Number System | ICSE Class 8 Maths Notes

This note covers rational numbers and their properties, number-line representation, numbers between rational numbers, word problems, integral exponents, squares and square roots, cubes and cube roots, estimation, generalised numbers, digit puzzles, divisibility tests and operations on sets.

What are rational numbers, and why are they needed?

Definition: A rational number can be written as p/q, where p and q are integers and q ≠ 0. Here p is the numerator, q is the denominator, the slash means division, and ≠ means “is not equal to”.

Natural numbers are the counting numbers starting at 1. Whole numbers include zero and all natural numbers. Integers include zero, positive whole numbers and their negatives. A negative number is less than zero; a positive number is greater than zero.

An equation states that two expressions have equal values. The sign = means “is equal to”. A letter can stand for an unknown number. In the following equations, x denotes the number to be found; 2x means 2 multiplied by x.

The equation x + 2 = 13 has the natural-number solution 11. The equation x + 5 = 5 requires zero. The equation x + 18 = 5 requires the integer −13. The equation 2x = 3 requires 3/2, which is rational but not an integer.

Every integer is rational because it can be written with denominator 1. Thus the move from integers to rational numbers keeps the earlier numbers and includes additional numbers needed for division by non-zero numbers.

Does every point represent a rational number?

Every rational number has a point on the number line, but the converse is not true. An irrational number cannot be expressed as p/q with integer numerator and non-zero integer denominator. Having infinitely many rational numbers does not mean they account for every point.

In the lowest form of a rational number, its numerator and denominator have no common factor other than 1. A factor divides a number exactly. Changing both numerator and denominator by the same non-zero factor gives an equivalent fraction, meaning the value remains unchanged.

Which properties hold for operations on rational numbers?

An operation is a calculation such as addition, subtraction, multiplication or division. These use +, −, × and ÷ respectively. A sum is an addition result, a difference a subtraction result, a product a multiplication result and a quotient a division result.

Property: Closure

A collection is closed under an operation when applying that operation to its members gives a member of the same collection. Rational numbers are closed under addition, subtraction and multiplication. Division needs a condition: the divisor, meaning the number being divided by, must not be zero.

For the general statements below, a, b and c denote rational numbers. Thus a + b, a − b and a × b are rational. The quotient a ÷ b is rational when b ≠ 0. Division by zero is undefined, so unrestricted division does not have closure.

Worked example 1. Add −3/8 and 5/7.

Answer: Use the common denominator 56, a denominator shared by equivalent forms of both fractions. Then −3/8 + 5/7 = −21/56 + 40/56 = 19/56. The result is rational because both 19 and 56 are integers and the denominator is non-zero.

Property: Commutativity

Commutativity means changing the order leaves the result unchanged. Addition and multiplication satisfy a + b = b + a and a × b = b × a. Subtraction and division are not commutative, so their order must be preserved.

Property: Associativity

Associativity means changing the grouping leaves the result unchanged. Brackets show which calculation to perform first. Rational numbers satisfy a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.

Subtraction and division are not associative. Commutativity changes order; associativity changes brackets. Use the correct name when explaining a rearrangement. Both properties can be useful when several rational numbers are being added or multiplied.

How do identities, inverses and distributivity simplify calculations?

An identity leaves a number unchanged under the specified operation. The additive identity is 0 because a + 0 = 0 + a = a. The multiplicative identity is 1 because a × 1 = 1 × a = a.

An additive inverse combines with a number to give zero. For rational a, its additive inverse is −a, because a + (−a) = 0. Subtracting a number can therefore be treated as adding its additive inverse.

A multiplicative inverse, or reciprocal, combines with a number by multiplication to give 1. For non-zero p/q, the reciprocal is q/p. Both p and q must be non-zero here. Zero has no reciprocal because no product of zero and a number equals 1.

Property: Distributivity

Distributivity connects multiplication with addition or subtraction: a × (b + c) = a × b + a × c, and a × (b − c) = a × b − a × c. Multiply every term inside the brackets by the outside factor.

Worked example 2. Evaluate (−3/4) × [2/3 + (−5/6)] using distributivity.

Answer: The expression becomes (−3/4 × 2/3) + (−3/4 × −5/6) = −1/2 + 5/8 = 1/8. Checking by doing the bracket first gives 2/3 − 5/6 = −1/6, followed by (−3/4) × (−1/6) = 1/8.

