Why Taking Away a Negative Number Makes the Answer Bigger
Learn to compare integers on a number line, add and subtract them with the token model, use additive inverses to turn subtraction into addition, and solve gain-and-loss problems.
Why does taking away a negative number make the answer bigger?
Because removing a debt leaves you better off. If a shopkeeper cancels ₹50 that you owed, your position improves by ₹50 — so .
That one idea explains the rule students find strangest, and this page builds up to it step by step. It covers everything in the CBSE Class 7 Mathematics chapter's first half: comparing integers on a number line, adding and subtracting with tokens, additive inverses, and expressions with brackets.
That one idea explains the rule students find strangest, and this page builds up to it step by step. It covers everything in the CBSE Class 7 Mathematics chapter's first half: comparing integers on a number line, adding and subtracting with tokens, additive inverses, and expressions with brackets.
How do you compare and order negative numbers?
On a number line, the number further to the right is always the greater one — and that settles every comparison, positives and negatives alike.
The integers are the whole numbers together with their negatives: Zero is neither positive nor negative.
So , because lies to the right of . This is where students slip: with negatives, the number that looks bigger is smaller. A temperature of in Leh is colder than , even though 7 is more than 2.
Every positive integer is greater than every negative integer, and greater than zero.
A lift panel is the everyday version. Floors run B2, B1, G, 1, 2, 3, which is really — and B1 is higher than B2 in exactly the way .
Ordering a mixed list means placing them left to right: is in increasing order. Sorting by size first and adding the signs afterwards is the mistake to avoid, because it reverses the negative half of the list.
The integers are the whole numbers together with their negatives: Zero is neither positive nor negative.
So , because lies to the right of . This is where students slip: with negatives, the number that looks bigger is smaller. A temperature of in Leh is colder than , even though 7 is more than 2.
Every positive integer is greater than every negative integer, and greater than zero.
A lift panel is the everyday version. Floors run B2, B1, G, 1, 2, 3, which is really — and B1 is higher than B2 in exactly the way .
Ordering a mixed list means placing them left to right: is in increasing order. Sorting by size first and adding the signs afterwards is the mistake to avoid, because it reverses the negative half of the list.
How do green and red tokens make adding integers easy?
Let a green token stand for and a red token for . One green with one red cancels to nothing, so you add by pairing off opposite colours and counting whatever survives.
Add . Take 5 green and 3 red. Three pairs cancel, leaving 2 green, so the answer is .
Add . Three pairs cancel from 8 red and 3 green, leaving 5 red, so the answer is .
The number line says the same thing with movement: start at the first number, then move right for a positive and left for a negative. From , moving 3 steps right lands on .
Two working rules come straight out of the tokens. With same signs, add the values and keep the sign: , since 10 red tokens have nothing to cancel against. With different signs, subtract the smaller value from the larger and take the sign of the larger: .
A cricket over is the everyday case — 12 runs conceded and 5 runs saved by good fielding nets runs against the bowler.
Add . Take 5 green and 3 red. Three pairs cancel, leaving 2 green, so the answer is .
Add . Three pairs cancel from 8 red and 3 green, leaving 5 red, so the answer is .
The number line says the same thing with movement: start at the first number, then move right for a positive and left for a negative. From , moving 3 steps right lands on .
Two working rules come straight out of the tokens. With same signs, add the values and keep the sign: , since 10 red tokens have nothing to cancel against. With different signs, subtract the smaller value from the larger and take the sign of the larger: .
A cricket over is the everyday case — 12 runs conceded and 5 runs saved by good fielding nets runs against the bowler.
What is an additive inverse, and how does it turn subtraction into addition?
The additive inverse of an integer is the number that cancels it to zero. So the additive inverse of is , and of is :
Zero is its own additive inverse, since .
The rule that matters is this: subtracting a number is the same as adding its additive inverse.
So becomes . And becomes — the case from the top of this page. Two minus signs together turn into a plus because the inverse of is .
In tokens it is just as clear. To subtract you must remove 3 red tokens; if none are there, add 3 red and 3 green first, which changes nothing, then take the 3 red away. What you are left with is 3 green — so subtracting added 3.
