Chapter 1 — Number System

What are natural numbers? Natural numbers are the counting numbers we use first when learning mathematics. They begin at 1 and continue indefinitely: 1, 2, 3, 4, 5... We use natural numbers to count objects, list positions (first, second, third), and to label quantities that are whole and positive. Because they are infinite, there is no largest natural number; you can always add one more.

Important properties Natural numbers follow simple laws that make arithmetic reliable. Closure means that if you add or multiply two natural numbers the result is again a natural number. Commutative property means you may swap the order in addition or multiplication without changing the result: a + b = b + a and a × b = b × a. Associative property allows grouping of terms: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). The distributive law links multiplication and addition: a × (b + c) = a × b + a × c. These rules help to rearrange calculations and to simplify expressions.

Identity elements For addition the identity is 0 because a + 0 = a, but note that many definitions of natural numbers exclude 0; in this chapter natural numbers start from 1, so 0 is treated separately as a whole number. For multiplication the identity is 1 because a × 1 = a. Understanding identity elements is useful when solving equations or simplifying expressions.

Limitations and care Subtraction and division of natural numbers do not always produce natural numbers: 3 − 5 or 2 ÷ 3 do not give natural numbers. This leads to extending the set of numbers to whole numbers, integers and rational numbers. When solving problems, do not assume subtraction will always result in a natural number. Using the properties above lets you reorganise calculations for speed and accuracy, and prepares you for higher topics such as prime factorisation and arithmetic of fractions and decimals.

Definition and role of whole numbers Whole numbers are the numbers 0, 1, 2, 3, ... . They include all natural numbers plus zero. The inclusion of zero is important because it represents the absence of quantity and acts as a placeholder in our number system. Whole numbers are used when counting can include 'none' as a valid amount: for example, zero apples means no apples.

Place value in the decimal system Our standard number system is base ten (decimal). Each digit in a whole number has a value depending on its position. From right to left the places are units (10^0), tens (10^1), hundreds (10^2), thousands (10^3), and so on. For example the number 4,582 has 4 thousands, 5 hundreds, 8 tens and 2 units. The value contributed by each digit is digit × place value. This positional system means digits stay the same but their value changes with position; thus 3 in 30 (3×10) is different from 3 in 3 (3×1).

Reading and writing large numbers For reading clarity, group digits in threes from the right: units, thousands, millions. A number like 1,234,567 is read as one million two hundred thirty-four thousand five hundred sixty-seven. When writing numbers in words follow the grouping method to avoid mistakes. In an examination, write commas correctly and label places in a place-value chart if asked.

Zero as placeholder and arithmetic Zero is essential for representing numbers like 105 where the tens place is empty. In written arithmetic align numbers by place value, especially when adding or subtracting. When multiplying or dividing by powers of ten, digits shift left or right: multiplying by 10 shifts digits one place to the left, dividing by 10 shifts them right. Correct placement of the decimal point depends on understanding place value.

Common errors and tips Common mistakes include misplacing zeros, misreading place names, or shifting digits incorrectly when working with decimals. Practice writing numbers in a place-value table and convert between words and figures regularly to build accuracy and confidence.

What are integers and why we use negatives Integers are the set of whole numbers extended to include their negative counterparts and zero: ..., −3, −2, −1, 0, 1, 2, 3, ... . Negative numbers arise naturally in contexts such as temperatures below zero, bank overdrafts, or levels below sea level. Integers allow subtraction to be always possible within the set if the result is integer (for example 2 − 5 = −3).

Number line representation The number line is the best way to visualise integers. Zero is in the middle; positive integers lie to the right and negatives to the left. The rule 'right is greater' helps compare any two integers: a number further right is larger. For example −2 is greater than −5 because −2 is to the right of −5. Distances on the number line show absolute values: |−4| = 4.

Rules for arithmetic with integers Addition and subtraction of integers use sign rules. When adding integers with same sign add absolute values and keep the sign; e.g., (−3) + (−6) = −9. When signs differ, subtract the smaller absolute value from the larger and take the sign of the larger absolute value: 7 + (−4) = 3. Subtraction can be treated as adding the opposite: a − b = a + (−b). Multiplication and division follow sign rules: same sign gives positive result, different signs give negative. So (−3) × (−2) = 6, and (−6) ÷ 3 = −2.

Applications and pitfalls Integers are used in everyday problems involving gains and losses, directions, and algebra. Students often confuse subtraction of negatives and forget to change signs; always convert subtraction into addition of the opposite to avoid mistakes. Also check answers by considering what makes sense in the problem context: for example negative number of items usually is not meaningful unless representing debt or loss.

