Number System | ICSE Class 6 Maths Notes

This note covers large numbers and place value, estimation, whole-number properties, brackets, factors and multiples, divisibility, highest common factor (HCF) and lowest common multiple (LCM), integers, sets, fractions and decimals.

How do place values help us read and compare large numbers?

A digit is one of the symbols 0 to 9 used to write numbers. Its face value is the digit itself. Its place value is the value it contributes because of its position in a number.

From right to left, the Indian places are ones, tens, hundreds, thousands, ten thousands, lakhs, ten lakhs and crores. Each place has ten times the value of the place immediately to its right. A crore is a number with eight digits.

In calculations, = means “is equal to”, + means addition, − means subtraction, × means multiplication and ÷ means division. A sum is an addition result; a difference is a subtraction result; a product is a multiplication result.

How do the Indian and International systems differ?

In the Indian system, separate the last three digits, then group the remaining digits in pairs from the right. In the International system, group digits in threes from the right. Commas make reading easier; changing their grouping does not change the number.

Worked example 1. Read 12,78,830 and identify the face and place values of its digit 7.

Answer: Twelve lakh seventy-eight thousand eight hundred and thirty. The face value of 7 is 7. Its place value is 70,000 because it occupies the ten-thousands place.

To compare numbers written using digits alone, with no initial zeros, first compare the number of digits. If the digit counts agree, compare corresponding digits from the left until they differ. If every digit agrees, the numbers are equal. The symbols < and > mean “is less than” and “is greater than”.

Ascending order runs from smaller to larger; descending order runs from larger to smaller. Convert number names into the same notation before comparing quantities expressed in lakhs, thousands or millions.

How can we calculate, estimate and convert large quantities?

Estimation gives an approximate value rather than an exact calculation. Rounding replaces a number by a nearby value at a chosen place. State that place because rounding to thousands and rounding to lakhs answer different questions.

To round, inspect the digit immediately to the right of the chosen place. If it is below 5, retain the chosen digit; if it is 5 or more, increase it by one. Replace the following digits by zeros. Carrying means transferring a complete group of ten into the next place; do this if necessary.

Worked example 2. Find 4,63,128 + 4,19,682 and judge whether 8,00,000 or 9,00,000 is the closer estimate.

Answer: Adding corresponding places gives 8,82,810. Its difference from 9,00,000 is 17,190; its difference from 8,00,000 is 82,810. Therefore, 9,00,000 is closer, and the exact sum is below it.

How do units affect a calculation?

A unit is a standard quantity used for measurement. The abbreviations km, m, cm and mm mean kilometre, metre, centimetre and millimetre. For mass, kg means kilogram and g means gram. Use the same mass unit throughout an addition or subtraction.

Use the relationships 1 km = 1,000 m, 1 m = 100 cm, 1 cm = 10 mm and 1 kg = 1,000 g. Conversion from a larger unit to a smaller unit increases the numerical count, while the measured quantity remains the same.

For addition and subtraction, first express quantities in the same unit. Align the place values and attach the unit to the answer. Regrouping exchanges one unit of a place for ten units of the next smaller place, or conversely. An estimate can expose a misplaced digit, but it does not replace the exact calculation.

What are natural numbers, whole numbers and their properties?

Natural numbers are the counting numbers 1, 2, 3 and so on. Whole numbers include these numbers and 0. The dots in a list such as 0, 1, 2, 3, … mean that the indicated pattern continues.

The successor of a whole number is obtained by adding 1. The predecessor is obtained by subtracting 1, when the result is still a whole number. Thus, 0 is the smallest whole number and has no whole-number predecessor.

A collection is closed under an operation when applying that operation to any two of its numbers gives another number in the collection. Whole numbers are closed under addition and multiplication. Subtraction and division do not have this property for whole numbers.

Property: Order and grouping in addition and multiplication

Let a, b and c stand for any whole numbers. Brackets, written ( ), group the calculation to be performed together. The commutative property permits changing order: a + b = b + a and a × b = b × a.

The associative property permits changing grouping: (a + b) + c = a + (b + c), and (a × b) × c = a × (b × c). Neither property can be applied generally to subtraction or division.

