Class 6 Mathematics Chapter 10: The Other Side of Zero

The tower with a basement

Ishani visits a tall fun tower in Surat. Shops sit above the entrance and shops sit below it. She walks in at the entrance level. Above it are a food court, an art room and a book shop. Below it are a toy shop and a games room. One lift serves the whole tower. It carries only two buttons, one marked plus and one marked minus. Pressing plus lifts her one floor. Pressing minus drops her one floor. How should the floors be numbered so that the lift makes sense?

Counting the presses

Every press moves the lift by exactly one floor. Two presses of plus are written as +2. Three presses of minus are written as -3. So a button press is a number with a sign in front of it. The sign says which way. The digits say how far.

Naming the entrance floor

Give the entrance level the number 0. It is where Ishani starts, so it stands for no movement at all. Every other floor is then named by the press that reaches it from the entrance. The food court needs one press of plus, so it is Floor +1. The toy shop needs one press of minus, so it is Floor -1.

Positive and negative

A number with a plus sign in front of it is a positive number. A number with a minus sign in front of it is a negative number. Floors above the entrance carry positive numbers. Floors below the entrance carry negative numbers. Ishani can now say exactly where she is with one signed number.

Zero belongs to neither side

Zero is the odd one out. We never put a plus or a minus in front of it. Zero is not positive and it is not negative. It is the mark that every other floor is measured from. Take zero away and the words above and below lose their meaning.

The floor number is also the journey

Here is something neat. Floor +3 is three floors above the entrance. Pressing +3 at the entrance also moves the lift three floors up. One signed number names a place, and the same signed number names a movement. Which of the two it means depends on the question you are asking.

Starting floor plus movement

This is the rule that runs the lift. Starting floor plus movement gives the target floor. Ishani stands at Floor +1 and presses +2. So (+1) + (+2) = +3, and she steps out at the book shop. The rule works downward too. From Floor +1 a press of -3 gives (+1) + (-3) = -2.

Adding a negative moves you down

Read that last line once more. Adding a positive number lifts the answer. Adding a negative number lowers it. That is what the minus sign means here. It is not an order to take something away. It is a direction of travel, and it points down the tower.

Two presses in a row

Pranav steps into the lift at Floor -1. He presses +4 and then presses -2. The two presses can be combined first. (+4) + (-2) = +2, so the whole trip is worth two floors up. Starting at -1, that lands him at (-1) + (+2) = +1. Combining the presses gives the same floor as making them one after the other.

Back to the entrance

Suppose Ishani presses +5 by mistake. Which single press brings her back? A press of -5. The two cancel each other, because (+5) + (-5) = 0. We call -5 the opposite of +5, and we call +5 the opposite of -5. Every number on the lift has one.

What an opposite looks like

A number and its opposite sit the same distance from the entrance, one above it and one below it. Floor +4 and Floor -4 are each four floors from 0. That is exactly why one press undoes the other. The opposite of 0 is 0, because the entrance is already the starting mark.

Writing the rule down

The rule is short and it never fails. A number added to its opposite gives zero. So (-7) + (+7) = 0 and (+12) + (-12) = 0. The digits stay the same and only the sign flips over. Test any pair by counting the floors between each number and the entrance.

What Ishani can do now

She can name every floor in the tower with a signed number. She can turn any trip into an addition. She can undo any press by using its opposite. The tower has taught her something about numbers and not only about lifts. The floors below zero are as real as the floors above it.

Scenario:

A class studies a building with three floors below ground. The lift panel reads 3, 2, 1, G, and then B1, B2, B3. Nita is asked which is lower, B2 or B3. She says B2, because 2 is less than 3. Their teacher asks the class to walk down the panel in order. Going down from B2 reaches B3, so B3 is lower. Nita sees her mistake. Below the ground the labels count how far down you are. A bigger number means further from the ground and so lower, which reverses the rule above.

Analysis:

Nita applied the rule from above ground. Below it the numbers count distance downward, so the order of size reverses.

Activity:

Draw a lift panel for a building with four basements. Mark which floor is highest and which lowest.

Title:

Build a paper lift shaft

What You Notice:

The clip returns to the box marked 0 every time a press is followed by its opposite. The boxes above zero and below zero are the same height, so Floor +5 and Floor -5 are the same distance from the middle.

