CBSE Class 6 Maths Advanced Knowing Our Numbers Notes Set 01

Knowing Our Numbers

Numbers

Natural numbers. The numbers we use for counting, 1, 2, 3, 4, 5, 6 and so on, are called natural numbers. For example, if a picture shows 1 bird, 2 persons and 3 trees, the counts 1, 2 and 3 are natural numbers.

Whole numbers. When 0 is added to the natural numbers, we get the whole numbers 0, 1, 2, 3, 4, ... Every natural number is a whole number, but 0 is a whole number that is not a natural number.

Predecessor and successor. The predecessor of a number is the number just before it (1 less). The successor is the number just after it (1 more). For example, the predecessor of 59028 is 59027, and the successor of 9999 is 10000.

Comparison of Numbers

1. The number with more digits is greater.
2. If two numbers have the same number of digits, compare them from the left. Find the first place where the digits differ; the number with the larger digit there is greater.

Illustration 1.1. Which is greater of 270346 and 48356?
Answer:
270346 has 6 digits and 48356 has 5 digits.
The number with more digits is greater, so 270346 > 48356.

Illustration 1.2. Find the greatest and the smallest numbers from the following group of numbers: 23787, 6895, 24569, 24659
Answer:
6895 has only 4 digits, so it is the smallest. The other numbers have 5 digits; comparing from the left, 24659 is the largest.
Greatest number: 24659. Smallest number: 6895.

Ascending and Descending Order

Ascending order means arranging numbers from the smallest to the greatest (like climbing up a ladder). Descending order means arranging them from the greatest to the smallest (like climbing down).

Illustration 1.3. Arrange the following numbers in ascending order: 257536, 38952, 385081, 365062
Answer:
38952 has the fewest digits, so it is the smallest. Of the 6-digit numbers, 257536 < 365062 < 385081.
Ascending order: 38952, 257536, 365062, 385081.

Illustration 1.4. Arrange the following numbers in descending order: 19710, 887151, 453212, 925473
Answer:
The greatest is 925473, followed by 887151 and 453212. The 5-digit number 19710 is the smallest.
Descending order: 925473, 887151, 453212, 19710.

Face Value and Place Value

Place value is the basis of our number system. The face value of a digit is the digit itself, wherever it stands in the number. The place value of a digit is its value because of the place it occupies:

Place value of a digit = face value of the digit × value of its place.

The face value of a digit never changes, but its place value changes with its position. Each place is worth ten times the place on its right. For example, in 5555 the place values of the four 5s, from the right, are 5, 50, 500 and 5000, while the face value is always 5. For the number 53,694:

Expanded Form

Writing a number as the sum of the place values of its digits gives its expanded form.

Illustration 1.5. Express (i) 3,64,029 (ii) 2,75,00,386 in expanded form.
Answer:
(i) 3,64,029 = 3 × 100000 + 6 × 10000 + 4 × 1000 + 0 × 100 + 2 × 10 + 9 × 1.
(ii) 2,75,00,386 = 2 × 10000000 + 7 × 1000000 + 5 × 100000 + 0 × 10000 + 0 × 1000 + 3 × 100 + 8 × 10 + 6 × 1.

Take the number 2,45,13,768 and put it in a place-value chart (C = crore, TL = ten lakh, L = lakh, T-Th = ten thousand, Th = thousand, H = hundred, T = ten, O = one):

Place values: 2 is 2,00,00,000 (two crore); 4 is 40,00,000 (forty lakh); 5 is 5,00,000 (five lakh); 1 is 10,000 (ten thousand); 3 is 3,000 (three thousand); 7 is 700 (seven hundred); 6 is 60 (sixty); 8 is 8 (eight).

In words: two crore forty-five lakh thirteen thousand seven hundred sixty-eight. In figures: 2,45,13,768. In expanded notation: 2,00,00,000 + 40,00,000 + 5,00,000 + 10,000 + 3,000 + 700 + 60 + 8.

Illustration 1.6. Write the following in expanded form and write the place value of the digit 3 in each case. (a) 6307825 (b) 80930090
Answer:
(a) 6307825 = 6000000 + 300000 + 7000 + 800 + 20 + 5. The place value of 3 is 300000.
(b) 80930090 = 80000000 + 900000 + 30000 + 90. The place value of 3 is 30000.

Illustration 1.7. Write the short form of 8000000 + 60000 + 500 + 30 + 9
Answer:
Put each part under its place (T-L, L, T-Th, Th, H, T, O) and use 0 for empty places:
8000000 → 8 0 0 0 0 0 0; 60000 → 0 6 0 0 0 0 (T-Th = 6); 500 → 5 in the hundreds place; 30 → 3 in the tens place; 9 → 9 in the ones place.
Short form: 8060539.

