Operations on Real Numbers
Once you understand what real numbers are and how they are classified into rational and irrational, the next natural question is: what happens when you add, subtract, multiply, or divide real numbers? Specifically, when you combine a rational number with an irrational number, or two irrational numbers, is the result rational or irrational? In Class 9, you will learn the precise rules that govern these operations. Some results are predictable — for instance, adding a rational number to an irrational number always gives an irrational result. Other results are surprising — the product of two irrationals can sometimes be rational. These rules are not just theoretical; they are essential for simplifying algebraic expressions involving surds (square roots), which appear frequently in coordinate geometry, trigonometry, and physics. This topic also introduces you to working with expressions like 3sqrt(2) + 5sqrt(2), sqrt(3) x sqrt(7), and (sqrt(5) + sqrt(2))(sqrt(5) - sqrt(2)). You will learn to add, subtract, multiply, and divide such expressions, and to recognise when a simplification is possible. Mastering operations on real numbers is a gateway to more advanced algebraic manipulation in higher classes.
What is Operations on Real Numbers?
Operations on real numbers follow specific rules depending on whether the numbers involved are rational or irrational.
Fundamental operations: The four basic operations — addition (+), subtraction (-), multiplication (x), and division (÷) — can be performed on any two real numbers (with division by zero excluded).
Rules for operations involving rational and irrational numbers:
Rule 1: The sum or difference of a rational number and an irrational number is always irrational.
Example: 5 + sqrt(3) is irrational. 7 - sqrt(2) is irrational.
Rule 2: The product or quotient of a non-zero rational number and an irrational number is always irrational.
Example: 4 x sqrt(5) = 4sqrt(5) is irrational. sqrt(7)/3 is irrational.
Rule 3: The sum, difference, product, or quotient of two irrational numbers may be rational or irrational — you must check.
Examples: sqrt(2) + sqrt(3) is irrational, but sqrt(8) + (-sqrt(8)) = 0 is rational.
sqrt(2) x sqrt(3) = sqrt(6) is irrational, but sqrt(3) x sqrt(3) = 3 is rational.
Working with surds (irrational square roots):
Like surds: Surds with the same radicand (number under the root) are called like surds. Like surds can be added and subtracted directly.
3sqrt(2) + 5sqrt(2) = 8sqrt(2) (just add the coefficients).
7sqrt(5) - 2sqrt(5) = 5sqrt(5).
Unlike surds: Surds with different radicands cannot be combined by addition or subtraction.
sqrt(2) + sqrt(3) cannot be simplified further.
Multiplication of surds: sqrt(a) x sqrt(b) = sqrt(a x b), for a >= 0, b >= 0.
Division of surds: sqrt(a) / sqrt(b) = sqrt(a/b), for a >= 0, b > 0.
Operations on Real Numbers Formula
Key formulas for operations on real numbers involving surds:
1. Addition/Subtraction of like surds:
m x sqrt(a) + n x sqrt(a) = (m + n) x sqrt(a)
m x sqrt(a) - n x sqrt(a) = (m - n) x sqrt(a)
2. Multiplication of surds:
sqrt(a) x sqrt(b) = sqrt(ab)
(m x sqrt(a)) x (n x sqrt(b)) = mn x sqrt(ab)
3. Division of surds:
sqrt(a) / sqrt(b) = sqrt(a/b)
(m x sqrt(a)) / (n x sqrt(b)) = (m/n) x sqrt(a/b)
4. Squaring a surd:
(sqrt(a))^2 = a
5. Algebraic identities with surds:
(sqrt(a) + sqrt(b))(sqrt(a) - sqrt(b)) = a - b
(sqrt(a) + sqrt(b))^2 = a + 2sqrt(ab) + b
(sqrt(a) - sqrt(b))^2 = a - 2sqrt(ab) + b
6. Simplifying surds:
sqrt(n) = sqrt(p x q) = sqrt(p) x sqrt(q), where p is the largest perfect square factor.
Example: sqrt(18) = sqrt(9 x 2) = 3sqrt(2)
sqrt(50) = sqrt(25 x 2) = 5sqrt(2)
sqrt(72) = sqrt(36 x 2) = 6sqrt(2)
7. Rationalisation identity:
1/sqrt(a) = sqrt(a)/a (multiply numerator and denominator by sqrt(a))
1/(sqrt(a) + sqrt(b)) = (sqrt(a) - sqrt(b))/(a - b)
Derivation and Proof
Proof that the sum of a rational and an irrational number is irrational:
Statement: If r is rational and s is irrational, then r + s is irrational.
