CLASS 8 MATHS · PART 1 · CHAPTER 3 · STUDY NOTES
A Story of Numbers
Understand a story 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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Match one object to one mark
Counting works only if each object gets exactly one mark or number name.Every apple gets one counter
- One-to-one matching
- Pairing each object in one collection with exactly one object in another, with none left over.
- Numeral
- A written representation of a number.
You can compare two collections without naming a number: pair their objects. Any unpaired objects show which collection is larger.
Counting needs a fixed order of number names, one name per object and no skipped or repeated objects. The last name tells the total.
Take a moment. What do you think?
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Question
Four balls are paired with five marks. What does the extra mark tell you?
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Tallies record one item at a time
A mark can remain after the object has gone. Groups of five make a tally easier to read.Group the marks so the total is visible
- Tally
- A record with one mark for each counted item.
Seven marks can be organised as a group of five and two more. This is more readable than one long ungrouped row.
People have used body positions, sticks, notches and spoken names for counting. An ancient marked object can be evidence of counting, but its exact purpose may be uncertain.
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Question
Make a tally for 12, then explain how you read it quickly.
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Grouping reduces repetition
Instead of writing one mark for every object, give a group its own name or symbol.Five pairs and one left over
- Group size
- The number of objects represented by one group.
A total of 11 can be described as five pairs and one more: 2 + 2 + 2 + 2 + 2 + 1.
Different communities developed different ways to group and name numbers. Similar patterns alone do not prove that communities had contact.
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Question
Why is grouping by tens useful when counting 83 beads?
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Roman symbols name landmark numbers
Combine familiar symbols, while following the permitted subtraction rules.Add the values in LXVIII
- Landmark number
- An easily recognised reference value with its own symbol.
- Roman numeral
- A representation using I, V, X, L, C, D and M for 1, 5, 10, 50, 100, 500 and 1000.
| Symbol | Value |
|---|---|
| I | 1 |
| V | 5 |
| X | 10 |
| L | 50 |
| C | 100 |
| D | 500 |
| M | 1000 |
LXVIII means 50 + 10 + 5 + 1 + 1 + 1 = 68. In IV and IX, I before V or X is subtracted.
Modern standard forms use IV, IX, XL, XC, CD and CM as subtraction pairs. Roman writing is not a decimal place-value system: moving I does not turn it into a digit worth ten times as much.
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Question
Write 94 using standard Roman notation.
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Egyptian symbols show powers of ten without place value
Each symbol has its own value wherever it appears. Repeat it to show how many groups you have.Picture from the NCERT book
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- Additive numeral system
- A system in which the values of symbols are added to form a number.
- Decimal base
- Grouping with powers of 10: 1, 10, 100, 1000 and so on.
For 243, use two hundred-symbols, four ten-symbols and three one-symbols. A hundred-symbol is not the digit 2; it contributes 100 each time.
Ten symbols of one value can be exchanged for one symbol of the next value. This is the grouping idea behind carrying in decimal addition.
- 1
For 243, remove two groups of 100: 243 − 200 = 43.
- 2
Remove four groups of 10: 43 − 40 = 3.
- 3
The remaining three units give 2 hundreds, 4 tens and 3 ones.
Let’s look a little closer.
Go deeper: Why the base helps arithmetic
In base 10, 10² × 10³ = 10⁵, another landmark value. Products can be distributed across groups. A calculating board can also use columns or lines for these powers; exchanging ten counters for one counter in the next place is carrying.
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Question
If an additive decimal representation has twelve ten-symbols and six one-symbols, how can it be regrouped?
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A base tells you when to exchange a group
In base 5, five units become one five; five fives become one twenty-five.Exchange five units for the next group
- Base b
- For an integer b ≥ 2, a place-value system with place values 1, b, b², … and digits 0 to b − 1.
The base-5 places are 1, 5, 25, 125, … . The ordinary number 38 equals 1 × 25 + 2 × 5 + 3, so it is written 123₅.
The subscript tells you which base is being used. 123₅ is not the same value as decimal 123.
Binary uses only 0 and 1
13 = 8 + 4 + 1 = 1101₂The positions are 8, 4, 2 and 1. Zero keeps the empty twos place.
- 1
Find the largest base-5 place that fits 38: 25.
- 2
Use one 25, leaving 13.
- 3
Use two 5s, leaving 3.
- 4
Read the coefficients as 123₅.
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Question
What is 24₅ in decimal?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 15, 16, 17, 18, 19, 20, 23, 33.
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Place value lets the same symbol do different jobs
A symbol group in the sixties position contributes sixty times its face value.Two sixty-groups, then eight units
- Positional system
- A numeral system in which a symbol’s value depends on its position.
- Sexagesimal
- Using base 60.
Use decimal labels for the positions to make the idea clear: [2, 8]₆₀ means 2 × 60 + 8 = 128. Each position must contain a value from 0 to 59.
Mesopotamian writing used groups of wedge symbols. Unmarked empty positions could be ambiguous; later placeholder signs reduced some of that ambiguity. Our bracket notation is a modern teaching aid, not an ancient inscription.
Go deeper: An influence still visible
Time uses 60 seconds in a minute and 60 minutes in an hour. This does not mean that an entire modern clock is a pure base-60 number system.
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Question
Read [1, 3, 5]₆₀ using place values 3600, 60 and 1.
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Maya dots and bars show place value in a different direction
A dot is 1, a bar is 5, and a shell-like zero marks an empty position.Read the stacked calendar positions
- Placeholder
- A symbol that keeps an empty position visible.
Two bars and three dots represent 13. Stack different positions vertically, with the smallest place at the bottom.
The calendar version shown in this chapter uses places 1, 20, 360, 7200, … . Its third place is 18 × 20, not 20². Distinguish this Long Count calendar convention from pure base-20 arithmetic.
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Question
In the chapter’s calendar convention, what does one dot in the 360 place, two dots in the 20 place and a zero in the units place represent?
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Rod numerals alternate direction to keep neighbours apart
In the rod system, place value is decimal, while the orientation of neighbouring digit patterns alternates.Orientation changes; place value remains decimal
- Rod numeral
- A number pattern made with counting rods, used in historical Chinese calculation.
Vertical patterns appear in units, hundreds and other alternating positions; horizontal patterns appear in tens, thousands and the positions between.
A blank place can represent no rods, but a written zero makes an empty place easier to preserve. Alternating direction helps avoid merging adjacent rod patterns.
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Question
Why could adjacent groups of upright rods be hard to read if no gap or alternating orientation were used?
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Zero keeps the place and works as a number
Ten digit symbols, place value and zero work together to make our everyday decimal notation compact.Leave the tens place empty, not missing
- Zero as a digit
- A symbol marking an empty place in a numeral.
- Zero as a number
- The value 0, which can participate in arithmetic.
504 means 5 × 100 + 0 × 10 + 4. Removing the zero gives 54, a different number.
The decimal place-value tradition developed in India and spread through scholars in the Islamic world to other regions. Digit shapes changed over time; this was not a single overnight invention.
Zero follows arithmetic rules
a + 0 = a; a × 0 = 0Division by zero is undefined. Zero is not merely a decorative placeholder.
Go deeper: Ideas, not a ladder of cultures
This chapter compares mathematical ideas, not a strict chronological ranking of communities. Counting systems developed in different contexts and periods. A place-value system needs reliable positions and a way to mark empty ones; knowing that explains its usefulness.
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Question
What is the difference between the zero in 205 and the zero in 9 + 0 = 9?
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