Learning Outcomes
What You Will Learn
Student Learning Outcomes (SLOs) as defined by FBISE for Unit 1.
Recall the history of numbers, from tally marks to the discovery of zero
Recall the set of real numbers as a union of sets of rational and irrational numbers
Depict real numbers on the number line
Demonstrate terminating and non-terminating recurring decimals on the number line
Give decimal representation of rational and irrational numbers
Know the properties of real numbers (closure, commutative, associative, identity, inverse, distributive)
Explain the concept of radicals and radicands
Differentiate between radical form and exponential form of an expression
Transform an expression from radical form to exponential form and vice versa
Recall base, exponent and value; apply the laws of exponents to simplify expressions
Math Unit 1 — Complete Notes
Full chapter with all properties, tables, worked examples & exercise solutions
Number System
The Real Number Family
Every real number belongs to one or more of these nested sets.
R
Real Numbers
Q ∪ Q′
Q
Rational
−5, 0, 9/11
Q′
Irrational
√11, −√7, π
Z
Integers
…,−1,0,1,…
W
Whole Numbers
0,1,2,3,…
N
Natural Numbers
1,2,3,…
Properties
Properties of Real Numbers
True for any real numbers a, b and c, with respect to addition and multiplication.
| Property | Addition | Multiplication | Quick Example |
|---|---|---|---|
| Closure | a+b ∈ R | a·b ∈ R | 6+4=10 |
| Commutative | a+b=b+a | a·b=b·a | 4+7=7+4 |
| Associative | a+(b+c)=(a+b)+c | a(bc)=(ab)c | — |
| Identity | a+0=a | a·1=a | 0 & 1 |
| Inverse | a+(−a)=0 | a·(1/a)=1 | a≠0 for × |
| Distributive | a(b+c) = ab+ac | 5(6+9)=75 | |
Indices
Laws of Exponents
For rational numbers m, n and non-zero real numbers a, b — these seven rules govern every simplification.
aᵘ·aⁿ
= a^(m+n)
Product Rule
aᵘ/aⁿ
= a^(m−n)
Quotient Rule
a⁻ⁿ
= 1/aⁿ
Negative Exponent
(aᵘ)ⁿ
= a^(mn)
Power of a Power
(ab)ⁿ
= aⁿbⁿ
Power of a Product
(a/b)ⁿ
= aⁿ/bⁿ
Power of a Quotient
(a/b)⁻ⁿ
= (b/a)ⁿ
Negative Power of Quotient
Quick Reference
Key Formula Cards
The most exam-tested formulas from radicals and rational exponents, at a glance.
Distributive Law
a(b+c) = ab+ac
The bridge between addition and multiplication. The number outside the bracket multiplies EVERY term inside — works for subtraction too: a(b−c) = ab−ac.
Product Rule (Radicals)
ⁿ√a · ⁿ√b = ⁿ√(ab)
Only valid when both radicals have the SAME index n. Different indices (e.g. ∛3 × √2) cannot be merged into one radical.
Quotient Rule (Radicals)
ⁿ√a / ⁿ√b = ⁿ√(a/b)
Same-index radicals can be combined under one division. Used to simplify fractions hidden inside roots.
Reducing the Index
ⁿ√(aᵏ) = a^(k/n)
If the index and the radicand's exponent share a common factor, divide both by it to rewrite the radical with a smaller index.
Rational Exponent
b^(1/n) = ⁿ√b
The denominator of a rational exponent is always the index of the equivalent radical. b^(m/n) = (ⁿ√b)^m.
Negative Exponent
a⁻ⁿ = 1/aⁿ
A negative exponent never makes a number negative — it sends the base to the denominator (or flips a fraction) instead.
PART 1
History & Classification of Numbers
History of Real Numbers
1. How did humans first start using numbers?
• What are tally marks?• Where is the oldest evidence of counting found?
The earliest numbers were not symbols — they were physical tally marks: one pebble, notch, or knot for every object counted (a one-to-one matching system). This gave rise to the Natural Numbers (1, 2, 3, …).
📝 The oldest confirmed tally evidence is a baboon leg bone with 29 notches, found in South Africa, dated before 30,000 BC.
2. Name civilizations that contributed to the numeral system but had no symbol for zero.
• What is the Babylonian base-60 system?• Did the Greeks and Romans use zero?
The Babylonians (sexagesimal, base-60 system — still used today for 60 seconds/minutes), Greeks (Attic numerals), and Romans (I, V, X, L, C, D, M) all understood "nothingness" conceptually but never gave it its own written symbol.
⚠️ The Mayans DID use a zero symbol, but only in their isolated calendar system — it never spread to the rest of the world.
