The World of Numbers · Lesson 3 of 7
Integers: Expanding the Horizon
“Negative numbers arrived when counting only what you had was no longer enough.”
• Understand why negative numbers are needed. • Represent integers on the number line. • Interpret positive and negative values through fortunes and debts. • Apply the arithmetic rules for signed numbers. • Explain why subtracting a negative is equivalent to adding a positive.
Integers
Natural numbers are useful when we count objects and answer questions such as “How many?” They begin with 1, 2, 3, and continue onward. When we include 0, we can also represent situations where there is no quantity at all. This gives us a larger set of numbers that works well for counting and many simple calculations.
However, subtraction introduces a new situation. For example, in (3 - 5), we are trying to take away more than we have. The result cannot be represented by 0 or by any positive number. To show such values, we extend the number line to the left of zero. The numbers found there are negative numbers, and together with zero and the positive whole numbers, they form the set of integers.
The set containing negative whole numbers, zero and positive whole numbers.
Fortunes and Debts
The chapter describes Brahmagupta’s use of a commercial interpretation: positive numbers as fortunes or assets, and negative numbers as debts. This makes signed numbers intuitive. A balance of ₹500 can be represented as +500, while owing ₹500 can be represented as −500.
An integer greater than zero, such as 1, 2, 3, ...
An integer less than zero, such as −1, −2, −3, ...
Adding Signed Numbers
If two quantities have the same sign, their magnitudes combine and the common sign remains. Two fortunes combine into a larger fortune. Two debts combine into a larger debt.
| Situation | Example | Result |
|---|---|---|
| Fortune + fortune | 5 + 4 | 9 |
| Debt + debt | (−5) + (−4) | −9 |
| Fortune + debt | 12 + (−7) | 5 |
| Debt + larger fortune | −8 + 13 | 5 |
Multiplying Signed Numbers
For multiplication, the sign depends on whether the two factors have the same sign or opposite signs.
| Signs | Resulting sign | Example |
|---|---|---|
| + × + | + | 3 × 4 = 12 |
| − × + | − | (−3) × 4 = −12 |
| + × − | − | 3 × (−4) = −12 |
| − × − | + | (−3) × (−4) = 12 |
Why Negative × Negative Is Positive
The chapter uses the language of debt. If −₹3 represents one debt of ₹3, then removing four such debts changes your financial position by +₹12. The idea 'remove a debt' corresponds to a negative action applied to a negative quantity, giving a positive change.
Problem
Interpret (−3) × (−4) using debt.
- 1.−3 represents a debt of ₹3.
- 2.The second negative can be interpreted as removing rather than adding.
- 3.Removing four debts of ₹3 improves the balance by ₹12.
- 4.Therefore (−3) × (−4) = +12.
Subtracting a Negative Number
Subtracting a negative means removing a debt. Removing a debt improves the balance, so subtraction of a negative becomes addition of the corresponding positive number.
Problem
Explain why 10 − (−5) = 15.
- 1.Start with a fortune of 10.
- 2.The quantity −5 represents a debt of 5.
- 3.Subtracting −5 means removing that debt.
- 4.Removing a debt of 5 improves the value by 5.
- 5.So 10 − (−5) = 15.
Using Integers in Temperature and Finance
Problem
The temperature is 4 °C and falls by 15 °C. Find the new temperature.
- 1.A fall of 15 is represented by adding −15.
- 2.4 + (−15) = −11.
- 3.The new temperature is −11 °C.
Problem
A trader begins with a debt of ₹850, then earns ₹1200, then loses ₹450. Find the final standing.
- 1.Debt = −850.
- 2.Profit = +1200.
- 3.Loss = −450.
- 4.Total = −850 + 1200 − 450.
- 5.−850 + 1200 = 350.
- 6.350 − 450 = −100.
- 7.The final standing is a debt of ₹100.
Practice Problems
- A hill station records 6 °C in the afternoon and the temperature falls by 18 °C at night. Find the night temperature.
- A trader owes ₹900, earns ₹1400, then loses ₹350. Represent the situation using integers and find the final balance.
- Calculate: (−12) × 5, (−8) × (−7), 0 − (−14), (−20) ÷ 4.
- Explain with a debt example why 7 − (−4) = 11.
- Write one real-world situation for each of +12, −12 and 0.
Key Takeaways
• Integers extend the number line to include zero and negative whole numbers. • Positive numbers can model fortunes; negative numbers can model debts. • Same-sign products are positive; opposite-sign products are negative. • Subtracting a negative is equivalent to adding the corresponding positive. • Signed numbers model temperature, money, elevation and many other real quantities.
Next, we fill the spaces between integers with fractions and develop the larger set of rational numbers.