CBSE Notes Class 6 Maths Chapter 10: The Other Side of Zero

Get the simplified notes of chapter 10, “The Other Side of Zero”, with complete explanations. Zero, a number, a concept, and an essential part of our everyday life, from Mathematics to philosophy, technology, and beyond. But have you ever wondered what lies on the other side of zero? The other side of zero is a fascinating realm: negative numbers. These numbers, though sometimes misunderstood or overlooked, hold immense significance in various fields, from mathematics to physics, economics, and even in our daily experiences. 

1.0Positive and Negative Number

• Positive Number: A number greater than zero, located to the right of zero on the number line. Examples include 1, 2, 3, etc.
• Negative Number: A number less than zero, located to the left of zero on the number line. Examples include −1,−2,−3, etc.

2.0Number Line

Representing integers on number line

• Integers are represented on a number line.
• Every integer has an opposite, except 0.
• Positive integers (+1, +2, +3, ...) are denoted by +Z.
• Negative integers (-1, -2, -3, ...) are denoted by -Z.
• Together with 0, they form the set Z or I.
• Thus, $Z=\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$.

3.0The Mathematical Side of Negative Numbers

In mathematics, negative numbers are just as essential as positive numbers. They provide balance and symmetry to the number system.

• Addition and Subtraction: When adding or subtracting negative numbers, you’re essentially dealing with direction. Adding a negative number moves you left on the number line, while subtracting a negative number moves you right. For example, 5+(−3)=25 + (-3) = 2, and 5−(−3)=85 - (-3) = 8.
• Multiplication and Division: Negative numbers also affect multiplication and division. The product of two negative numbers is positive, while multiplying a negative number by a positive number results in a negative product. For instance:
• • $(-3)\times (-2)$ = 6
  • $(-3)\times 2=-6$

This rule is important in a variety of mathematical operations, from basic arithmetic to more complex fields like algebra and calculus.

• The Concept of Absolute Value: Negative numbers are also integral to the concept of absolute value, which refers to the distance a number is from zero on the number line, regardless of direction. The absolute value of -7 is 7, just as the absolute value of 7 is also 7. 

4.0Rule of signs in Addition and Subtraction 

If a is any number, then 

• +(+a) = +a 
• +(–a) = –a 
• –(+a) = –a 
• –(–a) = +a

5.0Properties of Integers

Additive Inverse

• The sum of an integer and its opposite is 0. 
• Thus, if a is an integer then a + (–a) = 0 a and –a are called opposites or negatives or additive inverses of each other.

Successor and predecessor of an integer

• Let a be an integer then (a + 1) is called the successor of a and (a – 1) is called the predecessor of a.

6.0What Exactly Are Negative Numbers?

Negative numbers are simply numbers less than zero. On a number line, they are found to the left of zero. They represent quantities that are "below" a reference point, be it temperature, altitude, or a financial balance.

For instance:

• In finance, if you owe money, your balance might be negative.
• In geography, negative numbers are used to represent locations below sea level, such as the Dead Sea, which is approximately 430 meters below sea level.
• In science, negative values are used to describe concepts like velocity or charge, where direction or polarity matters.

7.0Negative Numbers in Everyday Life

1. Temperature: One of the most familiar real-world applications of negative numbers is temperature. In places that experience cold winters, temperatures can drop below zero. Whether it's -10°C or -30°F, negative numbers allow us to measure and understand temperatures that fall below freezing.
2. Financial Transactions: In the world of banking and accounting, negative numbers are crucial. A negative balance on your bank account indicates that you owe money, while negative profits in a business context signal a loss. Negative numbers are essential for keeping track of debts, income, and expenditures.
3. Geography and Altitude: In geography, negative numbers are used to represent locations below sea level. The Dead Sea, as mentioned earlier, is the lowest point on Earth’s surface, with an elevation of about -430 meters. The concept of negative elevation is essential for understanding the topography of our planet.
4. Electricity and Magnetism: Negative numbers play a vital role in physics, particularly when describing electric charges. Electrons, which carry a negative charge, are fundamental particles that make up electricity. The interactions between negative and positive charges form the foundation of electrical circuits and magnetism.

8.0The Birth of Negative Numbers

The concept of negative numbers has a rich history, beginning with ancient civilizations. Early number systems, such as those of the Babylonians and the Egyptians, did not feature negative numbers. It wasn't until the 7th century in India that negative numbers began to gain a more formal mathematical recognition. Indian mathematicians used negative numbers in their calculations and even recognized them as debts or losses in financial transactions.

However, it wasn't until the 16th and 17th centuries that negative numbers became more widely accepted in Europe. Mathematicians like René Descartes were initially hesitant to embrace negative numbers, referring to them as “fictitious” numbers. It was only through the works of later mathematicians, such as Carl Friedrich Gauss, that negative numbers found their place in formal mathematics.