Class 6 Mathematics · Chapter 10 NotesThe Other Side of Zero

We already know counting numbers 1, 2, 3, … and the number 0. But are there numbers less than 0? Imagine a building with floors below the ground. Bela's Building of Fun has a ground floor called Floor 0. Floors above it are numbered +1, +2, +3, … and floors below it are numbered –1, –2, –3, … The lift has a '+' button to go up and a '–' button to go down. Pressing '+' once is written as +1, and pressing '–' once is written as –1. A number with a '+' sign is positive, and a number with a '–' sign is negative. Zero is neither positive nor negative. Negative numbers let us describe quantities less than zero, such as temperatures below freezing or money owed.

• Positive numbers+1, +2, +3, …
• Negative numbers–1, –2, –3, …
• Zero is neither positive nor negative.

The numbers …, –4, –3, –2, –1, 0, 1, 2, 3, 4, … are called integers. They include all positive numbers, all negative numbers, and zero. On a number line, positive numbers lie to the right of zero and negative numbers lie to the left. The number line extends forever in both directions. You can imagine walking on it: moving right means adding a positive number, and moving left means adding a negative number. For example, starting at 5 and moving 4 steps right takes you to 9, so 5 + 4 = 9. Starting at 9 and moving 6 steps left takes you to 3, so 9 + (–6) = 3. The number line helps us visualise addition and subtraction of integers.

• Integers…, –4, –3, –2, –1, 0, 1, 2, 3, 4, …
• Positive integers1, 2, 3, …; Negative integers: –1, –2, –3, …
• On the number line, smaller numbers are to the left, larger numbers to the right.

We can compare integers using the number line or the Building of Fun. A floor higher up is greater, and a floor lower down is smaller. So, +3 < +4, and –4 < –3. All negative numbers are less than 0, and all positive numbers are greater than 0. For example, –2 < +5, –5 < –3, and 0 > –4. When comparing two negative numbers, the one with the larger digit without the sign is actually smaller. For instance, –10 < –7 because –10 is further to the left on the number line. You can also order integers from smallest to largest or largest to smallest by arranging them according to their position on the number line.

• –4 < –3 < –2 < –1 < 0 < 1 < 2 < 3 < 4
• All negative numbers are less than zero.
• All positive numbers are greater than zero.

Addition can be thought of as movement. Starting position + movement = target position. For example, if you start at Floor +1 and press +2, you reach Floor +3, so (+1) + (+2) = +3. If you start at Floor +2 and press –3, you reach Floor –1, so (+2) + (–3) = –1. You can also use tokens: a positive token and a negative token cancel each other to form a zero pair. After removing all zero pairs, the remaining tokens give the sum. Brahmagupta's rules summarise addition: the sum of two positives is positive; the sum of two negatives is negative; to add a positive and a negative, subtract the smaller number (without sign) from the greater and give the result the sign of the greater number. A number plus its additive inverse is zero, and adding zero leaves a number unchanged.

• Starting position + movement = target position.
• Additive inverse of a numberthe number that gives zero when added to it. For example, inverse of +7 is –7.
• Zero pairone positive and one negative token together have value zero.

Subtraction can be understood as finding the movement needed to go from a starting position to a target position: target position – starting position = movement needed. For example, to go from Floor +2 to Floor +5, you need to press +3, so (+5) – (+2) = +3. To go from Floor –2 to Floor –1, you need to press +1, so (–1) – (–2) = +1. You can also perform subtraction using tokens by taking away tokens, adding zero pairs if needed. A key rule is that subtracting a number is the same as adding its additive inverse. So, (+7) – (+5) = (+7) + (–5), and (+8) – (–2) = (+8) + (+2). This rule allows you to convert any subtraction into an addition problem.

• Target position – starting position = movement needed.
• Subtracting a number is the same as adding its additive inverse.
• Examples(+7) – (+5) = (+7) + (–5); (+8) – (–2) = (+8) + (+2).

Integers are used in many real-life situations. In banking, credits are positive and debits are negative. If you deposit money, your balance increases; if you spend or pay a bill, your balance decreases. A negative balance means you owe money. In geography, heights above sea level are positive and depths below sea level are negative. For example, Mount Everest is about 8848 m above sea level, while the Challenger Deep in the Mariana Trench is about –10994 m. In temperature, positive values are above 0°C and negative values are below 0°C. Water freezes at 0°C, so –4°C is colder than 0°C. These examples show how integers help us describe quantities that can go up or down from a reference point.

• Creditspositive numbers; Debits: negative numbers.
• Heights above sea levelpositive; below sea level: negative.
• Temperatures above 0°Cpositive; below 0°C: negative.

Negative numbers were first used in accounting in ancient Asia. In China, The Nine Chapters on Mathematical Art (completed by the 1st or 2nd century CE) used red and black rods for positive and negative numbers. In India, Kautilya's Arthaśhāstra (c. 300 BCE) discussed credits and debits, and the Bakśhālī manuscript (c. 300 CE) used a special symbol for negative numbers. Brahmagupta, in his Brāhma-sphuṭa-siddhānta (628 CE), gave the first general treatment of positive numbers, negative numbers and zero as equally valid numbers. He stated clear rules for addition and subtraction, such as: the sum of two positives is positive; the sum of two negatives is negative; subtracting a negative is the same as adding the corresponding positive. These rules are still used today.

• Brahmagupta's rules for addition and subtraction of integers.
• Zero and negative numbers were accepted in India long before Europe.
• Brahmagupta's work paved the way for modern algebra.