To divide by a non-zero rational number, multiply by its reciprocal. Keep the first number unchanged and invert the divisor. This converts a division into a multiplication, where cancellation of common factors can make the calculation easier.

Note: An inverse is different from an identity. The identity leaves the original number unchanged; an inverse combines with that number to produce the relevant identity, zero for addition or one for multiplication.

How are rational numbers placed and compared on a number line?

A number line is a straight line on which numbers are represented by points using a fixed unit length. Zero is the origin. Positive numbers lie to its right and negative numbers to its left. A point farther right represents a greater number.

For a fraction with positive denominator, divide a unit interval into as many equal parts as the denominator indicates. The numerator tells how many of those parts to count. Equal spacing matters: unequal divisions do not represent equal numerical steps.

Negative fractions use the same unit length on the left of zero. Comparison can also be made by expressing fractions with a common positive denominator and comparing their numerators. The sign < means “is less than”; > means “is greater than”.

How can another rational number be found between two given ones?

The arithmetic mean of two numbers is half their sum. For two distinct rational numbers a and b with a < b, the number (a + b)/2 is rational and lies strictly between them. Here “distinct” means that the numbers are different.

Repeat this process using either of the smaller intervals just formed. The process does not stop: between any two distinct rational numbers there are infinitely many rational numbers. This differs from whole numbers, where consecutive members have no whole number between them.

Do not confuse a drawing with the full collection of numbers. A diagram can label only a selection of points. An unlabelled interval is not empty, and finding one number between two endpoints does not make it the only possible answer.

How are rational-number word problems translated into calculations?

Begin by identifying what the quantities measure and what the question asks. Equal sharing calls for division; combining amounts calls for addition; finding an excess calls for subtraction. A question can require more than one operation, so organise the information before calculating.

A mixed number combines a whole-number part with a fraction. Convert it to a single fraction when that makes an operation easier. Keep measurement units attached to the result so that an area, length or quantity is not mistaken for a count.

Worked example 3. A farmer divides a field of 49 4/5 hectares equally among one son and two daughters. Find each share. A hectare, abbreviated ha, is an area of 10,000 square metres.

Answer: There are three equal shares. Convert 49 4/5 to 249/5. Each share is (249/5) ÷ 3 = (249/5) × 1/3 = 83/5 = 16 3/5 ha.

How should a two-operation problem be organised?

Worked example 4. One fruit salad recipe uses 1/2 cup of sugar. Another uses 2 tablespoons, with each tablespoon equivalent to 1/16 cup. How much more sugar does the first recipe require?

Answer: Convert the second amount to cups: 2 × 1/16 = 1/8 cup. Then subtract: 1/2 − 1/8 = 4/8 − 1/8 = 3/8 cup. The first recipe requires 3/8 cup more sugar.

First calculate any amount that must be converted or combined. Then use that result in the comparison or division requested. When dividing fractions, distinguish the dividend, the number being divided, from the divisor. Reversing them changes the question.

For an area problem involving equal shares, dividing the area gives an area, not a side length. A side length would require additional information about shape. Read the requested quantity carefully before choosing a formula or performing a root calculation.

Finally, check the result against the original situation. For equal sharing, multiplying one share by the number of recipients should recover the whole amount. This checks both the numerical calculation and the interpretation of the question.

How do integral exponents extend repeated multiplication?

An exponent or power tells how a base is used in a power expression. In aᵐ, a is the base and m is the exponent. For a positive integer m, aᵐ is the product of m copies of a. Thus 2⁵ = 2 × 2 × 2 × 2 × 2.

An integral exponent is an exponent that is an integer, so it may be positive, zero or negative. For non-zero a, a⁰ = 1. For positive integer m, a⁻ᵐ = 1/aᵐ. The negative exponent indicates a reciprocal, not a negative value.

Which laws apply?

In the following laws, a and b are non-zero integers, and m and n are integers. The notation mn means m multiplied by n. Brackets around a power show that the whole power is being raised again.

Worked example 5. Simplify 2⁵ ÷ 2⁻⁶.

Answer: The bases agree, so subtract the exponent of the divisor: 5 − (−6) = 11. Therefore 2⁵ ÷ 2⁻⁶ = 2¹¹. The subtraction brackets are essential because subtracting a negative integer increases the exponent.

Worked example 6. Express 4⁻³ as a power with base 2.

