Number Systems and Surds
FoundationThe number system expands from natural numbers through integers and rationals to the reals, the dividing line being whether a number can be written as a ratio of integers; mastering surd rules and rationalising denominators lets us simplify the irrational expressions common in the extended modules.
- Rationals are ratios of integers; irrationals have infinite non-recurring decimals; together they form the reals
- √(ab) = √a × √b and √(a/b) = √a/√b (a, b positive) are the basis of simplifying surds
- Factor out the largest square factor first; only simplest form reveals like surds and allows combining
文體小知識：數學概念說明文——由數系的分層講起，說明有理數與無理數的分別，再講解根式的化簡與分母有理化，是延伸單元二的代數基礎課。
§Text Passage
1The number system is the foundation of mathematics. The most basic numbers are the natural numbers — one, two, three, the numbers we count with; adding zero and negative numbers gives the integers. Numbers that can be written as a ratio of two integers are called rational numbers, such as one half or negative three quarters; integers, terminating decimals and recurring decimals are all rational, because each can be turned into a fraction.
2Not every number can be written as a ratio of integers. For example, when a square has side length 1, its diagonal has length equal to the square root of 2, about 1.4142, whose decimal digits go on forever without repeating — it cannot be written as a fraction, and such numbers are called irrational numbers; π is another famous example. Rational and irrational numbers together form the real numbers, matching the points on the number line one to one and filling it completely.
3Irrational numbers are often written with root signs, and expressions containing roots are called surds. Two properties matter most: the square root of a product equals the product of the square roots of the factors, and the square root of a quotient equals the quotient of the square roots. For example, 48 can be written as 16 times 3, so the square root of 48 equals the square root of 16 times the square root of 3, i.e. four root three. A surd whose radicand contains no square factor is said to be in simplest form.
4Surds with the same radicand combine just like like terms in algebra. For example, root 48 plus root 27 minus root 12 simplifies to four root three plus three root three minus two root three, giving five root three. At first the three surds look unrelated; once simplified to simplest form they reveal themselves as like surds — that is exactly the value of simplifying before operating.
5When a denominator contains a surd, we usually rewrite it without one — this process is called rationalising the denominator. For example, if the denominator is root five minus root two, multiply numerator and denominator by root five plus root two, and the denominator becomes five minus two, i.e. three. Two expressions differing only by the middle sign are conjugates: their product cancels the surd parts and leaves a rational number, a technique used constantly in the extended modules.
1Summary 重點摘要
The number system expands from natural numbers through integers and rationals to the reals, the dividing line being whether a number can be written as a ratio of integers; mastering surd rules and rationalising denominators lets us simplify the irrational expressions common in the extended modules.
2Vocabulary cards Vocabulary
The layered classification of numbers: natural numbers, integers, rationals, irrationals and reals, expanding layer by layer.
A number that can be written as a ratio of two integers; integers, terminating and recurring decimals are all rational.
A number that cannot be written as a ratio of integers, with an infinite non-recurring decimal expansion, such as the square root of 2 or π.
The collective name for rational and irrational numbers, matching the points on the number line one to one.
An expression containing a root sign; a non-exact square root is the common way of writing an irrational number.
A surd whose radicand has no square factor and whose denominator contains no surd — the standard form for answers.
Surds with the same radicand after simplification; they combine by adding or subtracting their coefficients.
Multiplying numerator and denominator by a suitable expression so that the denominator contains no surd.
The expression differing from the original only by the middle sign, e.g. √5 + √2 and √5 − √2; their product leaves only rationals via the difference of squares.
A decimal whose digits repeat in a fixed pattern; it can always be written as a fraction and is therefore rational.
3Step by step Steps
- Step 1: The layered number systemThe number system is built layer by layer. Natural numbers are the counting numbers; adding zero and negatives gives the integers; ratios of integers give the rationals, so integers, terminating decimals and recurring decimals are all rational. For example, 0.75 can be written as 34, and a recurring decimal like 0.333… can be written as 13, so both are rational. There is only one test — can the number be written as a ratio of two integers? When a question asks what type a number is, check layer by layer: first whether it is an integer, then whether it can be written as a fraction.
- Step 2: Irrational numbers and the realsA number that cannot be written as a ratio of two integers is irrational; its decimal expansion is infinite and non-recurring. For example, √2 is about 1.41421356, whose digits never repeat, so it is irrational; π, about 3.14159, is irrational too. Do not judge by the first few digits alone: 1.414 itself is a terminating decimal and therefore rational — it is merely an approximation of √2, not equal to it. Rationals and irrationals together form the real numbers, which correspond one to one with the points on the number line — the foundation for functions and limits later on.
