2025-07-07-algebraiclaws

5 Fundamental Algebraic Laws

1. Commutative Property

Changing order doesn’t matter

• Addition: $a+b=b+a$
• Multiplication: $ab=ba$
• Etymology: From Latin “commutare” = to exchange, switch places (like commuting between work and home)

2. Associative Property

Grouping doesn’t matter

• Addition: $(a+b)+c=a+(b+c)$
• Multiplication: $(ab)c=a(bc)$
• Etymology: From Latin “associare” = to join with, unite (elements associate/group together)

3. Distributive Property

Multiplication distributes over addition

• $a(b+c)=ab+ac$
• $(a+b)c=ac+bc$
• Etymology: From Latin “distribuere” = to divide up, spread out (one operation distributes over another)

4. Identity Property

“Do nothing” elements

• Additive identity: $a+0=a$
• Multiplicative identity: $a\cdot 1=a$
• Etymology: From Latin “identitas” = sameness (the element leaves others identical/unchanged)

5. Inverse Property

Every element has an opposite

• Additive inverse: $a+(-a)=0$
• Multiplicative inverse: $a\cdot \frac{1}{a}=1$ (where $a\ne 0$)
• Etymology: From Latin “inversus” = turned about, reversed (opposite/reverse operation)