Property: Additive and multiplicative identities

An identity leaves the other number unchanged under its operation. Zero is the additive identity: a + 0 = a. One is the multiplicative identity: a × 1 = a. Also, a − 0 = a and a × 0 = 0.

A dividend is the number being divided, a divisor is the number by which we divide, and a quotient is the result. Zero divided by a non-zero whole number is zero. Division by zero is undefined.

Property: Multiplication distributes over addition

The distributive property is a × (b + c) = a × b + a × c. Multiplication distributes over subtraction too. When working within whole numbers, use the subtraction form where the differences remain whole numbers.

Worked example 3. Given 53 × 18 = 954, find 63 × 18.

Answer: 63 × 18 = (53 + 10) × 18 = 53 × 18 + 10 × 18 = 954 + 180 = 1,134. The extra ten groups of eighteen account for the increase.

How should brackets and mixed operations be simplified?

An arithmetic expression combines numbers and operation signs to describe a calculation. Brackets identify a group that must be evaluated together. The value of an expression depends on its operation signs and grouping, not just on the numbers it contains.

Work inside brackets first. Then perform multiplication and division, moving from left to right where they occur together. Finally perform addition and subtraction from left to right. Nested brackets are groups inside other groups; evaluate the innermost group first.

Worked example 4. Evaluate 30 + 5 × 4.

Answer: First calculate 5 × 4 = 20. Then 30 + 20 = 50. Adding 30 and 5 first would change the grouping and produce 140, which is not the value of the expression given.

How do brackets preserve the meaning of a problem?

The symbol ₹ means rupees. If a payment must cover two purchases, add their costs before subtracting the total from the payment. Brackets make it clear that the entire combined cost is being removed.

Worked example 5. Irfan buys biscuits for ₹15 and toor dal for ₹56, and pays ₹100. Find his change.

Answer: Change = ₹100 − (₹15 + ₹56) = ₹100 − ₹71 = ₹29. The brackets group the two costs; subtracting one cost and then adding the other would describe a different calculation.

How are factors, multiples and prime numbers connected?

A factor of a positive whole number divides it exactly, leaving no remainder. A remainder is the amount left after taking out whole groups in division. A multiple is obtained by multiplying a number by a whole number.

In lists used to find a lowest common multiple, consider the positive multiples. A common factor divides each of the given numbers. A common multiple is a multiple of each of them. These two ideas answer different questions about divisibility.

An even number is divisible by 2; an odd number is not. A prime number is a whole number greater than 1 with exactly two factors, 1 and itself. A composite number has more than two factors.

The number 1 is neither prime nor composite. The number 2 is the only even prime. Two numbers are co-prime if their only common factor is 1. They do not both have to be prime: 4 and 9 are co-prime.

What does prime factorisation show?

Prime factorisation writes a whole number greater than 1 as a product of primes. Continue breaking composite factors into smaller factors until every factor is prime. Repeated prime factors must remain in the product; omitting a repetition changes its value.

Worked example 6. Use prime factorisation to decide whether 56 and 63 are co-prime.

Answer: 56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7. Both contain the prime factor 7, so they are not co-prime. The shared factor is visible even though the other prime factors differ.

A prime number is already its own prime factorisation. For a composite number, different starting factor pairs lead to the same prime factors, possibly in a different order. The number 1 has no prime factorisation.

Which divisibility tests can replace long division?

A number is divisible by another number when the division leaves remainder zero. Divisibility tests use patterns in digits to decide this without completing long division. Apply the appropriate test to the specified digits rather than using the same test for every divisor.

For the test for 11, take every other digit starting at one end, then the remaining digits, and subtract the smaller sum from the larger. For 4 or 8, use the whole number if it has fewer digits than the test specifies.

Worked example 7. Is 8,536 divisible by 4?

Answer: Its last two digits form 36. Since 36 ÷ 4 = 9, the number passes the divisibility test for 4. Looking at the final digit 6 alone would not establish divisibility by 4.

Note: Passing one part of the test for 6 is insufficient. Both evenness and divisibility by 3 are required.