Indian and International Systems of Numeration

A newspaper may say that Rs. 2500 crore has been given for building national highways, while another may say Rs. 25 billion. Both are the same amount, written in the Indian and the International systems, so we need to know both.

Indian system (periods: ones, thousands, lakhs, crores):

International system (periods: ones, thousands, millions, billions):

Reading and Writing Numbers

Indian system: start from the right. The first group (ones period) has 3 digits, and every next group (thousands, lakhs, crores) has 2 digits. Put a comma between the groups.

International system: make groups of 3 digits starting from the right and separate them with commas.

Example: the six-digit number 764325. In the Indian system 7 is in the lakh place, so it is written 7,64,325 and read as seven lakh sixty-four thousand three hundred twenty-five. In the International system it is written 764,325 and read as seven hundred sixty-four thousand three hundred twenty-five.

Illustration 1.8. Write the name of the numbers according to Indian system of numeration. (a) 1275834 (b) 30870209
Answer:
(a) 12,75,834 = Twelve lakh seventy-five thousand eight hundred thirty-four.
(b) 3,08,70,209 = Three crore eight lakh seventy thousand two hundred nine.

Illustration 1.9. Write the name of the number according to International system of numeration. (a) 7452283 (b) 48049831 (c) 699985102
Answer:
(a) 7,452,283 = Seven million four hundred fifty-two thousand two hundred eighty-three.
(b) 48,049,831 = Forty-eight million forty-nine thousand eight hundred thirty-one.
(c) 699,985,102 = Six hundred ninety-nine million nine hundred eighty-five thousand one hundred two.

Making Numbers

Without repeating digits: build the number from the left. For the greatest number put the largest digit first, then the next largest, and so on. For the smallest number put the smallest digit first, but 0 cannot be the first digit, so use the smallest non-zero digit first and then the remaining digits in ascending order.

Illustration 1.10. Write the greatest and the smallest 5-digit numbers by using each of the digits 8, 4, 7, 0, 2 only once.
Answer:
Greatest: arrange in descending order: 8, 7, 4, 2, 0 gives 87420.
Smallest: 0 cannot be first, so start with 2, then 0, then 4, 7, 8: 20478.

Illustration 1.11. Make the greatest and the smallest 5-digit numbers using any five different digits with 4 in the tens place.
Answer:
Fix 4 in the tens place. Greatest: 9 in the ten-thousands place, then 8, 7 and the largest left for the ones: 9, 8, 7, 4, 6 gives 98746.
Smallest: 1 in the ten-thousands place (0 cannot be first), 0 in the thousands, 2 in the hundreds, 4 in the tens, 3 in the ones: 10243.

With repeating digits: when a digit has to be repeated, it is easier to build the number from the right. For the greatest number the greatest digit is repeated; for the smallest number the smallest digit is repeated.

Illustration 1.12. Write the greatest and smallest numbers of 4 digits using all the digits 8, 0, 5.
Answer:
Greatest: the three digits are used and the greatest digit 8 is repeated: 8, 8, 5, 0 gives 8850.
Smallest: the smallest digit 0 is repeated and the first digit cannot be 0: 5, 0, 0, 8 gives 5008.

Quick rules: for the greatest number the greatest digit is repeated, and for the smallest number the smallest digit is repeated.

Estimation

News reports often use rounded numbers: '50,000 people took part in the marathon' or 'a loss of 80 crore rupees' do not give exact counts. Rounding is an approximation: when we say 50,000 people, the crowd was really somewhere between 45,000 and 55,000. Rounding helps us get approximate answers in addition, subtraction, multiplication and division. For example, if a DVD player costs Rs. 3847 and a TV costs Rs. 8348, you can say together they cost about Rs. 12,000. Planning food for a party, buying cement for a wall and paint for a house are all everyday uses of estimation.

Rounding to the nearest ten. Look at the ones digit. If it is 5 or more, increase the tens digit by 1 and make the ones digit 0. If it is less than 5, keep the tens digit and make the ones digit 0.

Rounding to the nearest hundred. Look at the tens digit. If it is 5 or more, increase the hundreds digit by 1; otherwise keep it. In both cases the tens and ones digits become 0.

Rounding to the nearest thousand. Look at the hundreds digit. If it is 5 or more, increase the thousands digit by 1; otherwise keep it. All the digits to the right become 0.

In general, to round to a place, look at the digit just to its right: 5 or more rounds up, less than 5 rounds down.

Estimation in Sums and Differences

Illustration 1.13. Estimate and compare with the actual sum. (a) 730 + 998 (b) 12904 + 2888
Answer:
(a) Round to the nearest hundred: 730 ≈ 700 and 998 ≈ 1000. Estimated sum = 700 + 1000 = 1700. Actual sum = 730 + 998 = 1728. The estimate is quite reasonable.
(b) Round to the nearest thousand: 12904 ≈ 13000 and 2888 ≈ 3000. Estimated sum = 16000. Actual sum = 12904 + 2888 = 15792. The estimate is reasonable.