Lift panels in many buildings show B1 and B2 for basement levels, which is another way of writing Floor -1 and Floor -2.

A car park under a mall is often signed downward from the entrance, so the deepest level carries the smallest number.

A lift panel is one of the few places where people meet numbers below zero every single day.

Reading the sign on a press — Plus climbs, minus dives — The sign in front decides the direction of the lift. The digits only decide how many floors it travels.

Finding an opposite — Same digits, other side — Keep the digits exactly as they are and swap the sign. The new number stands just as far from zero, but on the other side of it.

Who is standing higher?

Nitya is in the games room on Floor -2. Sameer is in the storeroom on Floor -5. They are arguing on the phone about who is higher up. Sameer says he must be higher, because 5 is bigger than 2. Nitya says he has it backwards. One of them is right, and the lift panel can settle it. The answer decides how we read every negative number from here on.

Higher on the lift means greater

Look at the two floors in the shaft. Floor -2 is two floors under the entrance. Floor -5 is five floors under it. So Sameer is three floors below Nitya. The higher floor is the greater number, so -2 is greater than -5. We write -2 > -5, and we may also write -5 < -2.

The trap of the bigger digit

This is the slip almost every class makes. The digits pull one way and the numbers go the other way. After a minus sign, bigger digits put the floor further down. Further down means smaller. So a bigger digit after the minus sign gives a smaller number, not a larger one.

Everything below zero is less than zero

Every negative floor lies under the entrance, so every negative number is less than 0. Every positive floor lies over the entrance, so every positive number is greater than 0. That gives a quick first check. A positive number always beats a negative one, however large the digits look.

Reading the order off the shaft

Put the whole shaft in order from the bottom upward and the numbers line up neatly. They run -4, then -3, then -2, then -1, then 0, then 1, then 2. Each step up adds one. So 0 > -20 is true, and so is +17 > -10.

A second meaning for subtraction

Subtraction has always meant take away. It has a second meaning, and this is the one the lift needs. Subtraction can answer the question how much must I add. If Meena has 6 rupees and Arun has 10, then 6 + 4 = 10, and we find that 4 by working out 10 - 6.

Target floor minus starting floor

Now the rule for the lift. Target floor minus starting floor gives the movement needed. Suppose you are at Floor +2 and you want Floor +5. You need three floors up, so the press is +3. Written out, (+5) - (+2) = +3. The answer is the press, not the place.

A trip that goes down

Try one that falls. You are at Floor +3 and you want Floor -1. Count the floors as you drop and you pass +2, +1, 0 and then -1. That is four floors down, so the press is -4. Written out, (-1) - (+3) = -4. A negative answer simply means press the minus button.

A trip that climbs out of the basement

Now start at Floor -2 and aim for Floor -1. Only one floor separates them and the target is the higher one. So the press is +1, and (-1) - (-2) = +1. The answer is positive even though both floors are below the entrance.

Taking away a negative

Here is the case that surprises people. You are at Floor -2 and you want Floor +2. Two floors bring you up to the entrance and two more carry you above it. That is four floors up, so (+2) - (-2) = +4. Notice that (+2) + (+2) = +4 as well. Taking away a negative number does the same work as adding the matching positive one.

The same rules in a mine

A mine has levels above and below the ground in just the same way. The ground level is 0, the levels above it are positive and the levels below it are negative. Nothing changes except the size of the numbers. So (-70) + (-45) = -115, and (+90) - (-60) = +150.

An imagined lift with no ends

Picture a lift shaft that carries on for ever, up and down, from Level 0. No building is tall enough, but nothing stops us imagining it. With that shaft, any two levels can be compared and any trip can be worked out. For instance, (+1500) - (-300) = +1800.

The whole collection has a name

The positive numbers, the negative numbers and zero together are called integers. They run away from zero in both directions and never come to an end. The positive integers are 1, 2, 3 and onward. The negative integers are -1, -2, -3 and onward. Zero sits between the two sets and belongs to neither.

Two checks before you stop

Check the sign first. If the target is the higher floor the press must be positive, and if it is the lower floor the press must be negative. Then check the size by counting the floors between them. If the count and the answer disagree, the working is wrong somewhere.

