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CBSE · Class 8 · Mathematics Part 1

Part 1, Chapter 3: A Story of Numbers

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Counting, tally marks and numerals

From daily life

A shopkeeper counts sacks by drawing one stroke for each sack and crossing every fifth stroke over four others. At the end he counts the bundles of five and then the extra strokes.

Simple definition

Tally marks use one stroke for each object, grouped in fives. A numeral is a symbol for a number: 7, VII and "seven" are three numerals for one number. In an additive system we add the values of the symbols, and their order does not matter.

In the figure: 3 bundles of five and 2 strokes: 3 × 5 + 2 = 17.

Rule 1: Read a tally — Total = 5 × (bundles) + (single strokes).

  • Easy: 3 bundles and 2 strokes: 15 + 2 = 17.
  • Medium: 6 bundles and 4 strokes: 30 + 4 = 34.
  • Exam-type: 7 bundles + 3 strokes, then 2 bundles + 4 strokes: 38 + 14 = 52.

Rule 2: Number or numeral? — The number is the idea of "how many"; a numeral is only the way we write it.

  • Easy: 7, VII and "seven" are numerals of one number.
  • Medium: XII, 12 and "twelve" all name the same number.
  • Exam-type: Changing the numeral never changes the number: 1 + 1 = 2 = II.

Rule 3: Additive symbols — Let ▲ = 1, ● = 10 and ■ = 100. Add the values of all the symbols; the order does not matter.

  • Easy: ■■●▲ = 200 + 10 + 1 = 211.
  • Medium: ■●●●●▲▲▲ = 100 + 40 + 3 = 143 (any order).
  • Exam-type: 1048 needs 10 ■, 4 ● and 8 ▲ = 22 symbols.
Remember

Tally marks suit small counts. For big counts we need symbols for bigger groups (10, 100, ...), which is how numeral systems began.

Common mistake

Multiplying the symbols: ▲● is 1 + 10 = 11, not 1 × 10 = 10.

Shortcut

Count the bundles of five first, then add the loose strokes. Memory clue: "Fives first, singles later".

Spot the difference

PointTally marksAdditive symbols
Symbols usedonly one strokedifferent symbols for 1, 10, 100
Writing 100100 strokes (20 bundles)1 symbol ■
Does order matter?nono

Practice questions

Easy
  1. 1A tally shows 4 bundles of five and 3 strokes. How many?
  2. 2Find the value of ■■▲▲▲ (▲ = 1, ● = 10, ■ = 100).
  3. 338 pebbles are grouped in tens. How many tens and how many ones?
Medium
  1. 49 bundles of five and 1 stroke: how many?
  2. 5How many symbols (▲ ● ■) are needed to write 250 with the fewest symbols?
  3. 6Are ▲▲●■ and ■●▲▲ equal? What is their value?
Challenging
  1. 7A herder records 8 bundles + 2 strokes in the morning and 3 bundles + 4 strokes in the evening. Total animals?
  2. 8Using at most 9 of each symbol (▲ ● ■), what is the greatest number that can be written?
Show answers (try them yourself first)
  1. 123
  2. 2203
  3. 33 tens and 8 ones
  4. 446
  5. 57 (2 ■, 5 ●)
  6. 6Yes, both are 112
  7. 742 + 19 = 61
  8. 8999

Roman numerals

From daily life

Clock faces, book chapters and the film years at the end of a movie often use Roman numerals: XII on a clock, "Chapter IV" in a book.

Simple definition

Roman symbols: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000. A smaller symbol AFTER a larger one is added (VI = 6); a smaller symbol BEFORE a larger one is subtracted (IV = 4). There is no zero and no place value.

Live figure

Roman numerals

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Rule 1: Reading a Roman numeral — Go from left to right: add each symbol, but subtract it if the next symbol is bigger.

  • Easy: XXVII = 10 + 10 + 5 + 1 + 1 = 27.
  • Medium: XLIV = (50 − 10) + (5 − 1) = 44.
  • Exam-type: CMXCIV = 900 + 90 + 4 = 994.

Rule 2: Repeating and subtracting limits — I, X, C, M repeat at most 3 times; V, L, D never repeat. Only I, X, C can be subtracted, and only from the next two bigger symbols.

  • Easy: III = 3 is allowed but IIII is not; 4 is IV.
  • Medium: VX is wrong; 5 is simply V.
  • Exam-type: 99 is XCIX, not IC.

Rule 3: Writing a number in Roman — Split the number into thousands, hundreds, tens and ones and write each part.

  • Easy: 38 = 30 + 8 = XXX + VIII = XXXVIII.
  • Medium: 49 = 40 + 9 = XL + IX = XLIX.
  • Exam-type: 2049 = 2000 + 40 + 9 = MM + XL + IX = MMXLIX.
Remember

The biggest number with symbols up to M and no subtraction trick is MMMDCCCLXXXVIII = 3888 in the ordinary rules. Roman numbers are hard to add or multiply because the symbols have no place value.

Common mistake

Writing 4 as IIII or 9 as VIIII. Correct: 4 = IV and 9 = IX. Also, there is no Roman symbol for zero.