3. Who discovered zero, and why is it called important in mathematics?
• What did al-Khwarizmi contribute to numbers?• How did the word "zero" originate?
Credit usually goes to the Indian mathematician Brahmagupta (628 AD). Later, al-Khwarizmi described a full base-10 number system using zero, calling the digit "Sifr." Fibonacci spread this system to Europe in 1202; "Sifr" became "zefero," then "zero."
Zero = additive identity (a + 0 = a) and the origin of the number line
4. Who discovered irrational numbers, and what does "irrational" actually mean?
• Who was Hippasus?• Does "irrational" mean a number is fake?
Hippasus of Metapontum, a follower of Pythagoras, found that √2 (the hypotenuse of a right isosceles triangle with sides 1) could not be written as a fraction a/b. He called such numbers "irrational" — meaning "unreasonable," NOT "fake" or "wrong."
⚠️ Common misconception: irrational numbers are perfectly real, exact quantities — they simply cannot be expressed as a ratio of two integers.
Classification of Real Numbers
1. Define the set of real numbers. What does R = Q ∪ Q′ mean?
• What are exhaustive and disjoint sets?• Can a number be both rational and irrational?
The set of Real Numbers (R) is the union of Rational (Q) and Irrational (Q′) numbers: R = Q ∪ Q′. Since Q ∩ Q′ = φ, the two sets never overlap (disjoint), and together they cover every real number with nothing left over (exhaustive).
R = Q ∪ Q′ | Q ∩ Q′ = φ
2. Describe the family tree of real numbers from R down to N.
• What is the difference between integers and whole numbers?• Is every integer a rational number?
R → splits into Q (rational) and Q′ (irrational). Q splits into Integers (Z) and Non-integer Rationals (e.g. 9/13). Z splits into Negative Integers and Whole Numbers (W). W splits into Zero and Natural Numbers (N).
📝 Every integer is rational (n = n/1), but not every rational number is an integer (e.g. 1/2).
3. What is a number line, and what is the "origin"?
• Why is the number line also called the "real line"?
A number line is a straight line with the point for 0 (the origin) at the centre, positive integers to the right, negative integers to the left, and equally spaced unit lengths. Every real number corresponds to exactly one point on this line — which is why it is also called the real line.
PART 2
Properties, Equality & Inequality
Properties of Operations
1. State the basic properties of real numbers with respect to addition and multiplication.
• What is the difference between commutative and associative?• What are the additive and multiplicative identities?
Closure (result stays a real number), Commutative (order doesn't matter), Associative (grouping doesn't matter), Identity (0 for +, 1 for ×), and Inverse (−a for +, 1/a for ×, a≠0).
⚠️ Subtraction and division are NEITHER commutative nor associative — only addition and multiplication allow reordering or regrouping.
2. What is the distributive property? Give an example.
• Verify a(b+c)=ab+ac for a=5, b=6, c=9.
The distributive property connects addition and multiplication:
a(b+c) = ab + ac
Example: 5(6+9) = 5(15) = 75, and 5(6)+5(9) = 30+45 = 75 — both sides match.3. Which real number has no multiplicative inverse?
• Why doesn't zero have a reciprocal?
Zero has no multiplicative inverse, because 1/0 is undefined. Every other non-zero real number a has a multiplicative inverse 1/a, since a × 1/a = 1.
Equality & Inequality
1. What is the Trichotomy Property?
• For any two real numbers a and b, which relations are possible?
For any two real numbers a and b, EXACTLY one of the following is true: a < b, a = b, or a > b. This guarantee is the Trichotomy Property.
2. State the properties of equality of real numbers.
• What is the reflexive property? Symmetric? Transitive?
Reflexive (a=a), Symmetric (a=b ⇒ b=a), Transitive (a=b & b=c ⇒ a=c), Additive (a=b ⇒ a+c=b+c), Multiplicative (a=b ⇒ ac=bc), and Cancellation (w.r.t. addition and multiplication).
3. What happens to an inequality sign when you multiply both sides by a negative number?
• State the multiplicative property of inequality for c<0 and c>0.• What is the most common exam mistake with inequalities?
If c < 0 and a > b, then ac < bc — the inequality sign REVERSES. If c > 0, the sign stays the same.
⚠️ This is the single most tested rule in this topic: multiplying or dividing both sides of an inequality by a NEGATIVE number flips the sign. Forgetting this is the #1 mistake examiners look for.
4. How do you graph an inequality like −2 ≤ x < 1 on a number line?
• What is the difference between a filled and a hollow circle?
A FILLED (solid) circle means the endpoint IS included (used for ≤ or ≥). A HOLLOW (open) circle means the endpoint is NOT included (used for < or >). For −2 ≤ x < 1: filled circle at −2, hollow circle at 1, with the line shaded between them.