How do we find and use HCF and LCM?

The highest common factor (HCF) is the greatest factor shared by the given positive whole numbers. The lowest common multiple (LCM), also called the least common multiple, is their smallest positive common multiple.

How does prime factorisation give each answer?

1. Write every given number as a product of prime factors.
2. For the HCF, select primes present in every factorisation, with the fewest occurrences available in any one.
3. Multiply these selected factors. If no prime is shared by all the numbers, the HCF is 1.
4. For the LCM, select every prime that occurs, taking the greatest number of occurrences required by any one factorisation, and multiply.

Worked example 8. Find the HCF of 45 and 75.

Answer: 45 = 3 × 3 × 5 and 75 = 3 × 5 × 5. One 3 and one 5 are shared. HCF = 3 × 5 = 15. Their complete list of common factors is 1, 3, 5 and 15.

Worked example 9. Find the LCM of 14 and 35.

Answer: 14 = 2 × 7 and 35 = 5 × 7. A common multiple needs 2, 5 and 7. Taking each once gives LCM = 2 × 5 × 7 = 70.

How does the division method work?

Divide both numbers by a common prime and write the resulting quotients below them. Continue until the two quotients have no common prime factor. Multiply the common divisors to obtain the HCF. For two numbers, multiply this HCF by both final quotients to obtain the LCM.

Worked example 10. Use common division to find the HCF and LCM of 300 and 150.

Answer: Dividing both by 2 gives 150 and 75; then by 5 gives 30 and 15; then by 5 gives 6 and 3; then by 3 gives 2 and 1. HCF = 2 × 5 × 5 × 3 = 150. LCM = 150 × 2 × 1 = 300.

Property: The HCF and LCM product

For two positive whole numbers, HCF × LCM = product of the two numbers. Use this as a check after finding both quantities. The statement concerns two numbers; it must not simply be extended to a longer list.

Choose HCF for the largest equal size that divides given quantities exactly. Choose LCM for the smallest positive quantity that is an exact multiple of each given size. Explain this choice from the situation before calculating.

For example, strips of length 6 cm and 8 cm can each make a length of 24 cm without cutting. Their positive multiples first coincide at 24. For the largest square tiles fitting a rectangular floor exactly, the tile side must divide both floor dimensions.

Why are negative numbers needed, and how are integers ordered?

Positive numbers are greater than zero; negative numbers are less than zero. They can represent opposite directions or quantities measured on opposite sides of a reference point. A reference point is the position chosen as the starting level for measurement.

Integers comprise zero, the positive whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …. Zero is neither positive nor negative. Usually, the + sign on a positive number is omitted.

How does position show order?

A number line represents numbers by positions along a straight line. Equal numerical differences use equal distances. Positive integers lie to the right of zero and negative integers to its left. Moving right leads to greater numbers; moving left leads to smaller numbers.

Thus, −5 < −3 because −5 lies farther left. Comparing the digits 5 and 3 without their negative signs would give the wrong ordering. Every negative integer is below zero, and every positive integer is above it.

A number and its additive inverse add to zero. They lie at equal distances on opposite sides of zero. For example, 2 + (−2) = 0. The additive inverse of zero is zero.

For a horizontal line, state which direction is positive. For a vertical setting, such as floors above and below ground, first identify the zero level. The sign describes position relative to that level, rather than the size of a count alone.

How are integers added and subtracted?

To add positive integers, add their values. To add two negative integers, add their values without signs and attach a negative sign. When the signs differ, subtract the smaller unsigned value from the larger and use the sign of the number with the larger unsigned value.

Unsigned value here means the distance of the integer from zero, ignoring its sign. Opposite integers have equal unsigned values and sum to zero. Adding zero leaves an integer unchanged, including when the integer is negative.

Worked example 11. Find −2 + (−3) and −5 + 3.

Answer: For −2 + (−3), add 2 and 3 and attach the negative sign, giving −5. For −5 + 3, the difference of the unsigned values is 2; the larger unsigned value belongs to −5, so the answer is −2.

How does the number line explain subtraction?

