Chapter 1 — Number System

Understanding place value is the first step for working correctly with numbers. Each digit in a number has a value determined by its position. For whole numbers, moving one place to the left multiplies value by ten; moving right divides by ten. For example, in 453, the digit 5 stands for fifty because it is in the tens place.

Decimals extend place value to the right of a decimal point. The first place after the point is tenths, the next is hundredths, then thousandths, and so on. For 3.476, the digit 4 is in the tenths place and represents 4/10; 7 is hundredths or 7/100, and 6 is thousandths or 6/1000.

Expanded form expresses a number as a sum of each digit times its place value. Writing a number in expanded form highlights the contribution of each digit. For example, 2,305 = 2×1000 + 3×100 + 0×10 + 5×1. For decimals, 6.204 = 6×1 + 2×1/10 + 0×1/100 + 4×1/1000.

Reading and writing numbers: practise converting between words, standard form and expanded form. For instance, eighteen thousand and fourteen has the digits placed accordingly and can be rewritten in expanded form which clarifies magnitude. For decimals practise reading numbers like 0.042 as "forty-two thousandths" and linking that to the fraction 42/1000.

Why this matters: place value and expanded form help with estimation, understanding size of numbers, aligning digits when adding and subtracting, and converting between forms. Students should practise reading numbers correctly, writing numbers in words and expanded form, and identifying place values for digits in varied examples. Precision in place value prevents common errors when performing arithmetic with multiple digits and decimals.

Key steps to practice:

• Read aloud numbers and point to the digit you say.
• Write several numbers in expanded form for whole numbers and decimals.
• Convert between words, standard notation and expanded form.

Comparing numbers means finding which number is larger, smaller or whether they are equal. For whole numbers begin by checking the number of digits: a number with more digits is larger than a number with fewer digits (ignoring leading zeros). If two numbers have the same number of digits, compare digits from leftmost to rightmost; the first differing digit determines which number is greater. For example, compare 3,405 and 3,495: the hundreds digit 4 is same, tens digit 0 and 9 differ, so 3,495 is larger.

Decimals require careful alignment by the decimal point. First compare integer parts; if equal, compare tenths, then hundredths and so on until a difference appears. Treat missing places as zeros: for example 2.5 equals 2.50, and 2.519 > 2.5 because at the hundredths place 1 > 0. When two decimals have a repeating pattern, compare their decimal expansions to sufficient places or convert to fractions for exact comparison.

Fractions may be compared by converting to like denominators, by converting to decimals, or by using cross-multiplication. Cross-multiplication is efficient: for a/b and c/d (with positive denominators), compute ad and bc; then a/b > c/d if ad > bc. For example compare 4/7 and 3/5: 4×5=20 and 3×7=21 so 4/7 < 3/5. For mixed numbers compare whole parts first then fractional parts.

Negative numbers reverse the usual order: more negative means smaller. On the number line negatives lie left of zero. To compare negatives, compare absolute values but reverse the inequality: -3 < -1 because 3 > 1 but negative sign flips order. Zero is greater than any negative number and less than any positive number.

Ordering arranges a list of numbers in ascending (small to large) or descending (large to small) order. A practical method is to convert all items to the same form — all decimals or all fractions with common denominators — then sort. Visual placement on a number line also helps: numbers to the right are larger. Remember that rational numbers are dense: between any two distinct rationals there exists another rational, so ordering may require precision in exams; when asked, present exact values (fractions in lowest terms) or correct decimal approximations to required places.

Fractions show parts of a whole or a collection. The numerator is the number of selected parts and the denominator shows how many equal parts the whole is divided into. A visual drawing such as a pie chart or rectangular bar divided into equal regions often clarifies the idea. Proper fractions have numerator smaller than denominator (e.g., 3/5) while improper fractions have numerator equal to or larger than denominator (e.g., 7/4). Mixed numbers combine a whole number and a proper fraction, for example 2 1/3.

Unit fractions are fractions with numerator 1 (1/2, 1/3, 1/4, ...). They are building blocks: many fractions are sums of unit fractions. Equivalent fractions represent the same amount with different numerators and denominators, for example 1/2 = 2/4 = 3/6. To produce equivalents multiply or divide numerator and denominator by the same non-zero integer. Recognising equivalent fractions helps when adding, subtracting or comparing fractions.

Fraction as division: a/b means a divided by b. Thinking of fraction as the result of division is useful for converting fractions to decimals by performing the division. For instance 3/8 means 3 ÷ 8 which gives 0.375. This viewpoint helps when solving measurement problems: 3/4 m is three parts of a metre divided into four equal parts.