PART 3
Radicals & Rational Exponents
Radicals & Radicands
1. Define square root, principal square root, radical, and radicand.
• What is the index of a radical?• Why is there no real square root of a negative number?
A square root of n is a number m such that m² = n. Every positive number has TWO square roots (additive inverses); the POSITIVE one is the principal square root. In ⁿ√b, ⁿ√ is the radical, n is the index, and b is the radicand.
⚠️ No real number is the square root of a negative number — e.g. √−16 ≠ 4, since 4² = 16, not −16.
2. State the product and quotient rules for radicals. When do they NOT apply?
• Can you combine ∛3 and √2 into one radical?
ⁿ√a · ⁿ√b = ⁿ√(ab) | ⁿ√a / ⁿ√b = ⁿ√(a/b)
These rules apply ONLY when both radicals have the SAME index. ∛3 · √2 cannot be combined, because the indices (3 and 2) are different.3. How do you reduce the index of a radical?
• Simplify ¹²√(9⁶) using a smaller index.
If the index and the radicand's exponent share a common factor, divide both by it. Example: ¹²√(9⁶) = 9^(6/12) = 9^(1/2) = √9 = 3.
Rational & Negative Exponents
1. Define b^(1/n) and b^(m/n).
• What does the denominator of a rational exponent represent?
b^(1/n) = ⁿ√b | b^(m/n) = (ⁿ√b)^m
The denominator n of the exponent is the index of the radical; the numerator m is the power to raise the result to.2. What is a negative rational exponent? Simplify 16^(−3/4).
• Does a negative exponent make the answer negative?
a^(−n) = 1/a^n. So 16^(−3/4) = 1/16^(3/4) = 1/(16^(1/4))³ = 1/2³ = 1/8.
📝 A negative exponent never makes a number negative — it sends the base to the denominator instead.
3. State all seven properties (laws) of exponents.
• What is the Power of a Power rule?
(i) aᵘ·aⁿ=a^(m+n) — Product Rule. (ii) aᵘ/aⁿ=a^(m−n) — Quotient Rule. (iii) a⁻ⁿ=1/aⁿ — Negative Exponent. (iv) (aᵘ)ⁿ=a^(mn) — Power of a Power. (v) (ab)ⁿ=aⁿbⁿ — Power of a Product. (vi) (a/b)ⁿ=aⁿ/bⁿ — Power of a Quotient. (vii) (a/b)⁻ⁿ=(b/a)ⁿ — Negative Power of a Quotient.
⚠️ Rules (i) and (ii) only work when the BASE is the same on both sides — disguise different numbers as powers of the same base first (e.g. 8 and 4 both as powers of 2).
PART 4
Real Numbers in Daily Life
Real-Life Applications
1. Give real-life examples of natural, integer, rational and irrational numbers.
• Where are irrational numbers used outside the classroom?
Natural numbers — counting stock, books, animals. Integers — temperature, gain/loss, rise/fall. Rational numbers — income, expenditure, wages, profit-sharing ratios. Irrational numbers — engineers use π when calculating the area, perimeter, or volume of circles, spheres and cylinders; they also appear in architecture, navigation and fluid mechanics.
2. How do you solve replenishment/inventory problems using percentages?
• If 20% of 5000 items are removed daily, how many days until 40% is removed?
Find the daily removal amount = percentage × total stock. Then divide the replenishment threshold (e.g. 40% of stock) by the daily removal amount to get the number of days.
Days = (40% of stock) ÷ (daily removal amount)
Glossary
Key Vocabulary
Every important term from Unit 1, in one place.
| Term | Meaning |
|---|---|
| Rational number (Q) | A number that can be written as a/b, where a, b are integers and b ≠ 0 |
| Irrational number (Q′) | A number that cannot be written as a ratio of two integers |
| Trichotomy property | For any a, b ∈ R, exactly one of a<b, a=b, a>b is true |
| Closure property | Adding or multiplying two real numbers always gives another real number |
| Distributive property | a(b+c) = ab + ac — multiplication distributes over addition |
| Multiplicative inverse | 1/a, the number that multiplies with a to give 1 (a ≠ 0) |
| Radical | An expression of the form ⁿ√b, used to denote a root |
| Radicand | The number or expression under the radical sign |
| Index | The small number n in ⁿ√b indicating which root to take |
| Principal square root | The positive square root of a positive number |
| Base | The repeated factor in an exponential expression, e.g. 3 in 3⁴ |
| Exponent | The number showing how many times the base is multiplied by itself |
| Rational exponent | A fractional power, e.g. b^(m/n), equivalent to (ⁿ√b)^m |
| Reciprocal | Another name for the multiplicative inverse of a number |
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