Answer: Since 4 = 2², write 4⁻³ = (2²)⁻³. Multiplying the exponents gives 2⁻⁶. This can also be written as 1/2⁶, using the rule for negative exponents.

Check the structure before choosing a law. Multiplying two powers with the same base differs from raising one power to another power. Keep brackets around a negative base so that the exponent applies to the complete base, including its sign.

How are squares and square roots found by factorisation?

The square of a number is the number multiplied by itself. A perfect square is a natural number that is the square of a natural number. A square root reverses squaring: its square gives the original number.

The symbol √ denotes the non-negative square root. For a positive perfect square, both a positive and a negative integer square to it, but the radical selects the positive one. Thus 9² = 81 and (−9)² = 81, while √81 = 9.

Why are prime factors paired?

A prime number is a natural number greater than 1 with exactly two positive factors, 1 and itself. Prime factorisation writes a number as a product of primes. In a perfect square, the prime factors can be grouped into identical pairs.

1. Express the given number as a product of prime factors.
2. Group identical factors into pairs, keeping every factor in the expression.
3. Take one factor from each pair and multiply these selected factors.
4. Square the result to check that it gives the original number.

Worked example 7. Find √324 by prime factorisation.

Answer: Write 324 = 2 × 2 × 3 × 3 × 3 × 3. There is one pair of 2s and two pairs of 3s. Taking one factor from each pair gives √324 = 2 × 3 × 3 = 18.

If a prime factor is left without a partner, the natural number is not a perfect square. For instance, 48 = 2 × 2 × 2 × 2 × 3 leaves an unpaired 3. Multiplying by 3 completes its pair and gives 144.

Square numbers end in 0, 1, 4, 5, 6 or 9, but an allowed last digit alone does not prove that a number is a square. Use factorisation or an exact root calculation to establish that conclusion.

How does the division method find whole-number and decimal square roots?

The division method builds a square root one digit at a time. Group digits in pairs starting from the units place and moving left. A single leftmost digit forms its own group. For decimals, group fractional digits in pairs moving right from the decimal point.

These calculations use numbers with no more than four total digits and no more than two decimal places. A decimal place is a digit position to the right of the decimal point. Pairing must begin at the decimal point, not at the far end of the fraction.

How is each new root digit chosen?

Worked example 8. Find √529 by the division method.

1. Group the digits as 5 | 29, where the vertical line separates digit groups.
2. The largest square not exceeding 5 is 2² = 4. Write the first root digit 2 and subtract 4 from 5, leaving remainder 1.
3. Bring down the complete next group, 29, beside the remainder to obtain 129.
4. Double the root obtained so far: 2 × 2 = 4. Append a trial digit to 4 and multiply the resulting number by that same digit.
5. Choose 3 because 43 × 3 = 129. Subtracting leaves zero, so the completed root is 23.

Answer: √529 = 23. A remainder is the amount left after the subtraction at a division stage.

Where does the decimal point go?

Worked example 9. Find √17.64.

Answer: Group the digits as 17 | 64 across the decimal point. Since 4² = 16, the first digit is 4 and the remainder is 1. Bring down 64 to make 164. Double 4 to get 8; choose 2 because 82 × 2 = 164. Put the decimal point before this root digit: √17.64 = 4.2.

For a positive perfect square, the number of digit groups in its whole-number part gives the number of digits in its whole-number square root. At each stage, choose the largest trial digit whose product does not exceed the current dividend. Then check the completed answer by squaring.

How are cubes, cube roots and root estimates understood?

The cube of a number is the product of three copies of it: a³ = a × a × a. A perfect cube is a natural number that is the cube of a natural number. The symbol ∛ denotes a cube root, which reverses cubing.

For cube roots by the factor method, use numbers containing at most three digits. Group equal prime factors into triples rather than pairs. Choose one factor from each triple and multiply the chosen factors to obtain the cube root.

Worked example 10. Find ∛216 by prime factorisation.

Answer: Write 216 = 2 × 2 × 2 × 3 × 3 × 3. There is one triple of 2s and one triple of 3s. Therefore ∛216 = 2 × 3 = 6. Cubing 6 gives 216, which checks the answer.

The grouping also tests whether a number is a perfect cube. For 243 = 3 × 3 × 3 × 3 × 3, two factors remain after one triple is formed. Therefore 243 is not a perfect cube.

What does estimating a root involve?

