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State Board · Class 6 · Mathematics

Lesson 3: In the World of Numbers

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

This lesson is also in: मराठी

Natural numbers and whole numbers

From daily life

We count fruits as 1, 2, 3, … and an empty basket has 0 fruits. Counting numbers and zero are the first numbers we learn.

Simple definition

The counting numbers 1, 2, 3, … are natural numbers (नैसर्गिक संख्या). Natural numbers together with 0 are whole numbers (पूर्ण संख्या).

Rule 1: Natural and whole numbers — Natural numbers: 1, 2, 3, … (never end; the smallest is 1). Whole numbers: 0, 1, 2, 3, … (the smallest is 0). Every natural number is a whole number, but 0 is a whole number that is not natural.

  • Easy : Easy: The smallest natural number is 1 and the smallest whole number is 0.
  • Medium : Medium: Is 0 natural? No. Is 0 whole? Yes.
  • Exam-type : Exam: 6 − 6 = 0 is a whole number but not a natural number.

Rule 2: Zero and its history — Zero was born in India. Outside India it was known as Void, Cipher, Zephyro and Zipra; Zephyro became "zero" in English. When 0 is put after a number, its value becomes 10 times: 7 → 70.

  • Easy : Easy: 9 with one zero after it becomes 90.
  • Medium : Medium: Tower heights are 120 m, 85 m and 64 m. The tallest is 120 − 64 = 56 m taller than the shortest.
  • Exam-type : Exam: Adding 0 on the right of 45 gives 450, which is 10 times 45.
Remember

Natural start at 1. Whole start at 0. Zero after a number makes it 10 times.

Common mistake

Calling 0 a natural number. It is a whole number only.

Shortcut

Memory clue: "नैसर्गिक = एक पासून; पूर्ण = शून्य पासून".

Spot the difference

SetStarts at
Natural numbers1
Whole numbers0

Practice questions

Easy
  1. 1What is the smallest natural number?
  2. 2What is the smallest whole number?
  3. 3Is 0 a natural number?
Medium
  1. 4Write the next three whole numbers after 7.
  2. 5What happens to 9 if we write a zero after it?
  3. 6Is every natural number a whole number?
Challenging
  1. 7Is 8 − 8 a natural number or a whole number?
  2. 8Where was zero born, and which name became "zero"?
Show answers (try them yourself first)
  1. 11.
  2. 20.
  3. 3No.
  4. 48, 9, 10.
  5. 5It becomes 90, which is 10 times 9.
  6. 6Yes.
  7. 70, a whole number but not natural.
  8. 8In India; Zephyro became "zero".

Large numbers: comparing, adding and subtracting

From daily life

The population of a state or the distance from the Earth to the Sun are very large numbers. We use the Indian system of lakhs and crores to read them easily.

Simple definition

In the Indian system, places go ones, tens, hundreds, thousands, ten thousands, lakhs (लाख), ten lakhs, crores (कोटी), ten crores. Commas are put after the first three digits from the right and then after every two digits.

Rule 1: Place value, reading and arranging digits — 3,05,40,208 is read three crore five lakh forty thousand two hundred eight. With digits 4, 0, 6, 8, 2 used once, the largest five-digit number is 86,420 and the smallest is 20,468 (a number cannot start with 0).

Place value chartPlace value chart of the Indian system for the number 20,61,84,350. ten crores: 2, crores: 0, ten lakhs: 6, lakhs: 1, ten thousands: 8, thousands: 4, hundreds: 3, tens: 5, ones: 0. Highlighted: crores.OnesThou.LakhsCroresTCr2Cr0TL6L1TTh8Th4H3T5O0

In the figure : In the figure the place value chart of 20,61,84,350, a 9-digit number. The highlighted 0 is in the crores place and 2 is in the ten crores place.

  • Easy : Easy: 4,56,000 is four lakh fifty-six thousand.
  • Medium : Medium: 7,08,09,010 is seven crore eight lakh nine thousand ten.
  • Exam-type : Exam: In 86,420 and 20,468 the digit in the hundreds place is 4, so its place value is 400 in both.