Converting between forms: to change an improper fraction to a mixed number divide numerator by denominator to get quotient (whole part) and remainder (fractional numerator). Example: 11/4 = 2 remainder 3 so 2 3/4. To convert a mixed number to improper multiply whole part by denominator then add numerator: 2 3/4 = (2×4 + 3)/4 = 11/4. Practise both conversions to avoid errors during arithmetic operations where one form is easier than another.

Visual and number-line models: draw bars or circles divided equally to show fractions. On the number line, mark unit intervals and divide each into the denominator number of equal parts to place fractions precisely. This helps understand order, size and subtraction or addition by counting steps. Emphasise careful drawing: equal parts must be equal, and shading must match the fraction described.

Equivalent fractions are different fractions that have the same value. They are formed by multiplying or dividing numerator and denominator by the same non-zero integer. For example, 2/3 = 4/6 = 6/9. Recognising these helps in comparing fractions, finding common denominators for addition or subtraction, and simplifying answers. Visually, equivalent fractions appear when the same part of a shape is shaded but the shape is divided into more or fewer equal pieces.

Lowest terms (simplest form) means the numerator and denominator share no factor greater than 1. A fraction in lowest terms cannot be reduced further. To reduce a fraction, find the greatest common divisor (GCD) of numerator and denominator and divide both by it. For instance, to simplify 42/56, GCD(42,56)=14 so divide to get 3/4. Simplifying keeps numbers small and makes comparisons clearer.

Methods to find GCD: list factors of each number to find the largest common one, use prime factorisation to cancel common primes, or use the Euclidean algorithm for larger numbers. Prime factorisation helps explain why cancellation works: common prime factors in numerator and denominator cancel leaving the simplest form. For classroom use, factor trees are useful to visualise this process.

Finding equivalent fractions with a specific denominator: when required to change denominator to a given value, multiply numerator and denominator by the same factor that converts the original denominator to the new one. If that factor is not an integer, first check whether the desired denominator is a multiple of the original. Alternatively, find the LCM of denominators when adding fractions to minimise the size of the common denominator.

Applications and cautions: always reduce final answers to lowest terms unless instructed otherwise. Avoid incorrect cancellation across addition or subtraction (for instance, you cannot cancel terms in 2/3 + 1/4 by removing common digits); cancellation applies only to factors multiplied across numerator and denominator. Regular practice prevents these common mistakes and speeds up calculations during exams.

Addition of fractions requires fractions to have the same denominator (like parts) so we can add the numerators directly. If denominators differ, find a common denominator. The least common multiple (LCM) of denominators is preferred because it keeps numbers smaller. Convert each fraction to an equivalent fraction with the common denominator, then add numerators and simplify the result to lowest terms. For example, to add 3/8 + 5/12, find LCM(8,12)=24; convert to 9/24 + 10/24 = 19/24.

Subtraction of fractions uses the same idea: convert to like denominators then subtract numerators. Be careful with signs: if subtracting leads to a negative result, record the negative sign. For mixed numbers, either convert to improper fractions and then subtract, or subtract whole parts and fractional parts with borrowing. Borrowing from the whole part means converting one whole into denominator-sized fraction pieces and adding to the fractional part before subtracting.

Working with mixed numbers: sometimes adding fractional parts may produce an improper fraction; convert this to a whole number plus a proper fraction by dividing numerator by denominator. For subtraction, if the fractional part of the minuend is smaller than that of the subtrahend, borrow one from the whole part of the minuend.

Efficient strategies: use cancellation where possible before converting to common denominator: factor denominators and cancel common factors to reduce work. Also consider converting both fractions to decimals when denominators are factors of powers of ten and when an exact decimal is acceptable. However in exams fractions in lowest terms are usually preferred unless instructed otherwise.

Common mistakes: do not add denominators when adding fractions; avoid cancelling across addition or subtraction; always simplify the final answer. Practise varied problems including adding three or more fractions and mixed number operations to build confidence.

Multiplication of fractions is done by multiplying numerators together to form the new numerator and denominators together to form the new denominator: (a/b)×(c/d) = (ac)/(bd). Before multiplying, simplify by cancelling any common factor between a numerator and another denominator to keep numbers small and reduce calculations. For example, (4/9)×(3/8) can be simplified by cancelling 4 and 8 by 4 giving (1/9)×(3/2) = 3/18 = 1/6. Converting mixed numbers to improper fractions first avoids confusion: 1 1/2 × 2 1/3 becomes (3/2)×(7/3) after conversion.

