Algebraic Rules and Number Theory Foundations - Comprehensive Notes
Equation Transformations and Equivalence
We can manipulate equations by applying identical algebraic transformations on both sides of an equality sign, which preserves the set of solutions. This is often summarized by saying the two equations are equivalent.
In particular, we can add or subtract the same quantity on both sides and maintain equivalence.
Example (solving for x): consider the equation x + 3 = 5 .
Step: subtract 3 on both sides (i.e., add \-3 to both sides): x + 3 = 5 \iff x = 5 - 3
This yields x = 2.
Verification: substitute back, 2 + 3 = 5, which is true.
The “if and only if” idea is often expressed as: A = B \iff A + C = B + C, i.e., the transformation preserves and reflects solutions.
This example illustrates the general technique: apply identical algebraic transformations to both sides to obtain a simpler equation while preserving the solution set.
The same principle extends to other operations such as multiplication, division, exponentiation, etc., which we’ll see in later parts of the material.
Additive Inverses and Equation Solving
If two integers (or numbers) satisfy a + b = 0, then b = -a and equivalently a = -b.
Proof sketch (step-by-step):
Start with the assumption a + b = 0.
Add \(-a\) to both sides: a + b - a = 0 - a.
By associativity and commutativity, this reduces to b = -a.
Therefore, the other side also shows a = -b.
Using this fact, we can derive the inverse-of-the-inverse property:
From a + (-a) = 0\,, replacing the role of a by -a gives (-a) + a = 0, which implies \(-(-a) = a\).
Hence the inverse of the inverse of a number is the number itself.
The transcript invites you to perform similar arguments on your own to see the parallel structure.
Multiplication Rules
Notation: multiplication can be written as a \times b, a \cdot b, or simply ab\(when no ambiguity arises).
Intuition: multiplication is repeated addition; e.g., a \times b means adding a to itself b times (or equivalently, adding b to itself a times).
Identity: the multiplicative identity is 1, i.e., a \cdot 1 = a and 1 \cdot a = a.
Associativity and Commutativity:
a \cdot b = b \cdot a (commutativity).
(a \cdot b) \cdot c = a \cdot (b \cdot c) (associativity).
Example: simplify the expression 2 \times x \times 3 \times y.
By commutativity, rearrange to 2 \times 3 \times x \times y.
By associativity and multiplication, this becomes (2 \cdot 3) \cdot (x \cdot y) = 6 \cdot (xy) = 6xy.
Distributivity (interaction with addition):
Left distributive law: a \cdot (b + c) = ab + ac.
Right distributive law: (b + c) \cdot a = ba + ca (equivalently ab + ac since multiplication is commutative).
Factorization perspective: distributivity allows turning a sum into a product by extracting a common factor, e.g., if ab + ac = a(b + c), we have factored out the common factor a.
Examples:
Left-to-right distributivity: 4 \cdot (3 + 2) = 4\cdot 3 + 4\cdot 2 = 12 + 8 = 20.
Right-to-left, identifying a common factor: if you have 4x + 4y, you can factor out the common factor 4 to get 4(x + y).
These rules let you move between sums and products and to rearrange terms to simplify expressions.
Exponents (Powers)
Exponent meaning: a power a^n represents multiplying the base a by itself n times.
Special case for zero exponent: if a \neq 0, then a^0 = 1.
Important caveat: the expression 0^0 is not defined because of ambiguity in the limiting/structural interpretation.
Rules for positive exponents:
a^m \cdot a^n = a^{m+n} (same base, add exponents).
(ab)^n = a^n b^n (exponent distributes over a product).
(a^m)^n = a^{mn} (power of a power).
Examples:
2^3 = 8 and (2x)^2 = 2^2 x^2 = 4x^2.
Exponentials and addition: the exponent does not distribute over addition in general.
(a+b)^n \neq a^n + b^n for general n \ge 2.
Binomial theorem provides the full expansion for such cases (e.g., (a + b)^2 = a^2 + 2ab + b^2 ).
Note: higher-order binomial expansions exist for n > 2 (to be covered later).
Divisibility, Even/Odd, and Primes
Divisibility (definition): For integers a and b with a \neq 0, we say a \mid b if there exists an integer c such that b = a c. Equivalently, dividing b by a yields an integer quotient.
Examples:
12 = 3 \cdot 4, so 3 \mid 12.
12 is not divisible by 5 (no integer c with 12 = 5c).
Basic facts:
Every integer is divisible by 1: n = 1 \cdot n.
Every integer is divisible by itself: n = n \cdot 1.
Zero is divisible by any nonzero integer: for any a \neq 0, 0 = a \cdot 0, hence a \mid 0\$.
Even and odd numbers:
An integer n is even iff n = 2m for some integer m.
An integer n is odd iff n = 2m + 1 for some integer m.
Examples: even: 2, 12, 0; odd: 3, 13, -1.
Prime numbers and the Fundamental Theorem of Arithmetic:
A prime is an integer p > 1 whose only positive divisors are 1 and p itself.
A composite number has additional divisors besides 1 and itself.
Fundamental Theorem of Arithmetic: Every integer greater than 1 can be written as a product of primes, and this factorization is unique up to the order of the factors.
Examples:
10 = 2 \cdot 5.
330 = 2 \cdot 3 \cdot 5 \cdot 11 (illustrates a product of several primes).
9780 = 2^2 \cdot 3 \cdot 5 \cdot 163$$ (example of a prime factorization with repetition).
Real-world relevance:
Primes are central to encryption schemes and many number-theory applications; these ideas form the foundation for topics covered in later weeks.
Notes on the content sequence: the next portion of the lecture would continue with related ideas and additional examples related to these foundations.