To add a positive integer, move right from the starting number. To add a negative integer, move left. Subtracting an integer means adding its additive inverse. In particular, subtracting a negative integer is equivalent to adding the corresponding positive integer.

One number line shows six leftward steps from 9 to 3. Another, labelled from −10 to 10, shows five leftward steps from 3 to −2. The steps illustrate addition of a negative integer.

Reference: NCERT Class 6, unnumbered diagrams, page 253

Worked example 12. Evaluate +2,000 − (−200).

Answer: Replace subtraction of −200 by addition of +200. Thus, +2,000 − (−200) = +2,000 + (+200) = +2,200. The brackets distinguish the subtraction operation from the negative sign belonging to 200.

The number-line movement from 3 to −2 is −5, giving 3 + (−5) = −2. Reversing the question, the required movement is target minus starting number: −2 − 3 = −5. Keep the order of subtraction clear.

What is a set, and how can it be represented?

Definition: A set is a well-defined collection of objects. “Well-defined” means that it is possible to decide whether a particular object belongs to the collection. Each object belonging to the set is an element or member.

Capital letters can name sets. In roster form, list the elements between braces, the symbols { }, separating them with commas. The order of listing does not matter. An element is not generally repeated when a set is written in this form.

For example, the set of positive even integers less than 7 is {2, 4, 6}. This description gives a definite membership rule. The braces show that the numbers form a collection; they are not instructions to add or multiply the numbers.

How does a rule describe a set?

In set-builder form, state the property that identifies the members. For example, A = {x : x is a natural number which divides 42}. Here A names the set, x stands for a possible member, and the colon means “such that”.

The corresponding roster is A = {1, 2, 3, 6, 7, 14, 21, 42}. Each listed number divides 42 exactly. A rule and a roster can therefore describe the same collection, using different forms of representation.

How are finite, infinite and empty sets different?

A finite set has a definite, limited number of elements, including possibly none. An infinite set is not finite, as with the natural numbers. An empty set has no elements and can be written { }.

The cardinality of a finite set is its number of distinct elements. The notation n(A) means the number of elements in the set named A. Count each distinct member once, regardless of the order in which the members are written.

Worked example 13. Classify {1, 2, 3, 4, 5} and find its cardinality. Then classify the set of natural numbers strictly between 1 and 2.

Answer: The first set is finite and has cardinality 5. The second set is empty because there is no natural number strictly between 1 and 2; its cardinality is 0.

How do fractions represent, compare and rename quantities?

A fraction represents a quantity using equal fractional units. In the notation a/b, the slash is the fraction bar: a is the numerator, counting the units taken, and b is the denominator, a positive whole number specifying equal parts per whole.

Thus, 3/4 means three units of size one-quarter. A fraction also expresses division: three wholes shared equally among four recipients gives 3/4 of a whole to each. Equal parts must have equal size, though their shapes need not be the same.

The pictures show a whole chikki, a separated quarter and a larger three-quarter piece. Below, two different divisions into six equal pieces illustrate one-sixth pieces with different shapes.

Reference: NCERT Class 6, unnumbered pictures, page 154

How are fractions classified?

A proper fraction has numerator smaller than denominator. An improper fraction has numerator greater than or equal to denominator. A mixed fraction combines a whole-number part and a proper fractional part. In 2 2/3, the gap means “two and two-thirds”, not multiplication.

Fractions can also be placed on a number line. Divide the interval from zero to one into as many equal parts as the denominator specifies. Count numerator-many parts from zero, continuing beyond one when necessary.

Equivalent fractions name the same quantity, as in 1/2 = 2/4 = 3/6. Multiply or divide numerator and denominator by the same non-zero whole number, using exact division where required, to obtain an equivalent fraction.

Worked example 14. Express 16/20 in its simplest form.

Answer: The HCF of 16 and 20 is 4. Dividing both by 4 gives (16 ÷ 4)/(20 ÷ 4) = 4/5. The numerator and denominator now have no common factor other than 1.

How can unlike fractions be compared?

Like fractions have equal denominators; unlike fractions have different denominators. With like fractions, compare the numerators. For unlike fractions, first choose a common denominator, meaning a shared denominator in equivalent forms, and then compare the new numerators.