Step 1: Assume, for contradiction, that r + s is rational. Let r + s = t, where t is rational.
Step 2: Then s = t - r.
Step 3: Since t is rational and r is rational, and the difference of two rational numbers is rational, we get s = t - r is rational.
Step 4: But s is irrational (given). This is a contradiction.
Step 5: Therefore, r + s is irrational.
Proof that the product of a non-zero rational and an irrational is irrational:
Statement: If r is a non-zero rational and s is irrational, then r x s is irrational.
Step 1: Assume r x s is rational. Let r x s = t, where t is rational.
Step 2: Then s = t/r.
Step 3: Since t is rational, r is non-zero rational, and the quotient of two rationals (with non-zero divisor) is rational, we get s is rational.
Step 4: But s is irrational. Contradiction.
Step 5: Therefore, r x s is irrational.
Why the product of two irrationals can be rational — understanding through examples:
Consider sqrt(2) x sqrt(8). We can simplify: sqrt(2) x sqrt(8) = sqrt(2 x 8) = sqrt(16) = 4.
Here, both sqrt(2) and sqrt(8) are irrational, but their product is 4 (rational). This happens because 2 x 8 = 16 is a perfect square. In general, sqrt(a) x sqrt(b) = sqrt(ab), and the result is rational only when ab is a perfect square.
Similarly, sqrt(3) x sqrt(12) = sqrt(36) = 6 (rational), because 3 x 12 = 36 is a perfect square.
Types and Properties
Operations on real numbers can be categorised by the types of numbers involved:
1. Rational + Rational:
Always gives a rational result. Examples: 3/4 + 1/2 = 5/4, 2.5 x 1.2 = 3.0, 7 - 3 = 4.
The set of rational numbers is closed under all four operations (except division by zero).
2. Rational + Irrational (or Rational - Irrational):
Always gives an irrational result (provided the operation is addition or subtraction).
Examples: 3 + sqrt(5) is irrational, 1/2 - sqrt(3) is irrational, pi + 1 is irrational.
3. Rational x Irrational (or Rational / Irrational):
Always gives an irrational result if the rational number is non-zero.
Examples: 5sqrt(2) is irrational, (2/3)sqrt(7) is irrational.
Exception: 0 x sqrt(5) = 0 (rational). The zero case is the only exception.
4. Irrational + Irrational:
May be rational or irrational — no general rule.
Irrational result: sqrt(2) + sqrt(3) = 3.1462... (irrational)
Rational result: (3 + sqrt(5)) + (3 - sqrt(5)) = 6 (rational)
5. Irrational x Irrational:
May be rational or irrational.
Irrational result: sqrt(2) x sqrt(3) = sqrt(6) (irrational)
Rational result: sqrt(5) x sqrt(5) = 5 (rational)
6. Operations involving conjugate surds:
Conjugate pairs like (sqrt(a) + sqrt(b)) and (sqrt(a) - sqrt(b)) have a special property:
Their product is always rational: (sqrt(a) + sqrt(b))(sqrt(a) - sqrt(b)) = a - b.
Their sum is always rational: (sqrt(a) + sqrt(b)) + (sqrt(a) - sqrt(b)) = 2sqrt(a) — wait, this is irrational if a is not a perfect square. Actually, the SUM of conjugates equals 2sqrt(a), which is irrational. But the PRODUCT equals a - b, which is rational. This is the basis of rationalisation.
Solved Examples
Example 1: Example 1: Adding and subtracting like surds
Problem: Simplify: (a) 3sqrt(5) + 7sqrt(5) (b) 8sqrt(3) - 2sqrt(3)
Solution:
(a) 3sqrt(5) + 7sqrt(5) = (3 + 7)sqrt(5) = 10sqrt(5)
(b) 8sqrt(3) - 2sqrt(3) = (8 - 2)sqrt(3) = 6sqrt(3)
Like surds (same number under the root) can be added or subtracted just like like terms in algebra — combine the coefficients.
Example 2: Example 2: Simplifying surds before adding
Problem: Simplify: sqrt(50) + 3sqrt(2)
Solution:
First simplify sqrt(50): sqrt(50) = sqrt(25 x 2) = sqrt(25) x sqrt(2) = 5sqrt(2).
Now add: 5sqrt(2) + 3sqrt(2) = (5 + 3)sqrt(2) = 8sqrt(2).
Key step: Before adding surds, always check if they can be simplified to like surds. sqrt(50) and sqrt(2) look unlike, but simplifying sqrt(50) reveals they are both multiples of sqrt(2).