This is why one rule replaces two: convert every subtraction into an addition and you never need a separate subtraction procedure again.
Zero is its own additive inverse, since .
The rule that matters is this: subtracting a number is the same as adding its additive inverse.
So becomes . And becomes — the case from the top of this page. Two minus signs together turn into a plus because the inverse of is .
In tokens it is just as clear. To subtract you must remove 3 red tokens; if none are there, add 3 red and 3 green first, which changes nothing, then take the 3 red away. What you are left with is 3 green — so subtracting added 3.
This is why one rule replaces two: convert every subtraction into an addition and you never need a separate subtraction procedure again.
How do you evaluate integer expressions with brackets?
Work out the brackets first, then read the remaining signs carefully, working left to right.
Worked example.
The bracket gave 7, the double minus became , and the rest is ordinary addition.
Another, with a bracket that turns negative:
A real-life example brings the pieces together. A lift starts on the ground floor, goes down 2 levels to the parking, then up 5 levels:
so it ends on the 3rd floor.
Temperature change works the same way. If the temperature in Srinagar falls from to , the change is
a drop of . Note the question's wording: change means final minus initial, in that order, and reversing it flips the sign of your answer.
Worked example.
The bracket gave 7, the double minus became , and the rest is ordinary addition.
Another, with a bracket that turns negative:
A real-life example brings the pieces together. A lift starts on the ground floor, goes down 2 levels to the parking, then up 5 levels:
so it ends on the 3rd floor.
Temperature change works the same way. If the temperature in Srinagar falls from to , the change is
a drop of . Note the question's wording: change means final minus initial, in that order, and reversing it flips the sign of your answer.
Exam tip
Exam tip: keeping track of two signs in a row
Most lost marks in this chapter come from a mishandled pair of signs, not from arithmetic.
Rewrite every subtraction as an addition before you calculate anything. Turn into on its own line, then evaluate. The extra line takes four seconds and removes the guesswork.
Then check the plausibility of the answer. Adding a negative must move you left on the number line, and subtracting a negative must move you right. If came out as 5, the direction is wrong.
And read "change", "difference" and "fall" carefully. A fall of from reaches , while a fall to is a different journey altogether.
Rewrite every subtraction as an addition before you calculate anything. Turn into on its own line, then evaluate. The extra line takes four seconds and removes the guesswork.
Then check the plausibility of the answer. Adding a negative must move you left on the number line, and subtracting a negative must move you right. If came out as 5, the direction is wrong.
And read "change", "difference" and "fall" carefully. A fall of from reaches , while a fall to is a different journey altogether.
Did you know
Why does one green and one red token cancel out?
Because and add to zero, so putting them together adds nothing at all.
That is what makes the token trick legal. You may drop in as many green-red pairs as you like without changing the value, which is exactly how you subtract red tokens that were not there to begin with.
The same freedom appears again later in algebra, where adding and subtracting the same quantity rearranges an expression without changing what it is worth.
That is what makes the token trick legal. You may drop in as many green-red pairs as you like without changing the value, which is exactly how you subtract red tokens that were not there to begin with.
The same freedom appears again later in algebra, where adding and subtracting the same quantity rearranges an expression without changing what it is worth.
Key takeaways
Adding and subtracting integers: quick revision
- On a number line the number further right is greater, so and every positive beats every negative.
- Green tokens are and red are ; pair off opposite colours and count what survives.
- Same signs: add and keep the sign. Different signs: subtract the smaller value from the larger and take the larger one's sign.
- The additive inverse of is , and .
- Subtraction is addition of the additive inverse, , which is why .
- Evaluate brackets first, and read "change" as final minus initial so the sign comes out right.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Green tokens are and red are ; pair off opposite colours and count what survives.
- Same signs: add and keep the sign. Different signs: subtract the smaller value from the larger and take the larger one's sign.
- The additive inverse of is , and .
- Subtraction is addition of the additive inverse, , which is why .
- Evaluate brackets first, and read "change" as final minus initial so the sign comes out right.
You will remember all of this far better after answering five questions on it than after reading it twice.