Understanding factors and multiples A factor of a number is an integer that divides the number exactly without leaving a remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 exactly. Factors come in pairs: a × b = n means a and b are a factor pair of n. A multiple of a number is obtained by multiplying that number by an integer: multiples of 5 are 5, 10, 15, 20, and so on. Every integer has infinitely many multiples but a finite number of positive factors.

Prime and composite numbers A prime number is a number greater than 1 that has exactly two distinct positive factors: 1 and itself. Examples include 2, 3, 5, 7, 11. Because 2 is divisible only by 1 and 2, it is the only even prime. Composite numbers have more than two factors; for example 12 is composite because it has factors 1, 2, 3, 4, 6 and 12. The number 1 is neither prime nor composite because it has only one factor.

Methods to find factors and test primes To find factors of a number efficiently, divide by integers up to the square root of the number; if a divides n then n ÷ a gives the paired factor. To test whether a number is prime try dividing by small primes (2, 3, 5, 7, 11...) up to its square root; if none divide it exactly, it is prime. For larger numbers more advanced methods exist, but for class 7 these simple trials work well.

Prime factorisation and its importance Every integer greater than 1 can be expressed as a product of primes — its prime factorisation. For example 84 = 2^2 × 3 × 7. Prime factorisation is useful to compute HCF and LCM, to simplify fractions, and to solve divisibility problems. Students should practise writing numbers as product of primes using factor trees or repeated division.

Common mistakes Confusing factors with multiples is common: factors divide the number, multiples are divisible by the number. Also forgetting that 1 is neither prime nor composite leads to errors. Practice by listing factors and multiples for small numbers and using factor pairs to see structure clearly.

Definitions and why they matter The Highest Common Factor (HCF) of two or more integers is the greatest integer that divides each of them exactly. The Lowest Common Multiple (LCM) is the smallest positive integer that is a multiple of each of the given numbers. HCF and LCM are widely used in arithmetic with fractions (simplifying and adding/subtracting), in solving problems involving repeated cycles (like bus schedules) and in partitioning objects into equal groups.

Finding HCF by prime factorisation Express each number as a product of primes. For HCF, identify the primes common to all numbers and take them with their smallest powers. Example: for 48 = 2^4×3 and 180 = 2^2×3^2×5, common primes are 2 and 3; smallest powers are 2^2 and 3^1, so HCF = 4×3 = 12. This method is systematic and helps avoid missed factors.

Finding LCM by prime factorisation Use the same prime factorisations but for LCM take each prime with its highest power appearing in any factorisation. Using above numbers 48 and 180, highest powers are 2^4, 3^2 and 5^1 so LCM = 16×9×5 = 720. This gives the smallest number divisible by both original numbers.

Euclid's algorithm for HCF Euclid’s division method is efficient for large numbers. Divide the larger number by the smaller, take the remainder, then replace the larger by the smaller and the smaller by the remainder and repeat until remainder is zero; the last non-zero remainder is the HCF. This algorithm relies on properties of divisibility and is fast even when prime factorisation is difficult.

Relation between HCF and LCM For two numbers a and b their product equals the product of HCF and LCM: a × b = HCF(a,b) × LCM(a,b). This formula can be used to find one when the other is known, and it often appears in exam questions. Practice both prime factor and Euclid methods so you can choose the faster one depending on the numbers given.

Definition and parts of a fraction A fraction represents a part of a whole and is written as a/b where a is the numerator and b is the denominator (b ≠ 0). The denominator shows into how many equal parts the whole is divided, and the numerator shows how many of those parts are taken. Fractions are essential for describing quantities less than one, sharing, and portions of measurements.

Types of fractions Fractions are classified into several types. Proper fractions have numerator smaller than the denominator (e.g., 3/7) meaning the value is less than one. Improper fractions have numerator equal to or larger than the denominator (e.g., 7/4); they are greater than or equal to one and can be converted into mixed numbers (1 3/4). Mixed numbers combine a whole number and a proper fraction, useful to represent everyday measurements like 2 1/2 hours. A unit fraction has numerator 1 (1/5), showing one part of equal division.

Visual interpretation and models Fractions can be shown using shapes (circles, rectangles) divided into equal parts and shading some parts. For example, to show 3/8 shade three of the eight equal parts. Using number line models, place fractions between 0 and 1 by subdividing the segment into equal parts; this helps compare fractions and understand size. Fraction bars and grids are useful teaching tools for visual learners.