Scenario:

A lift starts at floor 2 and goes to B1. A class is asked how far it travelled. Imran subtracts 1 from 2 and says one floor. Their teacher asks the class to count the doors it passes. From 2 to 1 is one, 1 to G is two, and G to B1 is three. So the answer is three floors. Imran sees that his subtraction skipped the ground floor entirely. The class works out why. The journey crosses zero, so the two distances have to be added rather than subtracted, because both are measured away from the ground.

Analysis:

Subtracting the labels gave one and counting the doors gave three. A journey across zero adds two distances rather than taking one from the other.

Activity:

Work out the floors travelled for three lift journeys that cross the ground floor. Check each by counting doors.

Some mines run several hundred metres below the ground, so their deepest level carries a large negative number.

Weather reports order cold places by the same rule, which is why a night at -2 degrees is warmer than a night at -9 degrees.

A lift panel in a deep car park is a ready-made ordered list of integers, printed from the bottom upward.

Comparing two negative numbers — Deeper is smaller — Read the two floors as places in a shaft, not as digits on a page. Whichever floor sits deeper underground is the smaller of the two numbers.

Finding the press to make — Target take away start — Write the floor you want first and the floor you are on second, then subtract. The sign of the answer tells you which button to press.

Turning the shaft on its side

A lift shaft is a straight line with zero somewhere along it. Lay that line down flat and something familiar appears. The floors above the entrance land on the right of zero. The floors below it land on the left. The old number line, which began at 0 and ran only to the right, is now complete in both directions.

The finished line

Mark 0 in the middle. To its right put 1, 2, 3 and onward. To its left put -1, -2, -3 and onward. The steps are all the same size, so the gap from -1 to 0 matches the gap from 0 to 1. Once the picture is drawn, every integer has a place of its own.

Dropping the plus sign

Above the entrance we wrote +1 and +2. On a number line we usually leave the plus off and write 1 and 2. The minus sign is never dropped, because it is the only thing marking a number as negative. So +7 and 7 are the same number, but -7 is a different one.

Right is forward, left is backward

Walking to the right is the positive direction. Walking to the left is the negative direction. That one sentence carries the whole of this topic. Every addition is a walk, and the sign of what you add tells you which way to face before you set off.

Bigger to the right

Numbers grow as you walk to the right and shrink as you walk to the left. So 2 < 5, and -3 < 2, and -5 < -3. There is no need to think about the signs separately. Whichever number sits further right is the greater of the two.

A walk that goes forward

Start at 4 and walk 3 steps to the right. You pass 5 and 6 and stop at 7, so 4 + 3 = 7. This is the same rule as before, now on a line instead of in a shaft. Starting number plus movement gives the target number.

A walk that goes backward

Start at 7 and add -5. The minus sign turns you around, so you walk 5 steps to the left. You pass 6, 5, 4 and 3, and stop at 2. That gives 7 + (-5) = 2. Adding a negative number moves you left every time.

Walking past zero

Start at 2 and add -6. Two steps to the left bring you to 0, and four more carry you past it. You stop at -4, so 2 + (-6) = -4. Nothing special happens at zero. It is one more mark on the line, and the walk carries straight through it.

Subtraction becomes addition

Every subtraction can be rewritten as an addition. To take a number away, add its opposite instead. So (+7) - (+5) = +2, and (+7) + (-5) = +2 gives the very same answer. Try it the other way round as well. Here (-3) - (+8) = -11, and (-3) + (-8) = -11.

A line with only zero marked

Large numbers will not fit on a ruled line. Draw a line with 0 marked and nothing else, then sketch the jumps roughly where they belong. To work out 74 + (-50), mark 74 on the right, jump 50 to the left and land on 24. The sketch keeps the direction straight even when the scale is only imagined.

Green tokens and red tokens

Now a second model. Let a green token stand for +1 and a red token stand for -1. A green and a red together are worth nothing at all, so we call the pair a zero pair. Throwing a zero pair away leaves the value of the whole pile unchanged.

Adding with tokens

Put out 5 green tokens and 3 red tokens. Match each red with a green and remove the three zero pairs. Two green tokens are left, so the pile is worth +2, and (+5) + (-3) = +2. The same method handles any addition. Put out both piles and cancel until one colour remains.

When the reds win

Put out 5 green and 8 red. Five zero pairs come out, and 3 red tokens are left behind. The pile is therefore worth -3, so (+5) + (-8) = -3. Whichever colour has tokens left over decides the sign of the answer.