Rule 2: Adding, subtracting and comparing — 63,48,125 + 25,17,360 = 88,65,485 and 76,40,000 − 28,35,500 = 48,04,500. To order big numbers, compare the number of digits first, then digit by digit from the left: 61,98,76,000 < 62,48,91,720 < 62,84,15,306 < 63,05,12,448.

Place value chartPlace value chart of the Indian system for the number 9,?6,10,475 (a hidden digit is shown as ?). crores: 9, ten lakhs: ?, lakhs: 6, ten thousands: 1, thousands: 0, hundreds: 4, tens: 7, ones: 5.OnesThou.LakhsCroresTCrCr9TL?L6TTh1Th0H4T7O59,?6,10,475

In the figure : In the figure one digit of 9,?6,10,475 is hidden. It is in the ten lakhs place; if the number is between 9,30,00,000 and 9,40,00,000, the hidden digit is 3.

  • Easy : Easy: 45,000 + 28,000 = 73,000.
  • Medium : Medium: 63,48,125 + 25,17,360 = 88,65,485 (add lakhs with lakhs, thousands with thousands).
  • Exam-type : Exam: A shop sold goods worth ₹ 76,40,000 in one year and ₹ 28,35,500 in another. The difference is ₹ 48,04,500.
Live figure

Large numbers: comparing, adding and subtracting

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Remember

Indian commas: 3 digits, then every 2. Compare digit counts first. Line up places when adding.

Common mistake

Putting commas in threes everywhere, like 12,345,678. In India write 1,23,45,678.

Shortcut

Memory clue: "तीन, मग दोन-दोन" (three, then two-two).

Spot the difference

PlaceValue of 1
Lakh1,00,000
Ten lakh10,00,000
Crore1,00,00,000

Practice questions

Easy
  1. 1Write five crore three lakh in figures.
  2. 2Read 8,45,000 in words.
  3. 3Make the largest four-digit number using 3, 7, 1, 5 once.
Medium
  1. 4Add 36,25,000 + 14,75,000.
  2. 5Subtract 90,00,000 − 45,50,000.
  3. 6Arrange in ascending order: 7,45,300; 7,40,530; 7,54,030.
Challenging
  1. 7Using 6, 0, 3, 9, 1 once, find the largest and smallest five-digit numbers.
  2. 8In 9,?6,10,475 the number lies between 9,50,00,000 and 9,60,00,000. Find the hidden digit.
Show answers (try them yourself first)
  1. 15,03,00,000.
  2. 2Eight lakh forty-five thousand.
  3. 37531.
  4. 451,00,000.
  5. 544,50,000.
  6. 67,40,530 < 7,45,300 < 7,54,030.
  7. 7Largest 96,310; smallest 10,369.
  8. 8The hidden digit is in the ten lakhs place and must make the number start with 9,5…, so it is 5.

Integers and the number line

From daily life

In Manali the temperature at night can be −5 °C, and the lift in a mall can go to the basement floor (−1). Numbers below zero are needed to describe such things.

Simple definition

Numbers to the left of 0 (−1, −2, −3, …) are negative numbers (ऋण संख्या) and to the right (1, 2, 3, …) are positive numbers (धन संख्या). Zero, positive and negative numbers together are integers (पूर्णांक).

Rule 1: Comparing on the number line — A number on the right is always greater than a number on the left: −4 < −1 < 0 < 2. A (+) sign is not needed for positive numbers, but (−) is needed for negative numbers.

Number lineNumber line from −6 to 6 in steps of 1. Point K at −5. Point L at −1. Point M at 3.−6−5−4−3−2−10123456KLM

In the figure : In the figure K = −5, L = −1 and M = 3. A number on the right is greater, so −5 < −1 < 3. Always check how much one mark is worth.

  • Easy : Easy: −3 < 2 because −3 is on the left of 2.
  • Medium : Medium: −8 < −5, because −8 is farther to the left.
  • Exam-type : Exam: Ascending order of −7, 3, −2, 0, −9 is −9, −7, −2, 0, 3.

Rule 2: Integers and whole numbers — 7 − 3 = 4 is a whole number, but 3 − 7 = −4 is not a whole number, it is an integer. Real-life uses: temperature below zero, floors below ground, a well water level below the ground (+2 > −3).