Shortcut

Learn the order I V X L C D M with the sentence "I Value Xylophones Like Cool Drinks Mildly". Memory clue: "I Value Xylophones Like Cool Drinks Mildly".

Spot the difference

PointRoman numeralsHindu-Arabic numerals
Zerononepresent
Place valuenoyes
49 is writtenXLIX49
Calculationhardeasy by columns

Practice questions

Easy
  1. 1Read XXIX.
  2. 2Write 18 in Roman numerals.
  3. 3Which can be repeated, V or X?
Medium
  1. 4Write 64 in Roman numerals.
  2. 5Read CDXL.
  3. 6XIX + XXVI = ? (answer in Roman).
Challenging
  1. 7Write 1999 in Roman numerals.
  2. 8Write 3888 in Roman numerals.
Show answers (try them yourself first)
  1. 129
  2. 2XVIII
  3. 3X (V never repeats)
  4. 4LXIV
  5. 5440
  6. 6XLV
  7. 7MCMXCIX
  8. 8MMMDCCCLXXXVIII

Number systems with different bases

From daily life

A shopkeeper packs eggs in trays of 6. For 40 eggs he gets 6 full trays and 4 loose eggs. Counting in groups of 6 instead of 10 is thinking in base 6.

Simple definition

In base b we use the digits 0 to b − 1, and the place values from the right are 1, b, b × b, ... The value of the number is the sum of (digit × place value). We write the base below the number: 1101₂.

16180412110

In the figure: 10110₂ = 16 + 4 + 2 = 22.

Rule 1: Other base to decimal — Multiply each digit by its place value and add.

  • Easy: 1101₂ = 8 + 4 + 0 + 1 = 13.
  • Medium: 243₅ = 2 × 25 + 4 × 5 + 3 = 73.
  • Exam-type: 345₈ = 3 × 64 + 4 × 8 + 5 = 229.

Rule 2: Decimal to another base — Divide by the base again and again; the remainders read from bottom to top are the digits.

  • Easy: 19 = 10011₂ (19 = 16 + 2 + 1).
  • Medium: 87 = 322₅ (87 = 3 × 25 + 2 × 5 + 2).
  • Exam-type: 100 = 144₈ (100 = 64 + 32 + 4).

Rule 3: Adding in binary — In base 2, 1 + 1 = 10 (write 0, carry 1). Add column by column from the right.

  • Easy: 11₂ + 10₂ = 101₂ (3 + 2 = 5).
  • Medium: 101₂ + 11₂ = 1000₂ (5 + 3 = 8).
  • Exam-type: 1011₂ + 110₂ = 10001₂ (11 + 6 = 17).
Remember

Base b never uses the digit b: base 5 has only 0 to 4. Computers use base 2 because a switch has two states, off (0) and on (1).

Common mistake

Reading 101₂ as "one hundred one" (it is 5) or using the digit 5 in base 5.

Shortcut

Read binary from the left: start with 0, double it and add the next bit, and repeat. Memory clue: "Double and add".

Spot the difference

PointBase 2Base 10
Digits used0, 10 to 9
Place values1, 2, 4, 8, ...1, 10, 100, ...
Meaning of 10111011₂ = 11one thousand eleven

Practice questions

Easy
  1. 1Write 101₂ as a decimal number.
  2. 2How many digits does base 4 use?
  3. 3Write 10 in base 2.
Medium
  1. 4Write 321₄ as a decimal number.
  2. 5Write 45 in base 6.
  3. 6Find 111₂ + 1₂ in base 2.
Challenging
  1. 7Write 2102₃ as a decimal number.
  2. 8What is the smallest 4-digit number in base 5, written in decimal?
Show answers (try them yourself first)
  1. 15
  2. 24 (0, 1, 2, 3)
  3. 31010₂
  4. 457
  5. 5113₆
  6. 61000₂
  7. 765
  8. 8125 (that is 1000₅)

Place-value systems (base 60 and base 20)

From daily life

Look at a clock: 3725 seconds is 1 hour 2 minutes 5 seconds. We count time in groups of 60, a habit inherited from the ancient Babylonians who used base 60.

Simple definition

In a place-value system the value of a digit = digit × value of its place. The Babylonians used base 60, the Maya base 20 and the Indians base 10. An empty place needs a marker (zero), otherwise numbers get confused.

36002601015

In the figure: (2, 10, 5) = 2 × 3600 + 10 × 60 + 5 = 7805.

Rule 1: Face value and place value — Face value is the digit itself; place value = digit × value of its place.

  • Easy: In 5372, the 3 has face value 3 and place value 300.
  • Medium: In 40,906, the 9 has place value 900.
  • Exam-type: In 7,070 the two 7s have place values 7000 and 70, differing by 6930.

Rule 2: Reading base 60 — Place values from the right: 1, 60, 3600, ...

  • Easy: (2, 10) = 2 × 60 + 10 = 130.
  • Medium: (4, 15, 30) = 4 × 3600 + 15 × 60 + 30 = 15330.
  • Exam-type: 5000 seconds = 1 h 23 min 20 s, because 3600 + 23 × 60 + 20 = 5000.