An estimate gives an approximate value rather than necessarily an exact one. Compare the given number with nearby perfect squares or cubes. Their roots locate an interval for the required positive root. Test numbers within the interval to move nearer to it.

For example, 2³ = 8 and 3³ = 27 show that the cube root of 9 lies between 2 and 3. This locates the root without making 9 a perfect cube. Distinguish such an interval estimate from an exact root found by complete prime-factor grouping.

For square roots, locating a number between nearby perfect squares likewise locates its root between their positive roots. Squaring a trial value inside that interval tells whether the trial lies below or above the required root, so the next trial can be closer.

How do generalised numbers help solve digit puzzles?

A digit is one of the symbols 0 to 9. In a two-digit number, let a denote the tens digit and b the units digit. The number has value 10a + b, where 10a means 10 × a. Its leading digit a must be non-zero.

For a three-digit number, let a, b and c denote the hundreds, tens and units digits respectively. Its generalised form is 100a + 10b + c. Each letter is a digit, and the hundreds digit is non-zero so that the number really has three digits.

This form keeps place value separate from the digit itself. Moving a digit to another position changes its contribution to the number. A row of digit letters in a puzzle represents those positions; it does not mean multiplication of the letters.

How are missing digits found?

Worked example 11. Find the digits A and B in 41A + B4 = 512. Here 41A means 410 + A and B4 means 10B + 4.

Answer: In the units column, A + 4 must end in 2, so A = 8 and 1 is carried to the tens column. The tens column then contains 1 + B + 1, giving B = 9. The completed addition is 418 + 94 = 512.

A carry transfers a complete group of ten to the next place. Work from the units column and include any carry before solving the next column. Keep a letter's value consistent throughout the puzzle, then substitute the digits back to check the entire calculation.

Why do the divisibility tests for 2, 3, 5, 9 and 10 work?

A number is divisible by another when division leaves no remainder. A multiple is obtained by multiplying by an integer. Divisibility tests use place value to decide whether a division will be exact without carrying out the full division.

For a three-digit number 100a + 10b + c, both 100a and 10b are multiples of 2, 5 and 10. Therefore divisibility by each of these depends on the units digit c. For two-digit numbers, the same reasoning applies to 10a + b.

Why does the digit sum work for 3 and 9?

Rewrite 100a + 10b + c as 99a + 9b + (a + b + c). The first two terms are multiples of 9 and also of 3. Hence the digit sum determines divisibility by 9 or by 3 respectively.

Similarly, 10a + b = 9a + (a + b). This proves the corresponding tests for a two-digit number. The argument explains why digits are added for these tests, whereas the units digit is inspected for 2, 5 and 10.

Worked example 12. The three-digit number 42x is divisible by 9, where x is its units digit. Find x.

Answer: The digit sum is 4 + 2 + x = 6 + x. Since x is a digit from 0 to 9, the only possible multiple of 9 for this sum is 9. Thus x = 3.

How do union, intersection, disjoint sets and complements work?

A set is a well-defined collection of objects, called its elements. Well-defined means membership can be decided clearly. Braces { } enclose the elements when they are listed. A subset contains only elements of another set. A universal set contains all elements under consideration.

Use A and B to name sets and U to name the universal set. The union A ∪ B contains elements in A or B or both. The symbol ∪ means union. List an element only once even when it belongs to both sets.

The intersection A ∩ B contains elements common to both sets. The symbol ∩ means intersection. These two operations answer different questions: union gathers everything included in either set, while intersection selects only the shared elements.

Worked example 13. For A = {2, 4, 6, 8} and B = {6, 8, 10, 12}, find A ∪ B and A ∩ B.

Answer: A ∪ B = {2, 4, 6, 8, 10, 12}. The common elements 6 and 8 appear once in this union. Both sets contain 6 and 8, so A ∩ B = {6, 8}.

How do Venn diagrams show these operations?

A Venn diagram represents sets by closed curves, usually circles. The universal set is usually represented by a rectangle. Overlapping regions show shared membership. The shading indicates the set selected by the operation.

A rectangle labelled U contains overlapping circles labelled A and B. Both circles, including their common region, are shaded. The label A ∪ B appears below the circles.

See Fig. 1.4 in your NCERT textbook

Overlapping circles A and B lie inside a rectangle labelled U. Only their shared central region is shaded, and the label A ∩ B identifies that intersection.