Worked example 15. Compare 4/5 and 7/9.

Answer: Use 45 as a common denominator. Then 4/5 = 36/45 and 7/9 = 35/45. Since 36 > 35, it follows that 4/5 > 7/9.

How are the four operations performed on fractions?

For addition and subtraction, the fractional units must agree. With like fractions, add or subtract the numerators and retain the denominator. With unlike fractions, rename them as equivalent fractions with a common denominator before operating on the numerators.

1. Identify the denominators and choose a common multiple for the common denominator.
2. Multiply each numerator and its denominator by the same suitable number.
3. Add or subtract the new numerators, keeping the common denominator.
4. Reduce the result to simplest form by removing any common factor of numerator and denominator.

Worked example 16. Find 1/6 + 1/3 and 3/4 − 2/3.

Answer: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2. For subtraction, 3/4 − 2/3 = 9/12 − 8/12 = 1/12. In each calculation the common denominator keeps the fractional unit unchanged.

What does multiplying fractions mean?

Multiplication by a fraction can mean taking that fraction of a quantity. Multiply the numerators to obtain the product's numerator and multiply the denominators to obtain its denominator. Then reduce the answer if numerator and denominator have a common factor.

Two-thirds of four-fifths is (2/3) × (4/5). Its value is (2 × 4)/(3 × 5) = 8/15. This differs from addition: multiplication does not require the fractions to have equal denominators before calculation.

How do reciprocals help with division?

The reciprocal of a non-zero fraction is obtained by interchanging numerator and denominator. Their product is 1. To divide by a non-zero fraction, multiply the dividend by the reciprocal of the divisor. Leave the dividend in its original form.

Worked example 17. Divide 2/3 by 3/5.

Answer: The reciprocal of 3/5 is 5/3. Therefore, (2/3) ÷ (3/5) = (2/3) × (5/3) = 10/9. The fraction being inverted is the divisor, 3/5.

Convert mixed fractions into improper fractions before multiplying or dividing. Use the whole-number part to count complete groups of denominator-many fractional units, then add the numerator. Keep the original denominator. This preserves the value while making the operation easier to organise.

How do decimals express fractions and measurements?

A decimal fraction has denominator 10, 100, 1,000 or a further power of ten, meaning a product of repeated tens. A decimal point separates whole-number places from fractional places. The first three places to its right are tenths, hundredths and thousandths.

One tenth means 1/10; one hundredth means 1/100; one thousandth means 1/1,000. Moving one place to the right divides the place's value by ten. The digit's position therefore determines its contribution on either side of the decimal point.

How are fractions and decimals converted?

To express a decimal as a fraction, use its place values and then simplify. To express a suitable fraction as a decimal, find an equivalent fraction with denominator 10, 100 or 1,000. Include zeros needed to preserve the correct places.

For example, 7.05 = 7 + 5/100. The zero tenths digit matters: it keeps the 5 in the hundredths place. When comparing decimals, compare whole-number parts first and then corresponding fractional places from left to right.

How are decimal quantities added and subtracted?

Arrange decimal points beneath one another so that like places align. Add or subtract as with whole numbers, regrouping between neighbouring places where necessary. Trailing zeros may be added after the fractional digits without changing a decimal's value.

Worked example 18. Priya needs 2.7 m of cloth and Shylaja needs 3.5 m. Find the total length and how much more Shylaja needs.

Answer: Total = 2.7 + 3.5 = 6.2 m. Difference = 3.5 − 2.7 = 0.8 m. For subtraction, regroup 3.5 as 2 units and 15 tenths; removing 2 units and 7 tenths leaves 8 tenths.

The same place-value method applies to money, mass, length and temperature when units agree. A two-operation calculation can combine the lengths and then remove Priya's share: (2.7 + 3.5) − 2.7 = 6.2 − 2.7 = 3.5 m.

Converting the total length to centimetres gives 6.2 × 100 = 620 cm. The decimal point and the unit must be read together: 6.2 m and 620 cm name the same length, despite having different numerical values.