Equivalent fractions and improper conversion Many fractions represent the same value: 1/2 = 2/4 = 3/6. Multiplying or dividing numerator and denominator by the same non-zero integer produces equivalent fractions. To convert an improper fraction to a mixed number divide numerator by denominator: the quotient is the whole part and the remainder forms the fractional part. Practising these conversions and visualising them helps students move comfortably between different representations.

Applications and caution Fractions are used in cooking, measurement, and dividing objects. A common error is treating denominators as independent of numerators when comparing fractions; instead, compare equal parts using common denominators or use visual models. Clear understanding of fraction types prepares students for operations on fractions which follow in later topics.

Equivalent fractions explained Equivalent fractions are different fractions that represent the same portion of a whole. They occur because multiplying or dividing both numerator and denominator of a fraction by the same non-zero number does not change its value. For example, multiplying numerator and denominator of 1/3 by 2 gives 2/6, and by 3 gives 3/9; all these express the same amount. Recognising equivalent fractions is a key skill for comparing, adding, subtracting and simplifying fractions.

Finding equivalent forms To produce equivalent fractions multiply numerator and denominator by any integer k ≠ 0. To reduce a fraction to an equivalent but simpler form divide numerator and denominator by a common factor. Rewriting fractions with common denominators is necessary when adding or subtracting—often LCM of denominators gives the least common denominator which simplifies the work and final simplification step.

Simplifying to lowest terms A fraction is in lowest or simplest terms when numerator and denominator share no common factor other than 1. To simplify, find the HCF of numerator and denominator and divide both by it. For example 56/98 has HCF 14; dividing both gives 4/7. Simplification makes arithmetic easier and final answers tidy. In exams always present fraction answers in simplest form.

Comparing fractions using equivalent forms To compare two fractions, convert them to equivalent fractions with a common denominator (often the LCM) and then compare numerators. Alternatively use cross-multiplication: a/b and c/d can be compared by checking whether ad > bc or ad < bc. Cross-multiplication avoids computing full LCM and is faster for exams, but both methods are valid and should be used correctly.

Practical tips Cancel common factors early when multiplying fractions to keep numbers smaller, and always check if the final fraction can be simplified further. Use visual models to confirm equivalent fractions, and practise converting fractions to several equivalent forms to become fluent in simplifying and comparing.

Understanding decimals Decimals extend the place-value system to represent numbers less than one, using a decimal point to separate the whole part from fractional parts. Each position to the right of the decimal point represents a negative power of ten: the first is tenths (10^-1), second is hundredths (10^-2), third is thousandths (10^-3), and so on. For example, 3.456 equals 3 + 4×10^-1 + 5×10^-2 + 6×10^-3 = 3 + 0.4 + 0.05 + 0.006 = 3.456.

Writing and reading decimals Read the whole-number part, say 'point', then read each digit separately: 0.205 is read as 'zero point two zero five'. When converting fractions with denominators 10, 100, 1000, write the numerator with the decimal point placed so that it occupies the correct number of decimal places: 57/100 = 0.57, 3/10 = 0.3. For fractions whose denominator is not a power of ten, perform division to get a decimal representation; it may terminate or repeat.

Terminating vs repeating decimals A decimal either stops after finite digits (terminating) or continues with a repeating pattern (recurring). A rational number in simplest form has a terminating decimal if its denominator has only 2 and/or 5 as prime factors. Otherwise, its decimal expands into a repeating pattern. For example 1/8 = 0.125 (terminating) while 1/3 = 0.333... (repeating).

Operations and shifting decimal point Multiplying by powers of ten shifts the decimal point to the right: 3.45×10 = 34.5, ×100 = 345. Dividing by powers of ten shifts it left: 345 ÷ 100 = 3.45. When adding or subtracting decimals align decimal points vertically, then operate on digits column-wise. For multiplication, ignore the point, multiply as integers and then place the decimal so that the total number of decimal places equals the sum of decimal places in the factors. For division, you may shift divisor and dividend to make the divisor an integer and then divide.

Comparing and converting To compare decimals, align decimal points and compare digits from left to right, adding trailing zeros if necessary. Convert decimals to fractions by writing the digits after the decimal as numerator over the corresponding power of ten and simplifying: 0.75 = 75/100 = 3/4. Practice is essential for correct placement of the decimal point in operations and understanding relationship between decimals, fractions and percentages.

Why conversions matter Fractions, decimals and percentages are three ways to express the same idea: parts of a whole. Different contexts use different forms—money often uses decimals, tests and statistics often use percentages, while recipes and ratios commonly use fractions. Being able to convert between these forms quickly helps in solving problems and checking answers.