See Fig. 1.5 in your NCERT textbook

What if sets have no common elements?

The empty set, denoted ∅, contains no elements. Sets are disjoint when their intersection is empty: A ∩ B = ∅. Merely having different lists does not make sets disjoint; check whether any element is shared.

The complement of A consists of elements in U that are not in A. It is written A′, read “A complement”. The universal set must be specified because the complement is taken within that collection.

For U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}, the complement is A′ = {2, 4, 6, 8, 10}. No elements outside U are added.

Glossary

• Rational number — A number expressible as a quotient of integers with a non-zero denominator.
• Irrational number — A number that cannot be expressed as an integer divided by a non-zero integer.
• Closure — The property that an operation on members of a collection produces another member.
• Commutativity — The property that changing the order of the numbers leaves the result unchanged.
• Associativity — The property that changing the grouping of numbers leaves the operation's result unchanged.
• Additive inverse — A number that gives zero when added to the original number.
• Reciprocal — The multiplicative inverse of a non-zero number, giving one when multiplied by that number.
• Integral exponent — An exponent that is an integer, including positive integers, negative integers and zero.
• Perfect square — A natural number obtained by multiplying a natural number by itself.
• Perfect cube — A natural number obtained as the product of three copies of a natural number.
• Prime factorisation — Expressing a natural number greater than one as a product of prime numbers.
• Union — The set containing elements in either of two sets, including their shared elements.
• Intersection — The set containing exactly the elements common to both of the given sets.
• Disjoint sets — Sets with no common elements, so that their intersection is the empty set.
• Complement — The set of elements in a specified universal set that are absent from the given set.

Common errors and misconceptions

• Misconception: Rational numbers exclude integers. Correct: Every integer can be written with denominator 1, so integers belong to the rational numbers.
• Misconception: Closure allows division by zero. Correct: A rational quotient requires a non-zero divisor. Division by zero is undefined.
• Misconception: Commutativity permits reversing a subtraction. Correct: Addition and multiplication are commutative; subtraction and division are not. Preserve their order.
• Misconception: A reciprocal means changing the sign. Correct: Changing the sign gives the additive inverse. A reciprocal gives a product of 1 with the original non-zero number.
• Misconception: A negative exponent makes a positive base negative. Correct: It indicates a reciprocal. The sign in the exponent does not change a positive base into a negative one.
• Misconception: √81 means both 9 and −9. Correct: Both numbers square to 81, but √81 denotes the positive square root, 9.
• Misconception: Cube roots use pairs of prime factors. Correct: Square roots use pairs; cube roots use triples. Account for every prime factor.
• Misconception: Different sets must be disjoint, and a complement includes everything outside a set. Correct: Disjoint sets share no elements. A complement includes only elements within the specified universal set.

Exam-style questions with model answers

Q1. State the additive and multiplicative identities for rational numbers and explain what each does. [2 marks]

1. The additive identity is 0. Adding zero to a rational number leaves that number unchanged.
2. The multiplicative identity is 1. Multiplying a rational number by one leaves that number unchanged.

Q2. Explain why infinitely many rational numbers lie between two distinct rational numbers a and b, where a < b. [3 marks]

1. The arithmetic mean (a + b)/2 is rational because addition of rational numbers and division by the non-zero number 2 give rational results.
2. This mean lies strictly between a and b, so it supplies a new rational number inside the given interval.
3. Repeat the mean construction in a smaller interval between an endpoint and the new number. The process never stops, giving infinitely many rational numbers.

Q3. Simplify 2⁵ ÷ 2⁻⁶ and express 4⁻³ as a power with base 2. Explain the exponent law used in each calculation. [4 marks]

1. For division of powers with the same non-zero base, subtract the exponent of the divisor from that of the dividend.
2. Here 5 − (−6) = 11, so 2⁵ ÷ 2⁻⁶ = 2¹¹. Retain the brackets around the negative exponent during subtraction.
3. For the second calculation, rewrite the base using 4 = 2². Thus 4⁻³ becomes (2²)⁻³.
4. For a power raised to another power, multiply the exponents. Hence (2²)⁻³ = 2⁻⁶.