Taking tokens away

Subtraction means removing tokens. To work out (+6) - (+9), put out 6 green tokens. You must remove 9 green, and there are not enough. So add 3 zero pairs, which changes nothing, and now 9 green are there to remove. What remains is 3 red, so the answer is -3.

Taking away what is not there

Try (+4) - (-6). Start with 4 green tokens. You must remove 6 red and the pile has none, so put down 6 zero pairs first. Remove the 6 red and 10 green tokens are left. So (+4) - (-6) = +10, which agrees with the rule that taking away a negative adds a positive.

Scenario:

A class models a bank account with tokens. A black token is one rupee in and a red token is one rupee owed. Rohit has five black and three red and says he has eight rupees. Their teacher asks him to pair them off. Each black cancels one red, so three pairs disappear and two black tokens remain. He has two rupees. Rohit sees that he had counted tokens rather than value. The class marks the position on a number line and finds two to the right of zero. They note that pairing off and moving along the line give the same answer every time.

Analysis:

Eight tokens and two rupees. A red and a black cancel, so counting pieces is not the same as counting value.

Activity:

Use two colours of paper tokens for money in and money owed. Model three situations and mark each on a number line.

Title:

Play with a box of two-colour tokens

What You Notice:

Pushing the pairs aside never changes the value of the pile, and the colour that survives decides the sign of the answer. Putting pairs down lets you take away tokens that were never on the tray, which is why subtracting a negative leaves more than you started with.

A ruler and a thermometer are both number lines, but only the thermometer usually shows the marks below zero.

A line with only zero marked on it is enough for most mental work, because the direction matters more than the exact scale.

Which way to walk — Plus walks right, minus walks left — Read the sign of the number being added before you move. It faces you in the right direction, and only then do the digits count out the steps.

Turning a subtraction round — Take away means add the opposite — Change the subtraction sign to a plus and flip the sign of the number that follows it. The answer stays exactly the same.

A passbook that dips below nothing

Ravi opens a savings account with 250 rupees. On Monday his uncle sends him 120 rupees and the bank prints it as a credit. On Tuesday he pays a repair bill of 90 rupees from the account, and the bank prints that as a debit. On Wednesday he pays 400 rupees for materials, which is more than the account holds. The passbook has to show something, and a counting number cannot show it.

Credits up, debits down

Treat a credit as a positive number and a debit as a negative one. The balance is then the total of every entry. Ravi starts at 250. The credit gives 250 + 120 = 370. The repair bill gives 370 + (-90) = 280. Each entry is a movement, and the balance is where the movement lands.

The balance goes under

Now the large payment. Adding a debit of 400 gives 280 + (-400) = -120, so the balance is 120 rupees below nothing. Some banks allow this for a short while and charge a fee or interest for it. A negative balance means the account owes money rather than holds it.

Climbing back out

On Thursday the work Ravi paid for earns him 500 rupees, which goes in as a credit. That gives (-120) + 500 = 380. The account is healthy again. It is generally better to keep a positive balance, because a negative one usually costs something extra.

A zero chosen by agreement

Heights on land are measured from sea level, and sea level is called 0 metres. A hill top may stand at +850 metres. A trench in the sea floor may lie at -300 metres. Nobody claims that sea level is nothing at all. It is simply the mark everyone has agreed to measure from.

Reading a cross section

A cross section is a side view, as though a slice had been cut out of the ground. Heights above the water line carry positive numbers and depths below it carry negative ones. The higher the point sits on the drawing, the greater its number, exactly as on a lift.

The gap between two places

How far is it from that hill top down to that trench? Use the rule from the lift. Target level minus starting level gives the movement, so (-300) - (+850) = -1150. The journey is 1150 metres downward, and the two places are 1150 metres apart.

Cold enough to go below zero

Temperature is measured in degrees Celsius. Zero degrees Celsius is the freezing point of water, and it is another agreed mark rather than an absence of heat. Warmer than freezing gives a positive reading. Colder than freezing gives a negative one.

A winter night in Leh

Leh, in Ladakh, becomes very cold in winter, and readings there fall below zero. Suppose an afternoon reading is 8 degrees and a reading late that night is 4 degrees below zero. The change is (-4) - (+8) = -12, so the temperature has fallen by twelve degrees.