Glossary

• Place value — The value contributed by a digit because of its position within a number.
• Whole numbers — The collection consisting of zero and all the positive counting numbers.
• Identity — A number that leaves another number unchanged under a specified operation.
• Factor — A number that divides a given positive whole number exactly, leaving no remainder.
• Prime number — A whole number greater than one with exactly two factors, one and itself.
• Co-prime numbers — Two positive whole numbers whose only common factor is the number one.
• HCF — Highest common factor, the greatest factor shared by all the given positive whole numbers.
• LCM — Lowest common multiple, the smallest positive multiple shared by all the given numbers.
• Integers — The numbers comprising zero, positive whole numbers and their corresponding negative numbers.
• Additive inverse — The number that gives zero when added to the original number.
• Set — A well-defined collection for which membership can be decided for each object.
• Cardinality — The number of distinct elements belonging to a given finite set.
• Equivalent fractions — Fractions that express the same quantity despite having different numerators and denominators.
• Reciprocal — A number that gives a product of one when multiplied by the original non-zero number.
• Decimal point — The separator between the whole-number places and the fractional places in decimal notation.

Common errors and misconceptions

• Misconception: Changing comma placement changes a number. Correct: Indian and International grouping change how a number is read, while its value stays the same.
• Misconception: Subtraction has the same order and grouping properties as addition. Correct: Reordering or regrouping subtraction generally changes the answer; follow the expression and its brackets.
• Misconception: Dividing by zero gives zero. Correct: Division by zero is undefined. Dividing zero by a non-zero number gives zero.
• Misconception: Co-prime numbers must both be prime. Correct: They need only have no common factor other than 1; 4 and 9 are co-prime.
• Misconception: HCF uses every prime factor found in either number. Correct: It uses shared primes with their minimum occurrences. LCM uses all required primes with their maximum occurrences.
• Misconception: −5 is greater than −3 because 5 is greater than 3. Correct: −5 lies left of −3 on the number line, so it is smaller.
• Misconception: Add fractions by adding their denominators too. Correct: First obtain equal denominators, then add the numerators and retain that denominator.
• Misconception: Decimal subtraction is aligned by the last written digit. Correct: Align decimal points so that corresponding place values are subtracted from each other.

Exam-style questions with model answers

Q1. In 12,78,830, give the face value and place value of the digit 7. [2 marks]

1. The face value of the digit is 7, independent of where it appears.
2. Its place value is 70,000 because it is in the ten-thousands place.

Q2. Evaluate 30 + 5 × 4. Explain the operation order and why 140 is incorrect. [3 marks]

1. Multiplication is performed before addition in this expression, so first calculate the product 5 × 4 = 20.
2. Add that product to the remaining number: 30 + 20 = 50. Therefore, the value of the given expression is 50.
3. The answer 140 comes from adding 30 and 5 before multiplying by 4. That inserts a grouping which the given expression does not have.

Q3. Use prime factorisation to find the HCF of 45 and 75, and list all their common factors. [4 marks]

1. Factorise the first number: 45 = 3 × 3 × 5. This records each prime factor, including the repeated 3.
2. Factorise the second number: 75 = 3 × 5 × 5. Here the prime 5 occurs twice.
3. The common part contains one 3 and one 5. Multiplying gives HCF = 3 × 5 = 15.
4. The complete list of common factors is 1, 3, 5 and 15. Each divides both original numbers exactly.

Q4. Use common division to find the HCF and LCM of 300 and 150, then check their product against the product of the original numbers. [5 marks]

1. Start with 300 and 150. Divide both numbers by the common prime factor 2 to obtain the pair 150 and 75.
2. Divide both new numbers by 5 to get 30 and 15. Divide both by 5 again to get 6 and 3.
3. Divide both by 3 to obtain 2 and 1, which have no common prime factor. Thus, HCF = 2 × 5 × 5 × 3 = 150.
4. Multiply the common divisors and the final two quotients to obtain the LCM: 150 × 2 × 1 = 300.
5. Check: HCF × LCM = 150 × 300 = 45,000. The original product, 300 × 150, is also 45,000, confirming the required relation.