Division of fractions uses reciprocals. To divide by a fraction, multiply by its reciprocal: (a/b) ÷ (c/d) = (a/b)×(d/c). This works because division by c/d is the same as multiplying by d/c. Cancel common factors between numerators and denominators before multiplication to simplify. For example, (5/6) ÷ (3/4) = (5/6)×(4/3) = (5×4)/(6×3) which simplifies to (5×2)/(3×3) = 10/9 = 1 1/9.

Applications: multiplication finds a fraction of a quantity (e.g., 2/5 of 150 is (2/5)×150 = 60). Division finds how many parts of a given fractional size fit into a whole (e.g., 6 ÷ 3/4 = 6×4/3 = 8). In measurement and recipes, these operations frequently occur. Practice cancellation and reciprocal identification to operate correctly and quickly.

Common pitfalls: do not flip incorrectly—reciprocal applies only to the divisor. Do not add numerators or denominators when multiplying or dividing. Always give the final answer in lowest terms or as a mixed number when appropriate. Work neatly showing cancellation to avoid arithmetic errors under exam time pressure.

Decimals are another way to write numbers that include parts less than one. The decimal point separates the integer part from the fractional part. Places to the right are tenths, hundredths, thousandths and so on; each place is ten times smaller than the previous. For example 12.304 has 3 tenths (0.3), 0 hundredths (0.00) and 4 thousandths (0.004). Reading decimals correctly links to place-value understanding: 0.05 is five hundredths, not five tenths.

Converting fractions to decimals is division: a/b means a ÷ b. When the denominator divides a power of ten exactly the decimal terminates; otherwise it repeats. For instance 1/4 = 0.25 terminates, while 1/3 = 0.333... repeats. It is useful to recognise denominators with factors only 2 and 5 lead to terminating decimals since 10 = 2×5.

Addition and subtraction with decimals require aligning decimal points vertically and filling missing places with zeros to avoid confusion. Work digit by digit from rightmost place to left, borrowing as necessary for subtraction. For multiplication, ignore the decimal points, multiply as whole numbers, then place the decimal point in the product so that the total number of decimal places equals the sum of decimal places in the factors. For example, 3.2×0.45: treat as 32×45=1440, then place decimal with three places → 1.440 = 1.44.

Division with decimals often requires making the divisor a whole number by multiplying both divisor and dividend by the same power of ten. For instance 4.5 ÷ 0.15 multiply both by 100 → 450 ÷ 15 = 30. When exact division is not possible the quotient may terminate or recur; show recurring digits with a bar in exact answers when required.

Rounding and estimation are important: round decimals to a suitable place to estimate results quickly and check answers for reasonableness. When dealing with money round to two decimal places. Know rules for rounding: if the next digit is 5 or more round up, otherwise round down. Practice many operations to build fluency and avoid misplaced decimal errors which are common in exams.

Two kinds of decimal expansions are important to know: terminating decimals, which end after a finite number of digits, and recurring (repeating) decimals, which continue forever with a repeating block of digits. Terminating decimals are those like 0.125 or 3.5; recurring decimals are those like 0.666... or 0.142857142857... where a group of digits repeats indefinitely.

Why some decimals terminate: consider a fraction a/b in lowest terms. The decimal expansion of a/b terminates if and only if the denominator b has no prime factors other than 2 and 5. This is because powers of 10 equal 2^n×5^n, so multiplying numerator and denominator by a suitable power of 2 or 5 will make the denominator a power of 10 and give a terminating decimal. For example 7/20 = 0.35 terminates because 20 = 2^2×5.

Why some decimals repeat: if the denominator has any prime factor other than 2 or 5 (for example 3,7,11), the decimal expansion cannot terminate and instead repeats. The repeating block length depends on the denominator and can be studied using long division: when a remainder repeats in the division process the corresponding sequence of quotient digits repeats. For example dividing 1 by 7 produces a repeating cycle of length 6 producing 0.142857 repeating.

Converting repeating decimals to fractions uses an algebraic trick. If x = 0.(r) where r is the repeating block of length n, multiply both sides by 10^n to shift the decimal point so the repeating parts align, then subtract to eliminate the repeating portion: (10^n x − x) is an integer equal to the repeating block, giving x = repeating block / (10^n − 1). For decimals with a non-repeating part before the repeat, multiply by two powers of ten and subtract to isolate the repeating portion. Practise with examples of single-digit repeats and longer blocks to become confident.

Notation and marking: represent repeating digits with a bar, for example 0.(3) or write dots above first and last digit of the block. In exams show workings: either long division demonstration of the repeat or algebraic conversion to fraction. Understanding termination and repetition is essential for recognising rational numbers and for exact answers in higher work.