Illustration 1.14. Give a rough estimate and also a close estimate of 439 + 334 + 4317
Answer:
Rough estimate: round each number to its greatest place: 400 + 300 + 4000 = 4700.
Close estimate: round each number to the nearest hundred: 439 ≈ 400, 334 ≈ 300, 4317 ≈ 4300. Sum = 400 + 300 + 4300 = 5000.

Illustration 1.15. Estimate 8325 − 491
Answer:
Rounding to thousands: 8325 ≈ 8000 and 491 ≈ 0, giving 8000, which is not a reasonable estimate.
Rounding to hundreds instead: 8325 ≈ 8300 and 491 ≈ 500, so the estimate is 8300 − 500 = 7800, which is reasonable.

Estimation in Products

Rules: (i) Round each factor to its greatest place. A 2-digit factor is rounded to the nearest ten, a 3-digit factor to the nearest hundred, and so on. (ii) Do not round a 1-digit factor.

Illustration 1.16. Estimate the following products: (a) 87 × 313 (b) 9 × 795 (c) 898 × 785
Answer:
(a) 87 ≈ 90 and 313 ≈ 300, so the product ≈ 90 × 300 = 27000.
(b) 9 is a one-digit number and is not rounded; 795 ≈ 800. The product ≈ 9 × 800 = 7200.
(c) 898 ≈ 900 and 785 ≈ 800, so the product ≈ 900 × 800 = 720000.

Estimation in Quotients

To estimate a quotient, round both the dividend and the divisor first, then divide.

Illustration 1.17. Estimate the following quotients: (a) 81 ÷ 17 (b) 7525 ÷ 365
Answer:
(a) 81 ≈ 80 and 17 ≈ 20. Divide 80 by 20 (or 8 by 2): estimated quotient = \( \frac{8}{2} \) = 4.
(b) 7525 ≈ 8000 and 365 ≈ 400. Divide 8000 by 400 (or 80 by 4): estimated quotient = 20.

Use of Numbers in Everyday Life

We use numbers to measure lengths (a pencil, the distance between two places), weights (an orange, a ship) and capacities (juice in a glass, water in a lake). Small lengths are measured in millimetres and centimetres, and large ones in metres and kilometres.

Length: 1 m = 100 cm = 1000 mm; 1 cm = 10 mm; 1 km = 1000 m = 1000 × 1000 mm = 1000000 mm.
Weight: 1 g = 1000 mg; 1 kg = 1000 g = 1000 × 1000 mg = 1000000 mg.
Capacity (volume): 1 L = 1000 mL; 1 kL = 1000 L = 1000 × 1000 mL = 1000000 mL.

Illustration 1.18. A tin of biscuits has 14 kg of biscuits. Express the weight in milligrams.
Answer:
1 kg = 1000 g and 1 g = 1000 mg, so 1 kg = 1000 × 1000 mg = 1000000 mg.
14 kg = 14 × 1000000 mg = 14000000 mg.

Illustration 1.19. The population of Rajasthan is 5,64,73,122, of Goa is 13,43,998 and of Karnataka is 5,27,33,958. What is the combined population of the three states?
Answer:
Total = 5,64,73,122 + 13,43,998 + 5,27,33,958 = 11,05,51,078.
In words: eleven crore five lakh fifty-one thousand seventy-eight.

Illustration 1.20. What must be added to 34,52,629 to make it equal to 6 crore?
Answer:
6 crore = 6,00,00,000.
Number to be added = 6,00,00,000 − 34,52,629 = 5,65,47,371.

Illustration 1.21. There are 785 students on roll in a residential public school. If the annual fee per student is Rs. 62,606, what is the total fee collected annually by the school?
Answer:
Total fee = 62,606 × 785 = Rs. 4,91,45,710.

Illustration 1.22. Find the number of pages in a book which has on an average 207 words on a page, and contains 201411 words altogether.
Answer:
Number of pages = 201411 ÷ 207 = 973.

Roman Numerals

The Roman system of writing numbers is still used, for example on some clock faces and for numbering lists. Unlike the Hindu-Arabic system, it uses only seven basic symbols:

Rules for Forming Roman Numerals

Rule 1 (repetition adds). If a symbol is repeated, its value is added as many times as it occurs. III = 1 + 1 + 1 = 3 and XX = 10 + 10 = 20.

Rule 2 (at most three times). The symbols I, X, C and M can be repeated at most three times in a row: I, II, III; X, XX, XXX; C, CC, CCC; M, MM, MMM (1, 2, 3; 10, 20, 30; 100, 200, 300; 1000, 2000, 3000).

Rule 3 (never repeated). The symbols V, L and D (5, 50, 500) are never repeated.