Q4. Use the division method to find √529, explaining the digit grouping, trial digit and final result. [5 marks]

1. Separate the digits into groups from the units place: 5 | 29. Begin with the leftmost group, 5, to find the first root digit.
2. The largest square not exceeding 5 is 2² = 4. Therefore write 2 as the first root digit and subtract 4, leaving remainder 1.
3. Bring down the whole next group, 29, to the right of this remainder. The new dividend is 129.
4. Double the root digit 2 to get 4. Append the trial digit 3 to form 43, because 43 × 3 = 129.
5. The subtraction leaves zero and no digit groups remain. The root digits are 2 and 3, so √529 = 23.

Q5. Find ∛216 using prime factorisation and verify your answer by cubing it. [3 marks]

1. Express 216 as a product of primes: 216 = 2 × 2 × 2 × 3 × 3 × 3. Retain every factor while making groups.
2. There is one complete triple of 2s and one complete triple of 3s. Taking one factor from each triple gives ∛216 = 2 × 3 = 6.
3. Check by cubing the answer: 6 × 6 × 6 = 216. This recovers the original number and verifies the cube root.

Q6. The number 41A has hundreds digit 4, tens digit 1 and units digit A. The number B4 has tens digit B and units digit 4. Find the digits A and B if 41A + B4 = 512, and check the addition. [4 marks]

1. In the units column, A + 4 must have units digit 2. Since A is a digit, A = 8, producing 12.
2. Write the units digit 2 and carry 1 into the tens column. That column now contains 1 + B + 1.
3. Its units digit must be 1, so B = 9. The tens sum is 11, giving another carry of 1 into the hundreds column.
4. The hundreds column becomes 4 + 1 = 5. Substitution checks the complete addition: 418 + 94 = 512.

Q7. Given A = {2, 4, 6, 8} and B = {6, 8, 10, 12}, find their union and intersection, and decide whether they are disjoint. [3 marks]

1. The union includes every element in either set, with repeated elements listed once. Therefore A ∪ B = {2, 4, 6, 8, 10, 12}.
2. The intersection contains only elements belonging to both sets. The shared elements are 6 and 8, so A ∩ B = {6, 8}.
3. The sets are not disjoint because their intersection contains elements. Disjoint sets would have an empty intersection, with no element common to them.

Q8. A farmer divides a field of 49 4/5 hectares equally among one son and two daughters. Find the area of each person's share. [3 marks]

1. The area is 49 4/5 hectares. Convert this mixed number to the fraction (49 × 5 + 4)/5 = 249/5 hectares.
2. One son and two daughters make three recipients. Equal sharing therefore requires division of the total area by 3, rather than subtraction of 3.
3. Each share is (249/5) ÷ 3 = (249/5) × 1/3 = 83/5 hectares, or 16 3/5 hectares. Each recipient receives this same area.

Key takeaways

• Rational numbers include integers and fractions expressible with integer numerator and non-zero integer denominator.
• Addition and multiplication are commutative and associative for rational numbers; subtraction and division are neither.
• Zero is the additive identity, one is the multiplicative identity, and zero has no reciprocal.
• The arithmetic mean gives a rational number between two distinct rational numbers, and the process can continue indefinitely.
• Negative integral exponents express reciprocals; apply exponent laws only after checking bases, brackets and non-zero conditions.
• Prime factors form pairs for square roots and triples for cube roots; ungrouped factors prevent a perfect power.
• Place value explains generalised numbers, digit puzzles and divisibility tests based on units digits or digit sums.
• Union includes either set, intersection selects shared elements, and a complement depends on the specified universal set.

Test yourself

Why must the denominator of a rational number be non-zero?

The denominator is a divisor, and division by zero is undefined.

What is the difference between commutativity and associativity?

Commutativity concerns changing the order of numbers; associativity concerns changing their grouping.

How do you find a rational number between distinct rational numbers a and b?

Calculate their arithmetic mean, (a + b)/2, which lies between the two given numbers.

What does a negative exponent mean for a non-zero base?

It indicates the reciprocal of the corresponding power with a positive exponent.

Why is 243 = 3 × 3 × 3 × 3 × 3 not a perfect cube?

Two factors of 3 remain after forming one triple, so complete grouping is impossible.

Why does a digit sum test divisibility of a three-digit number by 9?

Write the number as 99a + 9b + (a + b + c). The first two terms are multiples of 9.

When are two sets disjoint?

They are disjoint when they share no elements, so their intersection is empty.

For U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}, what is A′?

A′ = {2, 4, 6, 8, 10}, the elements in U that are not in A.