From fractions to decimals Divide numerator by denominator. If the denominator is a factor of a power of ten (its prime factors are only 2 and/or 5), the decimal will terminate. Otherwise it will repeat. For example 3/8 = 0.375 because dividing 3 by 8 ends after three decimal places. For denominators 10, 100, 1000 simply place the decimal point accordingly: 57/100 = 0.57.

From decimals to fractions Write the decimal digits after the point as numerator and put the appropriate power of ten as denominator, then simplify. For example 0.46 = 46/100 = 23/50. For repeating decimals use an algebraic trick: if x = 0.272727..., then multiply by an appropriate power of 10 to align repeats (100x = 27.2727...) and subtract to solve for x as a fraction.

From fractions/decimals to percentages Multiply by 100 and add the percent sign to convert decimals to percentages: 0.2 = 20%. For fractions multiply the fraction by 100%: (a/b)×100%. This is useful in calculating discounts, marks, or interest rates. To convert percentage to decimal divide by 100 (25% → 0.25) or to fraction write over 100 and simplify (25% = 25/100 = 1/4).

Practical strategies and checks Always simplify fractions after conversion. Use calculators for long repeating decimals but practise the algebraic method for repeating patterns to show exact answers. Visual checks on number line or converting back to the original form help confirm correctness. Learn common equivalents by heart: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75% to speed up problem solving.

Adding and subtracting like fractions When fractions have the same denominator (called like fractions), you can add or subtract by operating on numerators while keeping the common denominator the same. For example, 3/7 + 2/7 = (3+2)/7 = 5/7. Always reduce the result to simplest form if possible. If the result is an improper fraction, convert to a mixed number if the question requires that form.

Handling unlike denominators When denominators differ, first find a common denominator so the fractions refer to the same-sized parts. The least common denominator is often the LCM of denominators; convert each fraction into an equivalent one with that denominator. For example, to add 2/3 and 3/4, find LCM 12: 2/3 = 8/12 and 3/4 = 9/12, then add to get 17/12, which simplifies to 1 5/12.

Working with mixed numbers Convert mixed numbers into improper fractions before adding or subtracting. Alternatively, add whole parts and fractional parts separately, borrowing or carrying if needed. For subtraction that requires borrowing, convert one whole into fractional parts equivalent to the denominator before subtracting the fractional parts.

Sign rules and subtraction as addition Treat subtraction as adding the negative: a − b = a + (−b). This is helpful when fractions include negative signs or when combining with integers. Keep track of signs and convert all mixed numbers consistently to avoid errors.

Practical tips and checking Always simplify intermediate equivalents when possible to keep numbers small. After computing, simplify the final answer to lowest terms. For checking, convert the result to a decimal to see if the answer is reasonable, especially in word problems. Practice with diagrams such as fraction strips or pie charts to visualise why denominators must match before adding parts.

Multiplication of fractions To multiply two fractions, multiply their numerators to get the new numerator and their denominators to get the new denominator: (a/b)×(c/d) = (ac)/(bd). This rule follows from repeated portioning: a/b of c/d equals ac/bd. Before multiplying, simplify by cancelling common factors between any numerator and any denominator to keep numbers small; this is called cross-cancellation and helps avoid large intermediate products. For mixed numbers, first convert to improper fractions (for example 1 1/2 → 3/2) then multiply.

Division of fractions Dividing by a fraction means multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b)×(d/c), provided c ≠ 0. The reciprocal of c/d is d/c. Again convert mixed numbers to improper fractions first and use cancellation before multiplying. This rule comes from the idea of asking how many times the divisor fits into the dividend.

Interpreting results If fractions represent measurements, multiplication may change units; for example multiplying two lengths gives area. Division of fractions is used to split quantities into smaller equal parts or to find how many of a fractional portion fit into a whole. Always check whether the result should be a fraction, a mixed number, or a simplified improper fraction, depending on the context.

Strategies and checking Cancel factors before multiplying to simplify calculations. After finding a product or quotient, reduce the fraction to lowest terms and, if appropriate, convert to a mixed number. Estimate results by converting fractions to decimals to see if numerical answers are sensible. Practise with visual models: rectangle overlap can represent multiplication (e.g., 1/2 of 1/3 is 1/6) and grouping helps in understanding division.

Definition and examples Rational numbers are all numbers that can be expressed as a fraction a/b where a and b are integers and b ≠ 0. This set includes whole numbers, integers, terminating decimals and recurring decimals. For example 5 is rational because 5 = 5/1, 0.75 is rational because 0.75 = 75/100 = 3/4, and 0.333... is rational because it equals 1/3. Rational numbers describe exact ratios of integers and are useful in precise calculations.