  • Easy : Easy: Temperature −6 °C is colder than −2 °C.
  • Medium : Medium: Basement floors −1 and −2: floor −2 is lower.
  • Exam-type : Exam: A lift goes from floor 2 to floor −1: 2 − (−1) = 3 floors down.
Remember

Right is greater. Negative numbers are on the left of 0. Integers = negative, zero, positive.

Common mistake

Thinking −9 is greater than −2 because 9 is greater than 2. On the number line −9 is on the left.

Shortcut

Memory clue: "डावीकडे जाल तितकी लहान" (further left means smaller).

Spot the difference

TypeWhere on the number line
NegativeLeft of 0
ZeroIn the middle
PositiveRight of 0

Practice questions

Easy
  1. 1Which is greater, −2 or −6?
  2. 2Which is smaller, 0 or −4?
  3. 3Which of 5, −3, 2.5, 0 are integers?
Medium
  1. 4Arrange in ascending order: 2, −5, 0, −1.
  2. 5Is 3 − 8 a whole number?
  3. 6Temperatures are −6 °C and −2 °C. Which is warmer?
Challenging
  1. 7On a number line each mark is worth 2, and a point is 3 marks to the left of 0. What number is it?
  2. 8Why can't we say −9 > −2?
Show answers (try them yourself first)
  1. 1−2.
  2. 2−4.
  3. 35, −3 and 0.
  4. 4−5, −1, 0, 2.
  5. 5No, it is −5, an integer.
  6. 6−2 °C.
  7. 73 × 2 = 6 to the left, so −6.
  8. 8−9 lies to the left of −2 on the number line, so −9 < −2.

Opposite numbers and addition of integers

From daily life

Ravi gains ₹ 5 in one game and loses ₹ 5 in the next. His total is ₹ 0. A gain and a loss of the same amount cancel each other. Gains are positive and losses are negative.

Simple definition

Opposite numbers (विरुद्ध संख्या) like +4 and −4 are at the same distance from 0 on opposite sides. Their sum is 0.

Rule 1: Opposite numbers — The opposite of +6 is −6 and (−6) + (+6) = 0. Zero has no opposite number.

Number lineNumber line from −6 to 6 in steps of 1. A jump from −4 to 4 labelled +8. Point D at −4. Point E at 4.−6−5−4−3−2−10123456DE+8

In the figure : In the figure D = −4 and E = 4 are opposite numbers. The jump from D to E is 8 units long and crosses zero.

  • Easy : Easy: The opposite of +9 is −9.
  • Medium : Medium: (−7) + (+7) = 0.
  • Exam-type : Exam: If x + (−12) = 0, then x is the opposite of −12, so x = 12.

Rule 2: Adding integers by rules — Same signs: add the numbers and keep the sign: (+5) + (+2) = +7 and (−4) + (−3) = −7. Different signs: subtract the smaller number from the larger and keep the sign of the larger: (+6) + (−2) = +4 and (+2) + (−7) = −5.

Number lineNumber line from −6 to 6 in steps of 1. A jump from 1 to −3 labelled −4.−6−5−4−3−2−10123456−4

In the figure : In the figure, to add a negative number, jump to the left: 1 + (−4) = −3.

  • Easy : Easy: (−3) + (−4) = −7.
  • Medium : Medium: (+9) + (−4) = +5.
  • Exam-type : Exam: (+3) + (−8): 8 − 3 = 5 and 8 is larger and negative, so the answer is −5.

Rule 3: Several jumps — Positive numbers jump right and negative numbers jump left. (−5) + 7 = 2, then 2 + (−4) = −2.

Number lineNumber line from −6 to 6 in steps of 1. A jump from −4 to 2 labelled +6. A jump from 2 to −1 labelled −3.−6−5−4−3−2−10123456+6−3

In the figure : In the figure several jumps are made one after the other: (−4) + 6 = 2, then 2 + (−3) = −1.

  • Easy : Easy: 2 + (−5) = −3.
  • Medium : Medium: (−6) + 4 = −2.
  • Exam-type : Exam: (−6) + 9 + (−5) = 3 + (−5) = −2.
Live figure

Opposite numbers and addition of integers

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Remember

Opposites add to 0. Same signs: add. Different signs: subtract and take the sign of the larger.

Common mistake

Writing (−3) + (−2) = +5. Same negative signs give a negative answer: −5.