Inter-conversion links three common ways to present parts of a whole. Percent means per hundred. To convert a fraction to percent multiply by 100 and add the % sign: (a/b)×100%. To convert percent to fraction divide by 100 and simplify. To convert decimal to percent multiply by 100; percent to decimal divide by 100. For example 0.45 = 45% = 45/100 = 9/20 after simplification.

Fraction to decimal do a ÷ b. If you need percent, do a ÷ b × 100. Fractions with denominators that are factors of powers of ten (2,4,5,10,20,25,50) turn into terminating decimals easily. Others give repeating decimals requiring conversion methods or leaving answers as fractions for exactness.

Employing shortcuts: know common equivalents such as 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 1/5 = 0.2 = 20%, 3/4 = 0.75 = 75%, and 1/8 = 0.125 = 12.5%. These help solve quick problems in financial calculations, discounts and interest. To get a fraction with denominator 100 multiply numerator and denominator appropriately; then the numerator is the percentage directly (for example 3/25 = 12/100 = 12%).

Practical examples: to find 15% of 200 calculate 0.15×200 = 30. To express 7/8 as percent convert to decimal 7 ÷ 8 = 0.875 then multiply by 100 → 87.5%. When giving answers in exams choose the form requested: if asked for exact value, give fraction; if asked for percentage, show multiplication by 100 and simplify.

Accuracy and rounding: choose appropriate rounding: money typically to two decimal places, measurement answers to required precision. When converting between forms show intermediate steps so the examiner can follow reasoning; state recurring decimals with bar or as fraction for exactness. Regular practice with varied denominators improves speed and reduces mistakes in conversion tasks.

Rational numbers are numbers that can be expressed as a fraction a/b where a and b are integers and b ≠ 0. This includes integers (which have denominator 1), terminating decimals and recurring decimals since they can be written as fractions. The rational numbers form a large and familiar set used in everyday calculations.

Properties of rational numbers include closure under addition, subtraction, multiplication and division (except division by zero). That means the sum, difference, product or quotient of two rationals is rational. Rational numbers can be arranged on the number line and are dense there: between any two distinct rational numbers there exists another rational number. For example between 1/2 and 3/4 lies 5/8.

Standard form for rational numbers usually means a fraction in lowest terms, and when negative the negative sign is written with the numerator (e.g., -3/4). Being able to express rationals in different forms (fraction, decimal, percent) is useful. When working with mixed numbers convert to improper fractions for algebraic operations.

Comparisons and ordering: use common denominators or cross-multiplication to compare rationals exactly. When decimals are given, convert to fractions if exact comparison is needed. Remember sign rules: positives are to the right of zero and negatives to the left. The idea of distance on the number line relates to absolute value: distance of a rational a from zero is |a|.

Applications and density: rationals appear in measurements, probabilities, prices and ratios. Their density means you can always find a rational between any two others; this is an important idea leading to further study of real numbers. Practice converting decimals, simplifying fractions and ordering rationals to become fluent in exact arithmetic and reasoning.

Irrational numbers are numbers that cannot be written as a fraction of two integers. Their decimal expansions are non-terminating and non-repeating. Familiar examples include √2 and π. At this level we treat irrationals informally: students should recognise what makes a number irrational and be able to give examples and simple reasons.

How to tell: if a number's decimal goes on forever without settling into a repeating block, it is irrational. For instance, 0.101001000100001... inserts increasing numbers of zeros and never repeats, so it is irrational. In contrast, 0.666... repeats and is rational. Another way to see irrationality is when no pair of integers a and b can satisfy the equation number = a/b. For some numbers a simple contradiction argument shows impossibility: the classical proof that √2 is irrational assumes √2 = p/q in lowest terms, squares both sides and shows p and q would both be even, contradicting lowest terms.

Real numbers include both rationals and irrationals. The number line represents all real numbers: every point on the line corresponds to a real number. The reals fill the line continuously—the rationals, while dense, leave gaps that are filled by irrationals. This is why some geometrical lengths (like diagonal of a unit square) are irrational.

Practical understanding: students should know that many useful constants like π are irrational and so must be approximated by decimals when used in calculation. In exams give approximate decimal values for irrational quantities when required, and explain that exact expressions like √2 or π are preferred when an exact form is requested. Understanding this distinction prepares students for algebra and geometry problems where exact symbolic answers or decimal approximations may be needed.

The number line is a straight line used to represent numbers visually. Choose a point as zero, mark equal divisions to the right for positive integers and to the left for negative integers. Each point corresponds to one real number. Subdivide unit intervals into equal parts to place fractions and decimals precisely. The number line helps understand order, distance and operations like addition or subtraction as movements along the line.