Example 3: Example 3: Multiplying surds
Problem: Simplify: (a) sqrt(3) x sqrt(7) (b) 2sqrt(5) x 3sqrt(10) (c) sqrt(6) x sqrt(6)
Solution:
(a) sqrt(3) x sqrt(7) = sqrt(3 x 7) = sqrt(21) (irrational, since 21 is not a perfect square).
(b) 2sqrt(5) x 3sqrt(10) = (2 x 3) x sqrt(5 x 10) = 6 x sqrt(50) = 6 x 5sqrt(2) = 30sqrt(2).
(c) sqrt(6) x sqrt(6) = (sqrt(6))^2 = 6 (rational!).
Example 4: Example 4: Dividing surds
Problem: Simplify: (a) sqrt(45) / sqrt(5) (b) 12sqrt(6) / 4sqrt(3)
Solution:
(a) sqrt(45) / sqrt(5) = sqrt(45/5) = sqrt(9) = 3.
(b) 12sqrt(6) / 4sqrt(3) = (12/4) x sqrt(6/3) = 3 x sqrt(2) = 3sqrt(2).
Example 5: Example 5: Expanding (sqrt(a) + sqrt(b))^2
Problem: Expand and simplify: (sqrt(3) + sqrt(7))^2.
Solution:
Using (a + b)^2 = a^2 + 2ab + b^2, with a = sqrt(3), b = sqrt(7):
(sqrt(3) + sqrt(7))^2 = (sqrt(3))^2 + 2 x sqrt(3) x sqrt(7) + (sqrt(7))^2
= 3 + 2sqrt(21) + 7
= 10 + 2sqrt(21)
The result is irrational because it contains the term 2sqrt(21) and 21 is not a perfect square.
Example 6: Example 6: Using the difference of squares identity with surds
Problem: Simplify: (sqrt(11) + sqrt(3))(sqrt(11) - sqrt(3)).
Solution:
Using (a + b)(a - b) = a^2 - b^2, with a = sqrt(11), b = sqrt(3):
(sqrt(11) + sqrt(3))(sqrt(11) - sqrt(3)) = (sqrt(11))^2 - (sqrt(3))^2 = 11 - 3 = 8
The result is rational, even though both factors are irrational. This identity is the foundation of rationalising the denominator.
Example 7: Example 7: Determining whether the result is rational or irrational
Problem: State whether each result is rational or irrational. Justify.
(a) sqrt(2) + 5/3 (b) sqrt(7) x sqrt(7) (c) sqrt(3) x sqrt(5) (d) pi - pi
Solution:
(a) sqrt(2) + 5/3: Sum of irrational (sqrt(2)) and rational (5/3). By Rule 1, always irrational.
(b) sqrt(7) x sqrt(7) = 7: Product of two identical irrationals. Result is rational.
(c) sqrt(3) x sqrt(5) = sqrt(15): Since 15 is not a perfect square, sqrt(15) is irrational.
(d) pi - pi = 0: Difference of two identical irrationals. Result is rational.
Example 8: Example 8: Simplifying a complex surd expression
Problem: Simplify: sqrt(12) + sqrt(75) - sqrt(48).
Solution:
Simplify each surd:
sqrt(12) = sqrt(4 x 3) = 2sqrt(3)
sqrt(75) = sqrt(25 x 3) = 5sqrt(3)
sqrt(48) = sqrt(16 x 3) = 4sqrt(3)
Now all are like surds (multiples of sqrt(3)):
2sqrt(3) + 5sqrt(3) - 4sqrt(3) = (2 + 5 - 4)sqrt(3) = 3sqrt(3)
Example 9: Example 9: Multiplying conjugate surd expressions
Problem: Find the product: (2 + 3sqrt(5))(2 - 3sqrt(5)).
Solution:
Using (a + b)(a - b) = a^2 - b^2, with a = 2, b = 3sqrt(5):
(2 + 3sqrt(5))(2 - 3sqrt(5)) = (2)^2 - (3sqrt(5))^2
= 4 - 9 x 5
= 4 - 45
= -41
Even though each factor contains irrational terms, their product is a rational number (-41).
Example 10: Example 10: Expanding a product of binomial surd expressions
Problem: Expand: (sqrt(2) + sqrt(3))(sqrt(5) - sqrt(7)).