Rule 3: Reading base 20 — Place values from the right: 1, 20, 400, 8000, ...

  • Easy: (3, 12) = 3 × 20 + 12 = 72.
  • Medium: (2, 5, 7) = 2 × 400 + 5 × 20 + 7 = 907.
  • Exam-type: (6, 0, 3) = 6 × 400 + 0 × 20 + 3 = 2403.
Remember

Because 60 has 12 factors, halves, thirds, quarters, fifths and sixths of 60 are all whole numbers, which made base 60 handy for measuring time and angles.

Common mistake

Reading the base-60 group (1, 5) as 15. Correct: 1 × 60 + 5 = 65. And (1, 0, 5) = 3605 is very different.

Shortcut

Start from the left: multiply by the base, then add the next digit. For (4, 15, 30): 4 × 60 + 15 = 255, then 255 × 60 + 30 = 15330. Memory clue: "Multiply by base, then add".

Spot the difference

PointRoman numeralsPlace-value systems
Does position decide value?noyes
Need a zero or empty marker?noyes
Two equal symbolsXX: both X are 1022: the 2s are 20 and 2

Practice questions

Easy
  1. 1What is the place value of 4 in 8,431?
  2. 2Find (1, 30) in base 60.
  3. 3Find (2, 3) in base 20.
Medium
  1. 4Find (3, 0, 2) in base 60.
  2. 5Write 5000 seconds as hours, minutes, seconds.
  3. 6Find (1, 10, 0) in base 20.
Challenging
  1. 7In 5,050, by how much do the place values of the two 5s differ?
  2. 8Without a zero, (1, 0, 5) and (1, 5) could be confused in base 60. Find how far apart their values are.
Show answers (try them yourself first)
  1. 1400
  2. 290
  3. 343
  4. 410802
  5. 51 h 23 min 20 s
  6. 6600
  7. 74950
  8. 83605 − 65 = 3540

Hindu-Arabic system and zero

From daily life

A price tag reads ₹12,34,500. In India we group digits as lakh and crore; in many other countries the same amount is written 1,234,500. The ten digits 0 to 9 and the idea of zero came from India and spread through the Arabs to the world.

Simple definition

The Hindu-Arabic system is a base-10 place-value system with ten digits 0 to 9. Zero works as a placeholder AND as a number (a + 0 = a, a − a = 0, a × 0 = 0).

crore1ten lakh2lakh310 thou4thou5hundred0ten0one8

In the figure: Indian grouping: 1,23,45,008 = 1 crore 23 lakh 45 thousand 8.

Rule 1: Expanded form and place value — Write each digit times its place value; empty places (zeros) add nothing.

  • Easy: 3058 = 3 × 1000 + 5 × 10 + 8.
  • Medium: 40706 = 4 × 10000 + 7 × 100 + 6.
  • Exam-type: In 83,50,412 the ten-lakhs digit is 8 and the lakhs digit is 3.

Rule 2: Indian and international commas — Indian: first comma after 3 digits from the right, then after every 2. International: after every 3.

  • Easy: 12345678 is 1,23,45,678 (Indian) and 12,345,678 (international).
  • Medium: 1 crore = 100 lakh = 10 million.
  • Exam-type: 4 crore 7 lakh 2 thousand 9 = 4,07,02,009.

Rule 3: Making greatest and smallest numbers — Greatest: digits from big to small. Smallest: the smallest non-zero digit first, then the rest in increasing order.

  • Easy: Digits 3, 0, 7, 5, 1: greatest 75310.
  • Medium: Digits 3, 0, 7, 5, 1: smallest 10357.
  • Exam-type: Digits 4, 0, 9, 2, 0, 6: greatest 964200 and smallest 200469.
Remember

Removing or adding a zero changes the number a lot: 3405 without its zero is 345, a loss of 3060.

Common mistake

Putting Indian commas in groups of 3 (12,345,678) and calling it Indian. Correct Indian form: 1,23,45,678.

Shortcut

Indian commas: from the right, "Three, then two, two, two". Memory clue: "Three, then two-two-two".

Spot the difference

PointIndian systemInternational system
Grouping3, 2, 2, ...3, 3, 3, ...
One hundred thousand1,00,000 (1 lakh)100,000
Ten million1,00,00,000 (1 crore)10,000,000

Practice questions

Easy
  1. 1What is the place value of 7 in 5,07,324?
  2. 2Read 3,05,000 in words.
  3. 31 lakh equals how many thousands?
Medium
  1. 4Write 6 crore 3 lakh 8 with Indian commas.
  2. 5Write 25,00,000 with international commas.
  3. 6Greatest 4-digit number from the digits 4, 0, 8, 2 (each once).
Challenging
  1. 7Find 9,99,999 + 11.
  2. 8Write the smallest 7-digit number with Indian commas and name it.
Show answers (try them yourself first)
  1. 17,000
  2. 2Three lakh five thousand
  3. 3100
  4. 46,03,00,008
  5. 52,500,000
  6. 68420
  7. 710,00,010
  8. 810,00,000 (ten lakh)