Locating fractions: to plot a fraction a/b between 0 and 1, divide the interval into b equal parts and count a parts from 0. For example to place 3/5 divide the unit interval into five equal parts and move three steps from 0. For values greater than 1 place the whole part first then the fractional remainder. For negative fractions plot symmetric positions left of zero.

Locating decimals: decimals are located by subdividing the unit into tenths, hundredths, or finer as needed. For 0.37 divide the interval into 100 equal small parts and count 37 from zero (or take 3 tenths plus 7 hundredths). Drawing a fine division helps avoid errors when similar decimals are compared. Use consistent scale and label tick marks clearly.

Distance and midpoint: the distance between two numbers a and b equals |a − b| and is shown as the length of the segment joining their points. The midpoint is the point halfway: (a + b)/2. These ideas let you solve problems such as finding a point at a certain distance from a given number or dividing an interval into equal parts.

Ordering and visual checks: use the number line to order a set of numbers; those to the right are larger. The line helps check sign mistakes and compare magnitudes quickly. For exam sketches draw a clear scale and mark points accurately. Practice placing a mix of integers, fractions, terminating and recurring decimals and simple irrational approximations to build familiarity.

Powers of ten are useful when working with place value, scientific notation and large or small numbers. Ten to the power n, written 10^n, means 1 followed by n zeros for positive n. For negative powers, 10^{-n} equals 1 divided by 10^n and gives decimal places: 10^{-1} = 0.1, 10^{-2} = 0.01. These powers explain why moving one place left multiplies by ten and moving right divides by ten on the place-value chart.

Using exponents in decimals and fractions: represent decimals as multiples of powers of ten; for example 0.007 = 7×10^{-3} = 7/1000. Large numbers become easier to read as 5×10^3 instead of 5000. This notation helps when multiplying and dividing by powers of ten simply moves the decimal point: multiplying by 10^3 shifts the decimal three places to the right; dividing by 10^2 shifts it two places left.

Rules of exponents used here are: 10^a×10^b = 10^{a+b}; 10^a ÷ 10^b = 10^{a-b}; and (10^a)^b = 10^{ab}. Apply these when simplifying expressions involving powers of ten. For class 8 focus on moving decimal points and converting between fractional and decimal forms using powers of ten rather than advanced exponent manipulation.

Metric units and prefixes: relate powers of ten to metric prefixes used in everyday measurement: kilo = 10^3, centi = 10^{-2}, milli = 10^{-3}. Converting units uses multiplication or division by appropriate powers of ten: 3.2 km = 3.2×10^3 m = 3200 m; 45 mm = 45×10^{-3} m = 0.045 m.

Estimating and scientific notation: for very large or very small numbers scientific notation uses a×10^n where 1≤a<10. Though full scientific notation is beyond basic need, understanding that 5.6×10^{-4} means 0.00056 helps with reading measurements and calculator results. Practice by converting ordinary numbers to and from expressions with powers of ten to improve speed and reduce mistakes in decimal placement.

Divisibility tests are quick rules to determine whether a number is divisible by small integers without performing full division. For example, a number is divisible by 2 if its last digit is even; by 3 if the sum of its digits is divisible by 3; by 5 if the last digit is 0 or 5; by 9 if the sum of digits is divisible by 9; and by 11 using the alternating sum of digits rule. These rules save time when simplifying fractions or factoring numbers for GCD and LCM.

GCD and LCM: the greatest common divisor (GCD) or highest common factor (HCF) of two integers is the largest integer that divides both. The least common multiple (LCM) is the smallest positive integer divisible by each. Use prime factorisation or the Euclidean algorithm to compute GCD for larger numbers. The relation GCD(a,b) × LCM(a,b) = |a × b| for non-zero integers is helpful when both are needed.

Application to fractions: to reduce a fraction divide numerator and denominator by their GCD. For adding fractions find LCM of denominators to get a common denominator with smallest size which makes calculations simpler and reduces need for later simplification. For example add 1/12 + 1/18: LCM(12,18)=36 so convert to 3/36 + 2/36 = 5/36.

Strategies: begin factoring by checking divisibility rules for small primes (2,3,5) to find factors quickly. Use factor trees for clarity and cancellation. When handling multiple fractions find the LCM stepwise or via prime factors of all denominators to keep numbers manageable. Clarify cancellation rules: cancel factors only when they multiply across numerator and denominator, not across addition or subtraction.

Practice and exam tips: practise many factorisations and LCM/GCD problems to build speed. Show steps in exams: prime factors, cancelled factors, and final reduced fraction. This transparency helps examiners follow reasoning and awards method marks even if arithmetic slips occur.