Solution:
Use the distributive property (FOIL method):
(sqrt(2) + sqrt(3))(sqrt(5) - sqrt(7))
= sqrt(2) x sqrt(5) - sqrt(2) x sqrt(7) + sqrt(3) x sqrt(5) - sqrt(3) x sqrt(7)
= sqrt(10) - sqrt(14) + sqrt(15) - sqrt(21)
None of these surds can be combined (they are all unlike), so the answer is:
sqrt(10) - sqrt(14) + sqrt(15) - sqrt(21)
Real-World Applications
Operations on real numbers, especially involving surds, have wide applications across mathematics and the sciences:
Trigonometry: Standard trigonometric values involve surds: sin(45) = sqrt(2)/2, cos(30) = sqrt(3)/2, tan(60) = sqrt(3). When solving trigonometric equations, you frequently add, multiply, and rationalise such expressions. For example, finding the exact value of sin(75) = sin(45 + 30) requires computing sqrt(2)/2 x sqrt(3)/2 + sqrt(2)/2 x 1/2 = (sqrt(6) + sqrt(2))/4 — a series of surd operations.
Coordinate Geometry: The distance formula sqrt((x2-x1)^2 + (y2-y1)^2) often produces irrational results. When comparing distances, you need to simplify surd expressions. For instance, checking if three points form an equilateral triangle requires showing all sides equal sqrt(some value). The midpoint of two irrational coordinates also involves adding and dividing surds.
Physics — Free Fall and Motion: The formula for the velocity of an object falling from height h is v = sqrt(2gh). When comparing velocities at different heights, you operate on such expressions: sqrt(2 x 9.8 x 5) = sqrt(98) = 7sqrt(2) m/s versus sqrt(2 x 9.8 x 10) = sqrt(196) = 14 m/s. The pendulum period formula T = 2pi x sqrt(L/g) also involves surd operations when comparing pendulums of different lengths.
Architecture and Design: The golden ratio (1 + sqrt(5))/2 involves operations on irrationals. Calculations with the golden ratio in architectural proportions require adding and multiplying expressions with sqrt(5). For example, the reciprocal of the golden ratio is (sqrt(5) - 1)/2, and verifying that phi x (1/phi) = 1 requires multiplying conjugate surds: ((1 + sqrt(5))/2) x ((sqrt(5) - 1)/2) = (5 - 1)/4 = 1.
Electrical Engineering: AC circuit calculations involve sqrt(2) (RMS values), sqrt(3) (three-phase power), and combinations thereof. Engineers routinely simplify expressions like sqrt(3) x V_line / sqrt(2) = V_line x sqrt(3/2) = V_line x sqrt(6)/2. Power calculations in three-phase systems use the formula P = sqrt(3) x V x I x cos(phi), requiring multiplication of surds with other quantities.
Simplifying Radicals in Algebra: Advanced algebraic problems require combining, expanding, and factoring expressions with surds, building directly on the skills learned in this topic. Nested radicals like sqrt(5 + 2sqrt(6)) can be simplified to sqrt(3) + sqrt(2) using the techniques of expanding and comparing surd expressions.
Chemistry — Reaction Rates: The Arrhenius equation k = A x e^(-Ea/RT) involves operations on real numbers. Rate constants at different temperatures are compared using ratios and exponents, which require real number arithmetic including irrational values.
Key Points to Remember
- Rational + Irrational = always Irrational.
- Non-zero Rational x Irrational = always Irrational.
- Irrational + Irrational = may be Rational or Irrational (check each case).
- Irrational x Irrational = may be Rational or Irrational (check each case).
- Like surds (same radicand) can be added or subtracted by combining coefficients.
- Unlike surds cannot be combined by addition or subtraction.
- Always simplify surds first before trying to add or subtract: sqrt(50) = 5sqrt(2).
- sqrt(a) x sqrt(b) = sqrt(ab) for non-negative a and b.
- (sqrt(a) + sqrt(b))(sqrt(a) - sqrt(b)) = a - b (product of conjugates is rational).
- 0 x (any irrational) = 0, which is rational — the exception to the product rule.
Practice Problems
- Simplify: sqrt(27) + sqrt(12) - sqrt(75).
- Expand and simplify: (sqrt(5) + sqrt(2))^2.
- Find the product: (3sqrt(2) + 2sqrt(3))(3sqrt(2) - 2sqrt(3)).
- Simplify: (sqrt(6) + sqrt(3))(sqrt(6) + sqrt(3)) and state whether the result is rational or irrational.
- Show that (sqrt(3) + sqrt(2))^2 - (sqrt(3) - sqrt(2))^2 = 4sqrt(6).
- Simplify: 4sqrt(18) - 3sqrt(8) + sqrt(32).
