Chapter 10: The Other Side of Zero

Integers  |  Ganita Prakash — Grade 6

Numbers began with the counting numbers 1, 2, 3, 4, … Later we learnt about zero (0), which stands for nothing and comes just before 1. We also learnt about fractions like 1/2, 3/2 and 13/6 that sit between whole numbers.

The line we drew earlier (starting at 0 and going right) is really only a number ray. This chapter asks a big question: are there numbers to the left of 0, smaller than 0? The answer is yes — and finding them completes the ray into a full number line.

10.1 Bela's Building of Fun

Bela built a multi-storeyed building with some floors above the ground and some below the ground. A lift moves between floors using two buttons:

• '+' button → the lift goes up.
• '−' button → the lift goes down.

Numbering the floors in the Building of Fun

The ground floor (Welcome Hall) is taken as the starting point and called Floor 0. Floors above it use positive numbers and floors below use negative numbers. The floor number equals how many presses are needed from Floor 0.

Addition to keep track of movement

Movement in the lift is written as an addition:

Example: Start at Food Court +1 and press +2 → (+1) + (+2) = +3 (Book Store).

Combining button presses is also addition

If you press '+' twice then '−' three times, the total effect is one addition. Example: Gurmit presses +2 then −3 → (+2) + (−3) = −1, so he ends one floor below where he started.

Back to zero! (Inverse)

If Basant presses +3 by mistake, he can cancel it by pressing −3, since (+3) + (−3) = 0. We say −3 is the inverse of +3 (and vice-versa).

Comparing numbers using floors

A floor that is lower holds the smaller number. So +3 < +4 and, since Floor −4 is below Floor −3, we get −4 < −3.

Subtraction to find which button to press

Subtraction can mean "take away", but it can also mean "how much change is needed to make two quantities equal" — the missing number to be added. For integers we mostly use this second meaning.

Example: from Art Centre +2 to Sports Centre +5, movement is (+5) − (+2) = +3.

Adding and subtracting larger numbers (the mine)

A mine has levels marked above ground (positive) and below ground (negative), measured in metres from ground level 0. The same two formulas work here too.

Adding, subtracting and comparing any numbers (infinite lift)

We can imagine a lift that goes up and down forever from Level 0 — an infinite lift. It lets us add or subtract any integers.

Example: (+2000) − (−200). From −200 to +2000 the movement is +2200 (up 200 to reach 0, then up 2000 more). So (+2000) − (−200) = +2200. Notice (+2000) + (+200) also equals +2200.

Back to the number line

Rotate the infinite lift by 90° and it becomes the full number line. To the left of 0 are the negatives. We usually drop the '+' sign and just write positives as 1, 2, 3, …

On the number line, smaller numbers are to the left and bigger to the right. So 2 < 5, −3 < 2, and −5 < −3. Moving right is a positive (forward) movement; moving left is negative (backward).

Using the unmarked number line (UNL)

An unmarked number line shows only the position of 0. We imagine the scale and jump left or right to add or subtract. Example: 85 + (−60) = 25 (jump 60 back from 85).

Converting subtraction to addition (and back)

From 2 to −3 the movement is −3 − 2 = −5. Breaking the journey (2→0 gives −2, 0→−3 gives −3) gives (−3) + (−2) = −5 — the same answer with no subtraction!

10.2 The Token Model

The bored lift attendant keeps green (+) tokens and red (−) tokens. Each '+' press adds a green token, each '−' press adds a red token. A green and a red token together make a zero pair (their value is 0), so they cancel each other.

Using tokens for addition

To add, put down all tokens, remove every zero pair, and see what is left.

• 5 positives + 3 negatives → remove 3 zero pairs → 2 positives left → (+5)+(−3) = +2
• 5 positives + 8 negatives → remove 5 zero pairs → 3 negatives left → (+5)+(−8) = −3

Using tokens for subtraction

To subtract, take away the tokens being removed. If there are not enough tokens of that kind, add zero pairs first (this does not change the value), then take them away.

10.3 Integers in Other Places

Credits and debits

In a bank account, a credit (money in) acts like a positive number and a debit (money out) acts like a negative number. The balance is the total of all credits and debits, and it can be positive or negative.

Geographical cross sections

Heights are measured from sea level (0 m). Heights above sea level are positive; heights below sea level are negative.

Temperature

Temperature is measured in degrees Celsius (°C). 0 °C is the freezing point of water. Temperatures above 0 °C are positive; below 0 °C are negative. Cold places in India (usually high mountains, like Leh in Ladakh) can go below 0 °C.

10.4 Explorations with Integers

A hollow integer grid

In a special 3×3 grid, the two rows (top and bottom) and the two columns (left and right) each add up to the same number. This shared total is called the border sum.

An amazing grid of numbers!

Circle any number, strike out its whole row and column, then circle another unstruck number, and repeat until none are left. Adding the circled numbers always gives the same total — the magic is in the way the grid is built (each entry follows a row + column pattern), not the numbers themselves.

Figure it Out (mixed practice — p.265/266)

10.5 A Pinch of History

Negative numbers and zero were first thought of and used in Asia long ago. Their earliest known use was in accounting.

Brahmagupta's Rules for Addition

1. Positive + Positive = Positive. E.g. 2 + 3 = 5.
2. Negative + Negative = Negative. Add without signs, then put a minus. E.g. (−2)+(−3) = −5.
3. Positive + Negative: subtract the smaller (without sign) from the greater (without sign) and keep the greater number's sign. E.g. −5 + 3 = −2, 2 + (−3) = −1, −3 + 5 = 2.
4. A number + its inverse = 0. E.g. 2 + (−2) = 0.
5. A number + 0 = the same number. E.g. −2 + 0 = −2.

Brahmagupta's Rules for Subtraction

1. Smaller positive from larger positive → positive. E.g. 3 − 2 = 1.
2. Larger positive from smaller positive → negative. E.g. 2 − 3 = −1.
3. Subtracting a negative = adding the positive. E.g. 2 − (−3) = 2 + 3.
4. A number minus itself = 0. E.g. 2 − 2 = 0, −2 − (−2) = 0.
5. Number − 0 = same number; 0 − a number = its inverse. E.g. 0 − (−2) = 2.

Imp Formulas at a Glance