Arithmetic and closure Rational numbers are closed under addition, subtraction, multiplication and division (except division by zero). This means combining rational numbers by these operations always yields another rational number. For example the sum of two rationals a/b and c/d can be written as (ad + bc)/bd which is a rational number. Division by a rational c/d (non-zero) results in (a/b)×(d/c), again rational.

Decimal forms and termination Every rational number has a decimal expansion which either terminates or repeats a block of digits infinitely. The decimal terminates if, when the fraction is in lowest terms, the denominator's prime factors are only 2 and/or 5. If other prime factors are present (for example 3 or 7) the decimal repeats. Recognising this pattern helps in converting between fractions and decimals and in predicting whether a decimal will repeat or end.

Density and ordering Rational numbers are dense on the number line: between any two rational numbers there exists another rational number. This means there is no gap in rational numbers though there are irrational numbers between rationals. To compare rationals use common denominators or cross-multiplication. On number line, relative positions are used to order rationals. This concept of density is important for understanding limits and approximations later in mathematics.

Applications and caution Rational numbers model exact proportions in measurements, mixtures, finance and probability. Be careful to simplify fractions and to handle repeating decimals correctly in conversions. Practise problems that mix integers, fractions and decimals so you can recognise rational numbers in different forms quickly and use the most convenient form for computation.

General principles Comparing numbers means deciding which is larger, smaller or if they are equal. For whole numbers use place value: the number with more digits is larger, unless digits count is same then compare leftmost differing digit. For integers use a number line: the one to the right is larger. For mixed sets including fractions and decimals, convert to a common representation (decimals or fractions with common denominator) to compare reliably.

Comparing fractions If fractions have the same denominator compare numerators directly: the larger numerator means the larger fraction. If they have the same numerator, the one with smaller denominator is larger because the parts are bigger. For general fractions, cross-multiplication is a quick method: compare a/b and c/d by computing ad and bc; if ad > bc then a/b > c/d. This avoids finding LCM and is fast for written work.

Comparing decimals Align decimal points and compare digits from left to right. If necessary, add trailing zeros to make the number of decimal places equal (e.g., 0.5 = 0.50). For example to compare 0.309 and 0.291, compare tenths: 0 and 0 equal, hundredths: 3 vs 2 so 0.309 > 0.291. Many mistakes happen from ignoring trailing zeros or misaligning decimal points.

Ordering mixed types To order a mix of integers, fractions and decimals, choose a uniform form. Converting fractions to decimals often works but watch for repeating decimals; converting decimals to fractions or fractions to common denominators is another reliable approach. Use number line sketches to visualise ordering, particularly with negative numbers. In answering exam questions show clear steps of conversion and the method used so marks are awarded even if arithmetic slips occur.

Practical tips Practice comparing numbers of different types regularly. Use the cross-multiplication method for fractions and align decimals carefully. For estimation, round numbers to compare magnitudes quickly. Always re-check by converting two forms if unsure about the result.

Rounding rules and purpose Rounding replaces a number by a nearby value with fewer digits so calculations become easier and quicker. To round to the nearest ten, hundred or decimal place, look at the digit immediately right of the target place: if it is 5 or more round up, if less than 5 round down. For example 347 rounded to the nearest hundred becomes 300 since the tens digit 4 < 5. For decimals, round 4.678 to two decimal places by looking at the third decimal (8) and increasing the second decimal by one, giving 4.68.

Estimation techniques Estimation gives an approximate result quickly and helps check whether an exact answer is reasonable. Front-end estimation uses the leading digits: for 398 + 274 you might use 300 + 200 = 500 as a quick guide. Compatible numbers replace given numbers with close numbers that are easier to compute mentally: for division 198 ÷ 6, use 200 ÷ 5 ≈ 40 as a rough check. When multiplying many numbers, round each to one or two significant figures and multiply to get a rough idea.

Using estimation to check work After solving a calculation, estimate the expected size of the answer to detect mistakes. For addition of fractions, converting to decimals and approximating helps see if the result is sensible. Estimation is not a substitute for exact work in exams unless specifically asked, but it is a useful verification step.

Solving word problems step-by-step Read the problem carefully and identify known and unknown quantities and their units. Translate words into mathematical expressions using correct number types (integers, fractions, decimals). Plan the steps to solve, perform calculations showing work, and finally check units and reasonableness of the answer. For multi-step problems, label intermediate results and round only at the end unless told otherwise.

Practice tips Common exam tasks include rounding to specified places, estimating sums/products, and applying these skills in word problems involving money, measurement and time. Use flow steps: understand the problem, choose method, compute, and check. Regular practice with different contexts builds confidence and speed.