to it gives 0.
- In word problems, fix the positive direction, convert every quantity to a signed number, and state the answer with its sign, unit and meaning.
You will remember all of this far better after answering five questions on it than after reading it twice.