Shortcut

Memory clue: "सारख्या चिन्हांची बेरीज, वेगळ्यांची वजाबाकी".

Spot the difference

SignsExample
Same (+, +)(+5) + (+2) = +7
Same (−, −)(−4) + (−3) = −7
Different(+6) + (−2) = +4

Practice questions

Easy
  1. 1What is the opposite of −8?
  2. 2Find (−5) + (+5).
  3. 3Find (+6) + (+3).
Medium
  1. 4Find (−7) + (−2).
  2. 5Find (+8) + (−3).
  3. 6Find (+2) + (−9).
Challenging
  1. 7The temperature is −3 °C and it rises by 5 °C. What is the new temperature?
  2. 8Find (−6) + 9 + (−5).
Show answers (try them yourself first)
  1. 1+8.
  2. 20.
  3. 3+9.
  4. 4−9.
  5. 5+5.
  6. 6−7.
  7. 7(−3) + 5 = 2 °C.
  8. 8(−6) + 9 = 3, then 3 + (−5) = −2.

Subtraction of integers

From daily life

The temperature at night was −4 °C and in the afternoon it is 5 °C. How much did it rise? To find a change we subtract.

Simple definition

To subtract a number, add its opposite: a − b = a + (opposite of b).

Rule 1: Subtracting integers — (+7) − (+4) = +3. (−7) − (−4) = (−7) + (+4) = −3. (+5) − (−3) = (+5) + (+3) = 8. (−5) − (+3) = (−5) + (−3) = −8. More: 56 − (−24) = 80, (−30) − (−45) = 15, (−60) − 90 = −150, 120 − (−35) = 155.

  • Easy : Easy: 9 − 4 = 5.
  • Medium : Medium: (−8) − (−3) = (−8) + 3 = −5.
  • Exam-type : Exam: 6 − (−9) = 6 + 9 = 15. The night-to-afternoon rise is 5 − (−4) = 9 °C.

Rule 2: Word problems and a little history — A submarine at −30 m rises to −12 m: change = (−12) − (−30) = 18 m up. Kautilya's Arthashastra records negative numbers in accounts. The Bakhshali manuscripts show them with a special symbol. Brahmagupta's Brahmasphuta Siddhanta first explained operations on positive numbers, negative numbers and zero.

  • Easy : Easy: Temperature −2 °C in the morning and 6 °C at noon: rise = 6 − (−2) = 8 °C.
  • Medium : Medium: A diver at −15 m goes to −40 m: change = (−40) − (−15) = −25, which is 25 m down.
  • Exam-type : Exam: Kaprekar's constant 6174: from 4321, 4321 − 1234 = 3087; 8730 − 0378 = 8352; 8532 − 2358 = 6174.
Remember

Subtract = add the opposite. Change = final − starting.

Common mistake

Writing 5 − (−3) = 2. It is 5 + 3 = 8.

Shortcut

Memory clue: "वजा आणि वजा = मिळवा" (minus a minus is plus).

Spot the difference

SubtractionBecomes
5 − (+3)5 + (−3) = 2
5 − (−3)5 + (+3) = 8
(−5) − (−3)(−5) + (+3) = −2

Practice questions

Easy
  1. 1Find 9 − (+4).
  2. 2Find (+3) − (−5).
  3. 3Find (−4) − (−6).
Medium
  1. 4Find (−7) − (+2).
  2. 5Find 0 − (−8).
  3. 6Find (−12) − (−12).
Challenging
  1. 7The temperature falls from 4 °C to −3 °C. What is the change?
  2. 8Start with 5432: find 5432 − 2345, then continue the Kaprekar steps (8730 − 0378, 8532 − 2358).
Show answers (try them yourself first)
  1. 15.
  2. 23 + 5 = 8.
  3. 3(−4) + 6 = 2.
  4. 4(−7) + (−2) = −9.
  5. 58.
  6. 60.
  7. 7(−3) − 4 = −7, a fall of 7 °C.
  8. 83087 → 8352 → 6174.

Quick revision

  • Natural numbers start from 1; whole numbers start from 0; integers include negative numbers.
  • The sum of opposite numbers is zero.
  • Subtracting a number means adding its opposite.