The Floor Below Zero Still Counts The Other Side of Zero (Integers)

Bela owns a building, once an ice-cream factory, now converted with a Welcome Hall on the ground floor, a Food Court and Art Centre above it, and a Toy Store and Video Games floor below ground level. The lift has two buttons, '+' and '-', and pressing one moves the lift up or down that many floors from wherever it currently sits. Calling the Welcome Hall 'Floor 0', the floors above become +1, +2, +3, and the floors below become -1, -2. Gurmit wants to go down two floors from the Art Centre but presses '+' twice by mistake, ending up too high; three presses of '-' fixes it exactly: (+2) + (-3) = -1. Basant, starting at the ground floor, presses +3 by accident and needs exactly -3 to undo it back to 0 -- introducing the idea of an inverse, a number that exactly cancels another back to zero. Every number on this list of counting numbers used until now, 1, 2, 3, and so on, was really only ever a number RAY: a line starting at 0 and running off in one direction. A building with floors below ground, though, needs a mirror-image extension of that ray heading the other way, and that full two-sided line is exactly what this chapter builds.

The same above/below idea scales up using a mine, where ground level is 0, ore deposits above ground are positive, and tunnels below are negative, reaching numbers like -200 or +180 far larger than any real building would need. A mine descending at 3 metres per minute for 60 minutes moves 60 x (-3) = -180, exactly 180 metres below ground; starting 15 m above ground and descending for 45 minutes lands at 15 + (45 x (-3)) = 15 - 135 = -120. Imagining a building with infinitely many floors in both directions, then rotating the whole picture 90 degrees so the floors run left to right instead of up and down, turns Bela's lift shaft directly into the familiar number line, with every integer, positive, negative, and zero, sitting at its own exact point.

A bored lift attendant collects tokens: green for positive, red for negative, and whenever one green and one red token sit together, they form a zero pair and can be removed without changing anything, since +1 and -1 always cancel exactly. Addition simply combines two piles of tokens and cancels every zero pair that appears: (+5) + (-7) means 5 greens and 7 reds together, 5 zero pairs cancel, leaving 2 reds, so the answer is -2. Subtraction using tokens means physically taking tokens away: (+5) - (+4) takes 4 greens from a pile of 5 greens, leaving +1. But (+5) - (+6) hits a snag -- there aren't 6 greens to take away from only 5. The fix is to add extra zero pairs first, which changes nothing about the pile's true value: add one green-red zero pair, giving 6 greens and 1 red, now remove 6 greens, leaving exactly 1 red, i.e. -1. The same trick handles (+4) - (-6): 6 zero pairs are added first (giving 4+6=10 greens and 6 reds), then the 6 reds are removed as instructed, leaving 10 greens, i.e. +10 -- confirming that subtracting a negative really does behave like adding the matching positive.

Long before tokens or lifts, Brahmagupta's Brahmasphutasiddhanta (628 CE) gave the first known formal rules for arithmetic with fortunes (dhana, positive) and debts (rina, negative). His rules for addition: a fortune plus a fortune is a fortune (2+3=5); a debt plus a debt is a debt ((-2)+(-3)=-5); adding a positive and a negative means subtracting the smaller from the larger, keeping the sign of the larger (-5+3=-2); a number plus its inverse is zero (2+(-2)=0); and a number plus zero is unchanged. His rules for subtraction include the one that resists intuition longest: subtracting a negative number is the same as adding the matching positive (2-(-3)=2+3=5) -- explained through the debt-removal idea that taking away someone's debt of 3 makes them exactly 3 richer, the same effect as directly handing them 3. This formal treatment of positive numbers, negative numbers, and zero together, on equal footing, appears independently elsewhere too: The Nine Chapters on Mathematical Art (China, 1st-2nd century CE) used red counting rods for positive amounts and black rods for negative ones, and Kautilya's Arthashastra (around 300 BCE, India) already used credit/debit accounting resembling signed numbers. It took centuries for negative numbers to be fully accepted everywhere -- as late as the 18th century, the French mathematician Lazare Carnot still called them 'absurd'.

A bank balance starting at zero, credited 30+40+50 and debited 40+50+60, ends at (30+40+50)-(40+50+60) = 120-150 = -30, a genuinely negative balance, meaning money owed rather than money held. A geographical cross-section uses sea level as its zero: Mount Everest stands at +8,848 m above sea level, while the Mariana Trench's Challenger Deep reaches roughly -10,994 m, deeper below sea level than Everest stands above it. Temperature uses zero degrees Celsius as its own reference point entirely independent of the other two: a November day in Leh, Ladakh might swing from +14 degrees C in the afternoon down to -4 degrees C before dawn, the exact same 'other side of zero' idea showing up in a completely different unit and context.

A grid of integers, arranged so every row and every column sums to the exact same 'border sum', can be filled in more than one way for some target sums and in only one way for others -- worth noticing which happens when, and why. A related party trick: circle any number in a small grid, cross out its entire row and column, then repeat until every number has either been circled or crossed out; adding up only the circled numbers always gives the exact same total, no matter which numbers happened to get circled along the way. Both puzzles reward the same instinct this whole chapter has been building: that positive and negative numbers combine according to fixed, reliable rules, so a total worked out one way must match a total worked out any other equally valid way.