CLASS 9 MATHS · CHAPTER 3 · STUDY NOTES
The World of Numbers
Understand the world of numbers through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
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Numbers start by matching and counting
See the need that counting numbers answer.Match objects one for one
- Natural numbers
- The counting numbers 1, 2, 3, … in this chapter’s convention.
One pebble for each animal lets a person check whether the same number returned. This is one-to-one matching, even before using written numerals.
Tally marks, trade and calendars show why people needed numbers. An artefact can show a pattern of marks; a proposed use of that pattern is an interpretation, not something the marks alone prove.
Take a moment. What do you think?
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Question
Are natural numbers closed under subtraction?
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Zero is a number, not an empty gap
Use zero both as a placeholder and in arithmetic.A zero holds the tens place
- Zero
- The number representing no amount, and the result a − a.
In 305, the zero holds the tens place: 3 hundreds, 0 tens, 5 ones. Without it, 35 names a different number.
The chapter discusses Indian place-value mathematics and Brahmagupta’s rules: a + 0 = a, a − 0 = a, a × 0 = 0. Dividing by zero is undefined; a/0 is not zero.
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Question
Why is 0/4 allowed but 4/0 not allowed?
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Negative numbers extend the line left
Use direction and sign to explain an integer calculation.Signs tell the side of zero
Value (number)
- Number-line marks · Only horizontal position represents the number.
- 1-5 · (-5, 0)
- 20 · (0, 0)
- 33 · (3, 0)
The marks show the initial temperature, zero and final temperature.
Read the plotted values
| Value | Height |
|---|---|
| -5 | 0 |
| 0 | 0 |
| 3 | 0 |
- Integers
- …, −2, −1, 0, 1, 2, …: whole-number positions on the number line.
A debt can be represented by a negative amount and a credit by a positive amount. If you have −₹6 and receive ₹10, the net amount is ₹4.
Adding a negative moves left; subtracting a negative moves right. Multiplication with unlike signs is negative; with like signs it is positive. These rules also agree with distributivity.
Let’s look a little closer.
Go deeper: Why a negative times a negative is positive
Use 0 = (−3) × [2 + (−2)]. Distribute: 0 = −6 + (−3)(−2). The second product must therefore be +6. This explains the rule without relying only on a debt story.
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Question
A temperature is 3 °C and falls by 8 °C. What is it now?
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A fraction can name an integer too
Recognise numbers that can be written p/q.Different fractions, the same location
Value (number)
- Number-line marks · Only horizontal position represents the number.
- 1-0.75 · (-0.75, 0)
- 20 · (0, 0)
- 30.5 · (0.5, 0)
- 41 · (1, 0)
−6/8 and −3/4 coincide at −0.75; 1/2 and 2/4 coincide at 0.5.
Read the plotted values
| Value | Height |
|---|---|
| -0.75 | 0 |
| 0 | 0 |
| 0.5 | 0 |
| 1 | 0 |
- Rational number
- p/q, where p and q are integers and q ≠ 0.
5 = 5/1 and −2 = −2/1, so every integer is rational. Equivalent fractions name one position: 1/2 = 2/4 = 3/6.
Reduce a fraction by dividing its numerator and denominator by the same non-zero factor. A minus sign can sit in the numerator or denominator; two minus signs cancel.
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Question
Does −6/8 differ from −3/4?
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Use equal parts before adding
Choose the correct fraction operation.Two sixths plus one sixth
- Common denominator
- A shared denominator that puts fractions into equal-sized parts.
For 1/3 + 1/6, use sixths: 2/6 + 1/6 = 3/6 = 1/2. Adding the denominators would change the size of each part.
Multiply across for products. For division by a non-zero fraction, multiply by its reciprocal: (3/4) ÷ (1/2) = (3/4) × 2 = 3/2.
Subtract with a common denominator
5/8 − 1/4 = 5/8 − 2/8 = 3/8Convert quarters to eighths before subtracting.
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Question
Why is 1/2 + 1/2 equal to 1, rather than 2/4?
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Mark fractions between the integers
Locate a rational number and measure its distance from zero.Fractions have exact positions
Value (number)
- Number-line marks · Only horizontal position represents the number.
- 1-1.5 · (-1.5, 0)
- 2-1.25 · (-1.25, 0)
- 30 · (0, 0)
- 40.5 · (0.5, 0)
Equal subdivisions locate the fractions. Distance is horizontal separation.
Read the plotted values
| Value | Height |
|---|---|
| -1.5 | 0 |
| -1.25 | 0 |
| 0 | 0 |
| 0.5 | 0 |
- Absolute value |x|
- The non-negative distance of x from zero.
To mark −5/4, divide units into four equal parts and go five parts left from zero. It lies at −1.25, between −2 and −1.
The distance between a and b is |a − b|. It does not depend on which endpoint we name first.
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Question
What is the distance between −3/2 and 1/2?
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There is always another fraction between
Find a rational number between two different rational numbers.Zooming never runs out of fractions
Value (number)
- Number-line marks · Only horizontal position represents the number.
- 10.3333 · (0.33333, 0)
- 20.4167 · (0.41667, 0)
- 30.5 · (0.5, 0)
- 40.6667 · (0.66667, 0)
The positions are exact fractions; labels show rounded decimals only for readability.
Read the plotted values
| Value | Height |
|---|---|
| 0.3333333333333333 | 0 |
| 0.4166666666666667 | 0 |
| 0.5 | 0 |
| 0.6666666666666666 | 0 |
- Dense
- Between any two distinct rational numbers there is another rational number.
Between 1/3 and 2/3, their average is (1/3 + 2/3)/2 = 1/2. Repeat between 1/3 and 1/2 to get 5/12.
You can keep taking averages, so there are infinitely many rational numbers in that interval. This does not mean every point on the line is rational.
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Question
Find a rational number between 1/4 and 3/4.
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