Chapter 1: Number Systems

Chapter 1 Summary

Quick Reference Card

Number Types

TypeCan write as p/q?Decimal
RationalYESTerminates OR repeats
IrrationalNONever terminates, never repeats
Real-Both types combined

Operation Rules

OperationResult
Rational ± IrrationalIrrational
Rational × Irrational (≠0)Irrational
Irrational ± IrrationalCould be either!
Irrational × IrrationalCould be either!

Square Root Identities

IdentityExample
ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}12=23\sqrt{12} = 2\sqrt{3}
(a+b)(a−b)=a−b(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - bRationalizing
(a+b)2=a+2ab+b(\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{ab} + bExpansion

Laws of Exponents

LawFormula
Productam⋅an=am+na^m \cdot a^n = a^{m+n}
Power(am)n=amn(a^m)^n = a^{mn}
Quotientam/an=am−na^m / a^n = a^{m-n}
Same poweram⋅bm=(ab)ma^m \cdot b^m = (ab)^m

Rationalizing Denominators

TypeMultiply by
1/a1/\sqrt{a}a/a\sqrt{a}/\sqrt{a}
1/(a+b)1/(a + \sqrt{b})(a−b)/(a−b)(a - \sqrt{b})/(a - \sqrt{b})
1/(a+b)1/(\sqrt{a} + \sqrt{b})(a−b)/(a−b)(\sqrt{a} - \sqrt{b})/(\sqrt{a} - \sqrt{b})

🎉 Congratulations!

You have mastered Chapter 1: Number Systems with all exercises and detailed solutions!

Visualizer
Number System Hierarchy
Real Numbers (ℝ)
Rational (ℚ) | Irrational
Integers (ℤ)
Whole Numbers (W)
Natural Numbers (ℕ)
ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ

Interactive Visualization