GCSE/IGCSE 0580 Math Ext : Complete Guide to Number Section

Explore integers as the union of whole numbers, negatives, and zero, and use the number line to compare, order, and perform addition, subtraction, multiplication, and division with sign rules.

Understand how signs before integers determine the result: same signs produce a positive, different signs produce a negative, with even negatives yielding positive.

Learn the rules for powers on integers: negative bases with odd powers give negative results, even powers give positive results, and positive bases stay positive, illustrated with examples.

Master past paper fraction problems without a calculator by converting mixed and improper forms, multiplying, dividing, adding, and subtracting fractions, then converting results to mixed numbers using lcm and simplification.

Explore rational numbers defined as p/q with integers and terminating or recurring decimals. See how adding, multiplying, or dividing by rationals preserves irrationals.

Explore identifying rational and irrational numbers through stepwise evaluation of expressions, using identities, cancellations, and decimal forms.

Explore the distinction between rational and irrational numbers, illustrate infinite rational values between 4 and 6, and explain terminating vs non-terminating, non-repeating decimals, and how multiplying rationals and irrationals behaves.

Explore terminating decimals with a finite number of digits and non-terminating recurring decimals with repeating digits, using dot notation to indicate the start and end of the repeating block.

Learn to convert terminating decimals to fractions by counting decimal places, setting the denominator with zeros, and reducing to lowest terms using common factors.

Convert non-terminating repeating decimals to fractions by setting x to the decimal, multiplying by the needed power of ten, and subtracting to solve for x. Simplify to lowest form.

Learn to convert non-terminating decimals with non-repeating digits into fractions. Apply the tenfold method to build equations, subtract, and derive fractions like 11/90 and 7/30 using shortcut and long methods.

Convert non-terminating repeating decimals to fractions using the x method, align repeating digits, and apply shortcut forms like nines-based denominators (99, 999) for two or three repeating digits.

Explore powers and roots, including squares and cubes, using the radical sign for square roots and cube roots, higher order roots, and the prime factorization method with zero exponent rules.

Practice problems on powers and roots cover squaring and cubing integers, clarify the difference between (-12)^2 and -12 squared, and explain sign rules for even and odd powers.

Master square roots and cube roots using prime factorization, extracting pairs for squares and triples for cubes, with examples like 49, 121, 256, 196, and 729.

Learn to simplify square roots by multiplying and dividing under the same radical, using prime factorization to extract squares, and avoid adding or subtracting under radicals.

Tackle past paper problems on indices and exponents, converting decimals to powers of ten and applying base rules for negative powers, zero powers, and solving for x with same bases.

Use exponent rules to simplify expressions with the same base and multiply or divide exponents. Solve for x and m by equating exponents and using base conversions.

Master the order of operations (pemdas) from parentheses to exponents, and from multiplication and division to addition and subtraction, using left-to-right rules for mixed operations.

Practice problems on pemdas reinforce the order of operations: evaluate parentheses, exponents, then multiplication and division left to right, and addition and subtraction.

Explore why rounding numbers simplifies calculations and communicates approximate values, distinguishing exact figures from rounded estimates and covering rounding for repeating decimals and irrational numbers like pi.

Round numbers to the nearest ten, hundred, thousand, and whole numbers. Apply the next-digit rule: if less than five, keep; if five or more, increase and zero out following digits.

Learn to round decimals to a required decimal place by examining the next digit, rounding up when it is five or more and down when it is less, with examples.

Master rounding to significant figures by applying rules for whole numbers, decimals, and leading zeros, and practice rounding to various positions from units to thousandths.

Master significant figures and past paper problems by estimating expressions from one significant figure conversions and learn decimal and fraction techniques for accurate, quick comparisons.

Write numbers in standard form or scientific notation. Convert numbers to be between 1 and 10 and multiply by a power of ten, with positive or negative exponents.

Convert standard form to ordinary numbers by shifting the decimal according to the exponent in scientific notation, using positive exponents to move right and negative exponents to move left.

Learn to add and subtract numbers in scientific notation by aligning exponents, converting powers of ten, and applying distributive properties for positive and negative exponents.

Multiply and divide numbers in scientific notation using laws of indices to adjust exponents. Convert results to standard form with a number between 1 and 10 by shifting the decimal.

Calculate bounds for x and y from one significant figure and determine the greatest and least values of x, y, and x divided by y using standard form.

Explain googol and googolplex and estimate the time to write 100 googles and a googolplex at the speeds for writing a zero and a one.

Practice solving past paper problems of standard form, including squaring decimals, multiplying decimals, and adding or subtracting in standard form, and converting centimeter cubes to liters in standard form.

Learn how to determine lower and upper bounds when rounding to the nearest unit, using strict inequality and half of the unit, with examples from centimeters, millimeters, kilograms, and grams.

Learn to compute upper and lower bounds by adding or subtracting half the unit for rounded measurements, illustrated with door height, masses, distances, and times.

Apply upper and lower bounds to measurements rounded to the nearest centimetre, then compute maximum and minimum values for perimeters, areas, sums, differences, and ratios.

Apply upper and lower bounds to find x+y, y minus x, and x divided by y limits; determine the shaded area and the smallest p using V and R bounds.

Apply bounds to past paper problems in the number section by computing upper and lower bounds for eggs and hen counts, using nearest ten and nearest twenty adjustments.

Apply bounds to find a lower speed bound from 595 km over 8.75 h and a lower perimeter bound of 477 mm for a regular hexagon.

Explore bounds on wall measurements by calculating lower and upper bounds for length and height, determine maximum wall area, and estimate paint requirements using ten square metres per litre.

Apply upper and lower bounds to the timings and the packet dimensions to determine the maximum number of packets that fit along the shelf.

Compute the upper bound of a triangle's perimeter by adding upper bounds of its one-decimal sides, then determine the lower and upper bounds for the remaining ribbon after a cut.

Apply lower and upper bounds to problems such as estimating the minimum time for all people to leave a football match and lower bounds for speed from distance and time.