- Step 3: Two key surd propertiesSurds obey two basic properties: the square root of a product equals the product of the square roots of the factors, and the square root of a quotient equals the quotient of the square roots. In words, a root of a product or quotient can be split into roots of each part. For example, √36 = √4 × √9 = 2 × 3 = 6, and √(2549) = √25 ÷ √49 = 57. These properties are the key to shrinking big radicands: when simplifying, look inside the root for square factors such as 4, 9, 16, 25, 36 or 49, pull one out and take its square root.
- Step 4: Simplifying surds to simplest formThe standard method is to factor out the largest square factor before taking the root. For example, to simplify √72, write 72 = 36 × 2; since √36 = 6, we get √72 = 6√2. A surd whose radicand has no square factor and whose denominator has no surd is in simplest form, and answers in the extended modules are expected in this form, because simplest form makes size comparisons and combining possible. For example, √8 and √50 look unrelated, but they simplify to 2√2 and 5√2, revealing their relationship at a glance; stop half-way and the combining step goes wrong.
- Step 5: Combining like surdsOnce simplified, surds with the same radicand are like surds and combine like like terms — by adding or subtracting coefficients. For example, simplify √48 + √27 − √12: we have √48 = 4√3, √27 = 3√3 and √12 = 2√3, so the expression equals (4 + 3 − 2)√3 = 5√3. Note that surds with different radicands cannot be merged: √2 + √3 is not √5, just as two apples plus three oranges is not five apples. When surds are unlike, simplest form is the end of the road — never force a combination.
- Step 6: Rationalising the denominatorA fraction with a surd in the denominator is awkward for further work, so we rationalise: multiply numerator and denominator by the same expression, leaving the value unchanged. For example, for 3/√5, multiply top and bottom by √5 to get 3√55 with an integer denominator. If the denominator has two terms, such as 1/(√5 − √2), multiply by the conjugate √5 + √2; the denominator becomes 5 − 2 = 3 by the difference of squares, and the numerator becomes √5 + √2, completing the simplification. Writing out this multiplication step is a standard scoring point in the extended papers.
- Step 7: The power of conjugatesTwo expressions differing only by the middle sign are conjugates, such as √5 + √2 and √5 − √2. Their product uses the difference of squares: the surd parts cancel completely, leaving a rational number. For example, (√7 + √3)(√7 − √3) = 7 − 3 = 4, and (1 + √2)(1 − √2) = 1 − 2 = −1. The trick has wide use: rationalising two-term denominators, comparing surd sizes, and later rationalising numerators in limit problems. In practice, multiply by the conjugate to verify each exercise, until seeing a two-term surd instantly suggests which conjugate to use.
5Worked example Example
Example: (a) simplify √48 + √27 − √12; (b) simplify 1/(√5 − √2).
- (a) Simplify each surd first: √48 = √(16 × 3) = 4√3; √27 = √(9 × 3) = 3√3; √12 = √(4 × 3) = 2√3
- Combine like surds: the expression = 4√3 + 3√3 − 2√3 = (4 + 3 − 2)√3 = 5√3
- (b) Multiply numerator and denominator by the conjugate: 1/(√5 − √2) = (√5 + √2)/[(√5 − √2)(√5 + √2)]
- Expand the denominator by the difference of squares: (√5 − √2)(√5 + √2) = (√5)² − (√2)² = 5 − 2 = 3
- Hence 1/(√5 − √2) = (√5 + √2)/3
- Answer: (a) √48 + √27 − √12 = 5√3. (b) 1/(√5 − √2) = (√5 + √2)/3. Always reduce each surd to simplest form before combining; when rationalising, multiply both numerator and denominator by the conjugate and check with the difference of squares that the denominator really becomes rational.
6Common mistakes Mistakes
7Quick check Quick check
Q1What is a rational number? What is an irrational number? Give one example of each.
Q2Simplify √50.
Q3Simplify √32 + √18.
Q4Rationalise 6/√3.
Q5Why must numerator and denominator be multiplied by the same expression when rationalising?
8Recap Recap
- Rationals are ratios of integers; irrationals have infinite non-recurring decimals; together they form the reals
- √(ab) = √a × √b and √(a/b) = √a/√b (a, b positive) are the basis of simplifying surds
- Factor out the largest square factor first; only simplest form reveals like surds and allows combining
- Rationalise a denominator by multiplying top and bottom by the conjugate, turning the denominator rational via the difference of squares
- Check surd work by squaring the answer — it should return the original radicand