Putting cold nights in order

Which night was colder, one at 2 degrees below zero or one at 4 degrees below zero? Write them as -2 and -4. The lower reading is the colder night, so -4 is the colder one. Here too, -2 > -4, because a bigger digit after the minus sign means a smaller number.

A grid with a hollow middle

Here is a puzzle. Draw a three by three grid and leave the middle square empty. Fill the eight squares around the edge with integers. Then add the top row, add the bottom row, add the left column and add the right column. When all four totals agree, that shared total is called the border sum.

Working a border sum out

Take this grid. The top row reads 2, -5, -3. The middle row reads -6 and -4, with the hollow between them. The bottom row reads -2, -5, 1. The top row gives -6. The left column holds 2, -6 and -2, which also gives -6. The right column and the bottom row each give -6 as well, so the border sum is -6.

Filling a grid in

Puzzles like this usually have more than one answer. Suppose the top row must come to -6 and two of its three squares hold 2 and -5. The third square has to make up the rest, so it holds -3. Work along each edge in the same way, and check every line again at the end.

A grid that always agrees

Try another puzzle. Write some integers in a four by four grid, circle any one of them, and cross out its whole row and column. Circle another number that has not been crossed out and do the same. Keep going until nothing is left, then add the circled numbers. Whichever numbers you pick, the total comes out the same.

What each pair of signs can give

Some combinations are settled and some are not. Two positives added always give a positive. Two negatives added always give a negative. A positive added to a negative may be positive, negative or zero, and so may a positive take away a positive. A positive take away a negative is always positive.

Counting the years backwards

Years before the common era are counted downwards, so 700 BCE came before 300 BCE. Moving forward 400 years from 700 BCE brings you to 300 BCE. One warning is worth remembering. There was no year 0 at all, so counting across from BCE to CE needs extra care.

Scenario:

A school in Leh keeps a weather board by the office door. Two pupils, Tsering and Dolma, write the reading on it twice a day. On Monday the afternoon reading is 6 degrees. That night the board reads 3 degrees below zero. On Tuesday the afternoon reading is 2 degrees. That night it reads 7 degrees below zero. Their teacher asks them to work out how far the temperature falls each day. She also asks which of the two nights was colder. Then she sets them one more task. They must explain to the younger classes why a reading can sit below zero at all.

Analysis:

One agreed zero lets warm and cold readings share a single scale, and subtraction measures the change between any two of them.

Activity:

Keep your own board for a week, writing each reading with its sign, and work out the fall from afternoon to night.

Leh, in Ladakh, is one of the places in India where the temperature drops below zero degrees Celsius in winter.

A geographical cross section is a side view, as though a vertical slice had been cut out of the ground at one spot.

There was no year 0 in the calendar, so 1 BCE is followed straight away by 1 CE.

Reading a passbook — Credit climbs, debit drops — Write every credit with a plus and every debit with a minus. The balance is then one long addition, and its sign says whether the account holds or owes.

Comparing cold readings — Colder is lower — Put the readings on a thermometer scale, which is a number line stood upright. Whichever reading sits lower down is the colder night and the smaller number.

A question that took a very long time

We have used numbers below zero for a whole chapter without much trouble. That ease is recent. For hundreds of years, careful mathematicians argued about whether such numbers were real numbers at all. The story of how they were accepted runs from ancient account books in Asia to European classrooms of the eighteenth century, and India stands at the centre of it.

The idea begins in Asia

The whole set of integers, which takes in the negative numbers and zero, first came into use in Asia. That was thousands of years back, long before the idea travelled anywhere else. The earliest known uses of negative numbers were not in pure mathematics at all. They came out of accounting, where money owed has to be recorded beside money held.

Red rods and black rods

One of the most important mathematical works of China is The Nine Chapters on Mathematical Art, also called Jiuzhang Suanshu. It was finished during the first or the second century of the common era. In it, positive and negative quantities were shown by rods of two colours, red and black. That is very close to the green and red tokens of the last topic.

Account books in ancient India

India had a strong culture of accountancy in ancient times as well. Kautilya wrote about credit and debit at length in his Arthaśhāstra, which is dated to about 300 BCE. His writing includes the recognition that the balance of an account may be negative. That is the same idea Ravi met in his passbook, written down more than two thousand years ago.