Q5. Start at 3 on a number line and move five units left. State the movement, the endpoint and the addition statement. [3 marks]

1. Leftward movement is negative on the usual horizontal number line, so a movement of five units to the left is written as −5.
2. Starting at 3, move three units left to reach zero and then two more units left. The endpoint is −2.
3. The addition statement combines the starting number and the movement: 3 + (−5) = −2. The negative sign specifies the direction of travel.

Q6. Let A be the set of natural numbers dividing 42. Write A in roster form, state whether it is finite, find n(A), and classify the set of natural numbers strictly between 1 and 2. Here n(A) means the number of elements of A. [4 marks]

1. The roster form is A = {1, 2, 3, 6, 7, 14, 21, 42}; these are exactly the natural numbers dividing 42.
2. A is a finite set because its members form a definite, limited collection, rather than continuing without end.
3. There are eight distinct numbers in the roster, so the cardinality is n(A) = 8.
4. No natural number lies strictly between 1 and 2. That set is empty and therefore finite, with cardinality 0.

Q7. Compare 4/5 and 7/9 by converting both to equivalent fractions with denominator 45. Explain why this allows the comparison. [5 marks]

1. The denominators are different, so the fractions initially count different fractional units. Use the specified common denominator 45 to make their units agree.
2. For 4/5, multiply numerator and denominator by 9. This gives 4/5 = (4 × 9)/(5 × 9) = 36/45.
3. For 7/9, multiply numerator and denominator by 5. This gives 7/9 = (7 × 5)/(9 × 5) = 35/45.
4. The rewritten fractions both count units of one forty-fifth. Since 36 such units exceed 35 such units, 36/45 is the greater fraction.
5. Equivalent forms preserve the original quantities. Therefore, the same ordering holds for the original pair, giving the final comparison 4/5 > 7/9.

Q8. Priya needs 2.7 m of cloth and Shylaja needs 3.5 m. Find their combined requirement, how much more Shylaja needs, and the combined requirement in centimetres. Use 1 m = 100 cm, where m means metre and cm means centimetre. [4 marks]

1. Align the decimal points before calculating, so that metres and tenths of a metre occupy matching columns.
2. The combined requirement is 2.7 + 3.5 = 6.2 m. Regroup the twelve tenths as one metre and two tenths.
3. Shylaja needs 3.5 − 2.7 = 0.8 m more cloth. Regrouping one whole metre supplies the tenths needed for subtraction.
4. Using the given conversion, 6.2 × 100 = 620. Therefore, the combined requirement is 620 cm.

Key takeaways

• Place value depends on position; Indian and International comma grouping offer different ways to read the same number.
• Whole numbers include zero; addition and multiplication permit changes of order and grouping, while subtraction and division generally do not.
• Brackets preserve the intended grouping of operations, and multiplication and division are evaluated before addition and subtraction.
• Prime factorisation retains repeated factors; HCF uses the shared minimum occurrences, while LCM uses the maximum occurrences needed.
• Numbers increase towards the right of a number line; subtracting an integer means adding its additive inverse.
• A set needs a definite membership rule, and the cardinality of a finite set counts its distinct elements.
• Equivalent fractions preserve value; to add or subtract unlike fractions, first express them with a common denominator. A common denominator also provides a method for comparing fractions.
• Decimal calculations depend on matching place values, and unit conversions must preserve the measured quantity.

Test yourself

How many millions make one crore?

One crore is ten million; both names describe the number 10,000,000.

Which whole number is the additive identity?

Zero is the additive identity because adding it leaves the other number unchanged.

Why is 1 not a prime number?

It has only one factor, itself, whereas a prime number has exactly two factors.

Why do 4 and 9 count as co-prime?

Their only common factor is 1, although neither number is itself prime.

Which is greater, −5 or −3?

The greater integer is −3 because it lies to the right of −5.

What is the cardinality of an empty set?

Its cardinality is zero because an empty set contains no elements.

How do you simplify 16/20?

Divide numerator and denominator by their HCF, 4, to obtain the simplest form 4/5.

What is the reciprocal of 3/5?

The reciprocal is 5/3, obtained by interchanging the numerator and denominator.