- Determine whether each is rational or irrational: (a) sqrt(5) x sqrt(20), (b) sqrt(2) + (-sqrt(2)), (c) 3 + 2sqrt(7), (d) sqrt(3) x sqrt(6).
- If x = sqrt(5) + 2 and y = sqrt(5) - 2, find the value of x^2 + y^2.
Frequently Asked Questions
Q1. What happens when you add a rational and an irrational number?
The result is always irrational. For example, 3 + sqrt(2) is irrational. This is because if 3 + sqrt(2) were rational, then sqrt(2) = (3 + sqrt(2)) - 3 would be rational (difference of two rationals), contradicting the fact that sqrt(2) is irrational.
Q2. Is the sum of two irrational numbers always irrational?
No. The sum of two irrational numbers can be rational. For example, sqrt(5) + (-sqrt(5)) = 0, which is rational. Similarly, (2 + sqrt(3)) + (2 - sqrt(3)) = 4, which is rational. However, many sums of irrationals are indeed irrational, like sqrt(2) + sqrt(3). You must check each case individually.
Q3. What are like surds and unlike surds?
Like surds have the same number under the square root sign. Examples: 3sqrt(5) and 7sqrt(5) are like surds. Unlike surds have different numbers under the root. Examples: sqrt(2) and sqrt(3) are unlike surds. Like surds can be added and subtracted (combine coefficients), but unlike surds cannot be simplified further when added or subtracted.
Q4. How do you simplify sqrt(72)?
Find the largest perfect square that divides 72. Since 72 = 36 x 2, and 36 = 6^2, we get sqrt(72) = sqrt(36 x 2) = sqrt(36) x sqrt(2) = 6sqrt(2). Always look for the largest perfect square factor: 72 = 4 x 18 would give 2sqrt(18), which needs further simplification. Using 36 directly gives the fully simplified form.
Q5. What is the product of conjugate surds?
Conjugate surds are pairs of the form (sqrt(a) + sqrt(b)) and (sqrt(a) - sqrt(b)). Their product equals a - b, which is always rational. This uses the identity (x + y)(x - y) = x^2 - y^2. For example, (sqrt(7) + sqrt(3))(sqrt(7) - sqrt(3)) = 7 - 3 = 4. This property is the basis for rationalising denominators.
Q6. Can sqrt(2) + sqrt(3) be simplified?
No, sqrt(2) + sqrt(3) cannot be simplified further because sqrt(2) and sqrt(3) are unlike surds (different numbers under the root). The expression is already in its simplest form. It is an irrational number approximately equal to 3.146. Do not try to write it as sqrt(5) — that would be incorrect since sqrt(2) + sqrt(3) is not equal to sqrt(5).
Q7. Is sqrt(a + b) equal to sqrt(a) + sqrt(b)?
No! This is a very common mistake. sqrt(a + b) is NOT equal to sqrt(a) + sqrt(b). For example, sqrt(4 + 9) = sqrt(13) = 3.606..., but sqrt(4) + sqrt(9) = 2 + 3 = 5. These are clearly not equal. The square root of a sum is NOT the sum of the square roots. The correct multiplication rule is sqrt(a) x sqrt(b) = sqrt(a x b), which works for products, not sums.
Q8. How do you multiply expressions like (2 + sqrt(3))(4 - sqrt(3))?
Use the distributive property (FOIL): (2 + sqrt(3))(4 - sqrt(3)) = 2 x 4 + 2 x (-sqrt(3)) + sqrt(3) x 4 + sqrt(3) x (-sqrt(3)) = 8 - 2sqrt(3) + 4sqrt(3) - 3 = 5 + 2sqrt(3). Combine rational parts (8 - 3 = 5) and irrational parts (-2sqrt(3) + 4sqrt(3) = 2sqrt(3)) separately.
Q9. Why is 0 times an irrational number rational?
Because 0 times any number equals 0, and 0 is rational (0 = 0/1). The rule that 'rational x irrational = irrational' only applies when the rational number is non-zero. If the rational number is 0, the product is 0 regardless of the other factor. This is the only exception to the rule.
Q10. What is the closure property for real numbers?
The closure property means that performing an operation on real numbers always gives a real number. Real numbers are closed under addition (any two reals added give a real), subtraction, multiplication, and division (except by zero). This is important because it means you never 'leave' the real number system when doing arithmetic. Note: rational numbers alone are also closed, but irrational numbers alone are NOT closed (sqrt(2) x sqrt(2) = 2, which is not irrational).