A sign after the number

Negative numbers appear in several ancient Indian works on accounting. One of them is the Bakśhālī manuscript, which is placed at around the year 300 of the common era. It marked a negative number with a special symbol, and that symbol was written after the number rather than in front of it. Our own habit of writing the sign first came later.

Brahmagupta gathers it together

In the year 628 the Indian mathematician Brahmagupta completed a work called the Brāhma-sphuṭa-siddhānta. In it he treated positive numbers, negative numbers and zero as equally valid numbers. All three could be added, subtracted, multiplied and even divided. Nobody before him had given zero the rank of an ordinary number beside the other two kinds.

His first two rules for adding

Brahmagupta set out his rules clearly and explicitly. The first says that two positive numbers added together give a positive number, as in 4 + 5 = 9. The second deals with two negatives. Add the numbers without their signs, then put a minus in front of the answer. That is why (-4) + (-5) = -9.

Adding a positive to a negative

The third rule handles a mixed pair, and it is the one worth learning carefully. Ignore the signs and subtract the smaller number from the greater one. Then give the answer the sign of the greater number. So (-8) + 3 = -5, because 8 is the greater and it is negative. In the same way, (-3) + 7 = 4.

The two rules about zero

Brahmagupta's fourth rule says that a number added to its inverse gives zero, so 9 + (-9) = 0. The inverse is what we have been calling the opposite. His fifth rule says that a number added to zero stays the same, so (-6) + 0 = -6. Two short rules settle every question involving zero.

Taking one positive from another

His rules for subtraction begin with the familiar case. Take a smaller positive from a larger positive and the answer is positive, as in 8 - 3 = 5. Take a larger positive from a smaller one and the answer is negative. So 3 - 8 = -5, which a counting number alone could never express.

Taking away a negative

The next rule is the one we met with tokens. Subtracting a negative number gives the same answer as adding the matching positive number. So 7 - (-4) is worked out as 7 + 4, and either way the answer is 11. Brahmagupta stated this plainly, centuries before it was accepted elsewhere.

Subtracting from and to zero

Two rules finish the list. Taking a number away from itself always gives zero, whether the number is positive or negative, so 5 - 5 = 0 and (-5) - (-5) = 0. Taking zero away from a number leaves it alone, so (-7) - 0 = -7. Taking a number away from zero gives its inverse, so 0 - (-7) = 7.

The long journey westward

These ideas did not spread quickly. Zero and the negative numbers were carried to the Arab world, where they were accepted and studied further by the ninth century. From there they reached Europe by the thirteenth century. Several hundred years had passed between Brahmagupta writing his rules and European scholars reading them.

Doubts that lasted for centuries

Even in the eighteenth century many European mathematicians were unwilling to accept negative numbers. Lazare Carnot, a French mathematician of that period, described them as absurd. In time they proved impossible to do without, in mathematics and in science alike. They are now treated as being every bit as important as the positive numbers, which is exactly what Brahmagupta had described in 628.

Scenario:

A school in Ahmedabad is holding a mathematics exhibition. Ishani and Farhan are given one corner of a hall. Their subject is the story of numbers below zero. They plan a timeline running down a long sheet of paper. On it they will mark the ancient account books, then Brahmagupta, then the journey westward. Beside the timeline they will set out a tray of green and red counters. Visitors may work a sum out with the counters and then read the matching rule. Their teacher asks them to keep every claim to what the chapter records. She also asks them to name each work and each person on the card beside it.

Analysis:

The same rules that a class uses today were written out explicitly in 628, and their first use came from practical record keeping.

Activity:

Draw your own timeline of the six stops, then write one sentence under each saying what was new about it.

Zero was given the rank of an ordinary number by Brahmagupta, who set it beside the positive and the negative numbers.

The rods used in the Chinese work were red and black, which is very close to the two coloured tokens we used in this chapter.

Brahmagupta's way of treating all numbers under one set of rules helped open the way to algebra.

Adding a positive to a negative — Take the smaller from the greater, keep the greater sign — Drop both signs and subtract the smaller number from the greater one. Then give the answer whichever sign belonged to the greater of the two.

The order of the story — Ledgers, then rules, then the world — Account books came first, Brahmagupta's written rules came next, and the journey to the Arab world and to Europe came long afterwards.