Rule 4 (smaller on the right adds). A smaller symbol written to the right of a larger one is added to it. XII = 10 + 2 = 12, LX = 50 + 10 = 60, DCCCX = 500 + 300 + 10 = 810. In the same way VI = 5 + 1 = 6, XI = 10 + 1 = 11 and DC = 500 + 100 = 600.

Rule 5 (smaller on the left subtracts). A smaller symbol written to the left of a larger one is subtracted from it. IV = 5 − 1 = 4, IX = 10 − 1 = 9, XL = 50 − 10 = 40, XC = 100 − 10 = 90, CD = 500 − 100 = 400, CM = 1000 − 100 = 900.

Rule 6 (which symbols may be subtracted). V, L and D are never subtracted. I can be subtracted from V and X only. X can be subtracted from L and C only (XL = 40, XC = 90). C can be subtracted from D and M only.

Rule 7 (bar). A bar over a numeral multiplies its value by 1000. \( \overline{\mathrm{V}} \) = 5000 and \( \overline{\mathrm{L}} \) = 50000.

Illustration 1.23. Write the Roman numerals for (a) 105 (b) 213 (c) 4592 (d) 5839
Answer:
(a) 105 = 100 + 5 = CV.
(b) 213 = 200 + 10 + 3 = CCXIII.
(c) 4592 = 4000 + 592 = \( \overline{\mathrm{IV}} \) + D + XC + II = \( \overline{\mathrm{IV}} \)DXCII.
(d) 5839 = 5000 + 839 = \( \overline{\mathrm{V}} \) + DCCC + XXX + IX = \( \overline{\mathrm{V}} \)DCCCXXXIX.

Illustration 1.24. Write in Hindu-Arabic numerals: (a) CXXXV (b) CXLI (c) \( \overline{\mathrm{VI}} \)CMXXI (d) \( \overline{\mathrm{XLI}} \)CCVI
Answer:
(a) CXXXV = C + XXX + V = 100 + 30 + 5 = 135.
(b) CXLI = C + XL + I = 100 + 40 + 1 = 141.
(c) \( \overline{\mathrm{VI}} \)CMXXI = 6000 + 900 + 20 + 1 = 6921.
(d) \( \overline{\mathrm{XLI}} \)CCVI = 41000 + 200 + 6 = 41,206.

Add to Your Knowledge

1. After a million come, in order: billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion and undecillion.
2. If you spell out the numbers from 0 to 1000, the letter 'A' appears only in 'one thousand'.
3. Prime numbers are natural numbers greater than 1 that are divisible only by 1 and by themselves, for example 2, 3, 5, 7, 11, 13, 17, 19, 23.
4. If the ones digit of a two-digit number is x and its tens digit is y, the number is 10y + x.
5. Division algorithm: dividend ÷ divisor = quotient + remainder ÷ divisor, which gives \( \frac{\text{Dividend}}{\text{Divisor}} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}} \), or Dividend = (Divisor × Quotient) + Remainder.

Concept Map: Quick Review

1. Face value and place value of 2423: the face value of 2 is 2 at both the thousands and the tens place; its place value is 2000 at the thousands place and 20 at the tens place.
2. Making numbers: without repetition, the smallest number uses the digits in ascending order (0 not first) and the greatest uses descending order; with repetition, the smallest digit is repeated for the smallest number and the greatest digit for the greatest number.
3. Estimation in a sum: 267 + 132 ≈ 300 + 100 = 400; 59 + 238 ≈ 60 + 240 = 300.
4. In a difference: 53 − 18 ≈ 50 − 20 = 30; 409 − 48 ≈ 410 − 50 = 360.
5. In a product: 42 × 58 ≈ 40 × 60 = 2400; 387 × 31 ≈ 400 × 30 = 12000.
6. In a quotient: 94 ÷ 34 ≈ 90 ÷ 30 = 3; 627 ÷ 23 ≈ 600 ÷ 20 = 30.
7. Roman numerals: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000.

Points to Remember

1. The set of natural numbers is 1, 2, 3, ... and the set of whole numbers is 0, 1, 2, 3, ... Both sets go on forever, so there is no greatest number.
2. The smallest natural number is 1 and the smallest whole number is 0.
3. The successor of a whole number is 1 more than it (successor of 25 is 26); the predecessor is 1 less (predecessor of 100 is 99).
4. Every digit has a face value, which never changes, and a place value, which depends on its position. Each place is ten times the place to its right.
5. There are two systems of numeration: the Indian system (ones, thousands, lakhs, crores) and the International system (ones, thousands, millions, billions).
6. To round a number to the nearest 10, 100, 1000 and so on, look at the digit one place to the right: less than 5 rounds down, otherwise rounds up.
7. The Roman system has seven basic symbols: I for 1, V for 5, X for 10, L for 50, C for 100, D for 500 and M for 1000.