1.3 Real-Number Structure and Number Theory
Key Takeaways
- Know the hierarchy of number systems and the difference between rational, irrational, real, and complex values.
- The real numbers form a field: addition and multiplication follow closure, associativity, commutativity, identities, inverses where defined, and distributivity.
- Prime factorization is the fastest route to divisibility, greatest common factor, least common multiple, and many number-theory proofs.
- Complex-number operations depend on i squared = -1 and often become simpler when using conjugates.
Number systems as structure, not labels
The official Domain I profile includes real-number structure, complex numbers, number properties, operations, prime factorization, greatest common factor, and least common multiple. These topics reward precision. Many multiple-choice errors come from confusing a familiar label, such as rational or prime, with its exact definition.
A rational number can be written as a/b where a and b are integers and b is not zero. Terminating decimals and repeating decimals are rational. An irrational number is real but cannot be written as a ratio of integers; its decimal expansion is nonterminating and nonrepeating. The square root of 49 is rational because it equals 7, while the square root of 2 is irrational. A complex number has the form a + bi, where a and b are real and i squared = -1.
Core structure of the real numbers
The real numbers under addition and multiplication form a field. For test purposes, know the properties and know where they fail when a set or operation changes.
| Property | Addition example | Multiplication example | Common trap |
|---|---|---|---|
| Closure | real + real is real | real times real is real | Division by zero is not closed. |
| Identity | a + 0 = a | a times 1 = a | 0 is not a multiplicative identity. |
| Inverse | a + (-a) = 0 | a times 1/a = 1 for a not 0 | Zero has no multiplicative inverse. |
| Commutative | a + b = b + a | ab = ba | Matrix multiplication is not generally commutative. |
| Associative | (a + b) + c = a + (b + c) | (ab)c = a(bc) | Subtraction and division are not associative. |
| Distributive | a(b + c) = ab + ac | connects operations | Forgetting distribution over every term creates false simplifications. |
If an item asks about a group, check the set and the operation. The real numbers under addition form an abelian group. The nonzero real numbers under multiplication form an abelian group. The real numbers under multiplication do not form a group because 0 is included and 0 has no multiplicative inverse.
Prime factorization, GCF, and LCM
The Fundamental Theorem of Arithmetic says every integer greater than 1 has a unique prime factorization, apart from the order of factors. That uniqueness powers many fast solutions. For 84 and 120, write 84 = 2 squared times 3 times 7 and 120 = 2 cubed times 3 times 5. The greatest common factor uses the smaller shared exponents: 2 squared times 3 = 12. The least common multiple uses the larger exponents appearing in either number: 2 cubed times 3 times 5 times 7 = 840.
A useful identity for positive integers a and b is GCF(a,b) times LCM(a,b) = ab. Use it as a check, not as a substitute for understanding prime factors. With 84 and 120, 12 times 840 = 10080, and 84 times 120 = 10080.
Complex-number operations
Add and subtract complex numbers componentwise: combine real parts with real parts and imaginary parts with imaginary parts. Multiply by distributing and replacing i squared with -1. To divide, multiply numerator and denominator by the conjugate of the denominator. The conjugate of a + bi is a - bi, and (a + bi)(a - bi) = a squared + b squared.
Worked example: (3 + 2i)/(1 - i). Multiply by (1 + i)/(1 + i). The numerator becomes 3 + 3i + 2i + 2i squared = 1 + 5i. The denominator is 1 - i squared = 2. The quotient is 1/2 + (5/2)i.
Common traps
Do not call 1 prime; a prime number has exactly two positive factors, 1 and itself. Do not assume every square root is irrational; perfect-square radicands produce rational values. Do not cancel terms across addition, as in (a + b)/a = b; cancellation applies to factors, not separate added terms. Do not forget that complex answers may be equivalent even when written in different forms, such as 0.5 + 2.5i and (1 + 5i)/2.
For AEPA practice, write prime factorizations vertically when comparing several integers. The visual layout prevents missing a prime factor and makes divisibility arguments easier to explain, which matches the exam's emphasis on mathematical communication.
Divisibility and modular habits
Divisibility rules are more than shortcuts; they are compact statements about remainders. An integer is divisible by 3 when its digit sum is divisible by 3 because powers of 10 all leave remainder 1 modulo 3. An integer is divisible by 9 for the same reason with remainders modulo 9. An integer is divisible by 4 when its last two digits form a number divisible by 4 because every hundred contributes a multiple of 4.
Modular thinking also helps with proof and answer elimination. If a question asks whether an expression can be even, odd, or divisible by a certain number, test remainders rather than expanding large values. For instance, the square of any odd integer has remainder 1 modulo 8. That fact quickly eliminates claims that an odd square could have remainder 3, 5, or 7 modulo 8.
Number-system closure is another common source of traps. Integers are closed under addition and multiplication but not division. Rational numbers are closed under addition, subtraction, multiplication, and division by a nonzero rational. Irrational numbers are not closed under addition or multiplication: sqrt(2) + (-sqrt(2)) = 0 is rational, and sqrt(2) times sqrt(2) = 2 is rational.
Which set-operation pair forms a group?
The greatest common factor of 84 and 120 is 12. What is their least common multiple?
What is (3 + 2i)/(1 - i) in standard a + bi form?