Strand 3 — Number

Number Systems & Surds

5th Year · 6th Year (Leaving Cert)

✓By the end of this lesson students will be able to distinguish between rational and irrational numbers.
✓By the end of this lesson students will be able to simplify and manipulate surds using the properties of roots.
✓By the end of this lesson students will be able to apply the laws of indices to simplify algebraic and numerical expressions.
✓By the end of this lesson students will be able to rationalise the denominator of a fraction containing surds.
✓By the end of this lesson students will be able to prove that √2 is an irrational number (HL).

Key concepts

Rational Numbers

A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. Examples include 3/4, -5, 0.75, and 0.333... (which is 1/3).

Irrational Numbers

An irrational number is a real number that cannot be expressed as a simple fraction p/q. Its decimal representation is non-terminating and non-repeating. Examples include √2, √3, π, and e.

Surds

A surd is an irrational number that is the root of an integer. For example, √2, √3, ³√5 are surds. If the root of an integer is rational (e.g., √4 = 2), it is not a surd. Surds can be simplified using the property √(ab) = √a × √b.

√(ab) = √a × √b; √(a/b) = √a / √b

Laws of Indices

Indices (or exponents) are used to show that a number is multiplied by itself a certain number of times. The laws of indices provide rules for simplifying expressions involving powers.

a^m × a^n = a^(m+n) a^m ÷ a^n = a^(m-n) (a^m)^n = a^(mn) (ab)^n = a^n b^n (a/b)^n = a^n / b^n a^0 = 1 (for a ≠ 0) a^(-n) = 1/a^n a^(1/n) = n√a a^(m/n) = (n√a)^m = n√(a^m)

Rationalising the Denominator

Rationalising the denominator is the process of eliminating any surds from the denominator of a fraction. If the denominator is a single surd (e.g., √a), multiply the numerator and denominator by √a. If the denominator is of the form (a + √b) or (√a + √b), multiply the numerator and denominator by its conjugate (a - √b) or (√a - √b) respectively.

Proof that √2 is irrational (HL)

This proof uses the method of contradiction. We assume the opposite of what we want to prove and show that this assumption leads to a contradiction, thus proving our original statement. 1. Assume, for contradiction, that √2 is a rational number. 2. If √2 is rational, it can be written as a fraction a/b, where a and b are integers, b ≠ 0, and a/b is in its simplest form (meaning a and b have no common factors other than 1). So, √2 = a/b. 3. Square both sides: (√2)² = (a/b)² => 2 = a²/b². 4. Rearrange to get a² = 2b². 5. This equation implies that a² is an even number (since it is 2 times an integer b²). 6. If a² is even, then a itself must be an even number (because the square of an odd number is always odd, e.g., 3²=9). 7. Since a is even, we can write a = 2k for some integer k. 8. Substitute a = 2k into the equation a² = 2b²: (2k)² = 2b² 4k² = 2b² 9. Divide both sides by 2: 2k² = b². 10. This equation implies that b² is an even number (since it is 2 times an integer k²). 11. If b² is even, then b itself must be an even number. 12. So, we have deduced that both a and b are even numbers. 13. This contradicts our initial assumption that a/b was in its simplest form (i.e., a and b have no common factors other than 1). If both a and b are even, they have a common factor of 2. 14. Since our initial assumption (that √2 is rational) led to a contradiction, the assumption must be false. 15. Therefore, √2 is an irrational number.

Key facts to remember

1A rational number can be written as p/q where p, q ∈ Z, q ≠ 0.
2An irrational number cannot be written as p/q; its decimal is non-terminating and non-repeating.
3Surds are irrational roots of integers, e.g., √3, ³√7.
4Key surd properties: √(ab) = √a × √b and √(a/b) = √a / √b.
5To add or subtract surds, they must be 'like surds' (have the same root part), e.g., 2√5 + 3√5 = 5√5.
6Laws of indices: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), (a^m)^n = a^(mn).
7Negative indices: a^(-n) = 1/a^n. Fractional indices: a^(m/n) = n√(a^m).
8To rationalise a denominator of the form (a + √b), multiply by its conjugate (a - √b).

Worked examples

Example 1

Simplify the expression: √75 - √12 + √48

IBreak down each surd into its prime factors to find perfect square factors:

II√75 = √(25 × 3) = √25 × √3 = 5√3

III√12 = √(4 × 3) = √4 × √3 = 2√3

IV√48 = √(16 × 3) = √16 × √3 = 4√3

VSubstitute these simplified surds back into the original expression:

VI5√3 - 2√3 + 4√3

VIICombine the like surds (treat √3 as a common variable):

VIII(5 - 2 + 4)√3

Answer

7√3

Always look for the largest perfect square factor to simplify surds efficiently.

Example 2

Simplify: (x^3 y^-2)^2 / (x^-1 y^4)

IApply the power of a power rule (a^m)^n = a^(mn) to the numerator:

II(x^3 y^-2)^2 = x^(3*2) y^(-2*2) = x^6 y^-4

IIIRewrite the expression with the simplified numerator:

IV(x^6 y^-4) / (x^-1 y^4)

VApply the division rule a^m / a^n = a^(m-n) for each variable:

VIFor x: x^6 / x^-1 = x^(6 - (-1)) = x^(6+1) = x^7

VIIFor y: y^-4 / y^4 = y^(-4 - 4) = y^-8

VIIICombine the simplified terms:

9x^7 y^-8

10Express with positive indices (a^-n = 1/a^n):

Answer

x^7 / y^8

Remember to be careful with negative signs when subtracting exponents.

Example 3

Rationalise the denominator of the fraction: 3 / (√5 - √2)

IIdentify the conjugate of the denominator. The denominator is (√5 - √2), so its conjugate is (√5 + √2).

IIMultiply both the numerator and the denominator by the conjugate:

III[3 / (√5 - √2)] × [(√5 + √2) / (√5 + √2)]

IVMultiply the numerators:

V3(√5 + √2) = 3√5 + 3√2

VIMultiply the denominators using the difference of two squares formula (a-b)(a+b) = a² - b²:

VII(√5 - √2)(√5 + √2) = (√5)² - (√2)² = 5 - 2 = 3

VIIICombine the simplified numerator and denominator:

9(3√5 + 3√2) / 3

10Factor out 3 from the numerator and cancel with the denominator:

Answer

√5 + √2

The difference of two squares formula is crucial when rationalising denominators of the form (a ± √b) or (√a ± √b).

Common mistakes

✗Incorrectly applying index laws, especially with negative or fractional exponents (e.g., (x^2)^3 ≠ x^5, it's x^6).
✗Adding or subtracting unlike surds (e.g., √2 + √3 ≠ √5). Surds must be simplified first to check if they are like surds.
✗Errors when rationalising the denominator, particularly forgetting to multiply the entire numerator by the conjugate, or incorrectly applying the difference of two squares.
✗Assuming that √a + √b = √(a+b) or √a - √b = √(a-b). These are incorrect.
✗In the √2 irrationality proof, failing to clearly state the initial assumption of rationality and that a/b is in simplest form.

Exam tips

★For surd questions, always simplify each surd to its simplest form first before attempting addition or subtraction.
★When working with indices, write out each step clearly to avoid errors, especially with multiple operations or negative/fractional powers.
★For rationalising denominators, remember to multiply both the numerator and the denominator by the conjugate to maintain the value of the fraction.
★For the √2 irrationality proof (HL), ensure you understand the logical flow of the proof by contradiction and can clearly articulate each step, especially how the contradiction arises.

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