Number Theory and Fundamentals

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A whole number greater than 1 that has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23. 2 is the only even prime. 1 is NOT prime.

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A whole number greater than 1 that has more than two factors (divisible by something other than 1 and itself). Examples: 4, 6, 8, 9, 10. Every composite number has a unique prime factorization.

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Every integer greater than 1 can be written as a unique product of prime numbers (unique prime factorization). Example: 60 = 2² × 3 × 5.

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÷2: ends in 0,2,4,6,8. ÷3: digit sum divisible by 3. ÷4: last two digits divisible by 4. ÷5: ends in 0 or 5. ÷6: divisible by both 2 and 3. ÷9: digit sum divisible by 9. ÷10: ends in 0.

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The largest number that divides exactly into two or more numbers. Find using prime factorization — multiply shared prime factors. Example: HCF(12,18) = 2×3 = 6.

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The smallest number that is a multiple of two or more numbers. Find using prime factorization — multiply all prime factors at their highest power. Example: LCM(4,6) = 12.

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HCF(a,b) × LCM(a,b) = a × b. Example: HCF(4,6)=2, LCM(4,6)=12. 2×12 = 4×6 = 24. ✓ Useful shortcut when one is known.

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Rational: Can be expressed as p/q where p,q are integers and q≠0. Decimals terminate or repeat. Examples: ½, 0.75, 3. Irrational: Cannot — decimals are non-terminating, non-repeating. Examples: π, √2, e.

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Irrational roots that cannot be simplified to whole numbers or fractions. Example: √2, √3, √5 are surds. √4=2 is NOT a surd. Surds are left in root form for exact answers.

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√(ab) = √a × √b. √(a/b) = √a / √b. √a × √a = a. (√a + √b)(√a − √b) = a − b (rationalizing the denominator).

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Eliminating surds from the denominator. Multiply numerator and denominator by the conjugate or the surd. Example: 1/√2 = √2/2. 1/(√3−1) = (√3+1)/2.

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Natural (ℕ): 1,2,3... Whole: 0,1,2,3... Integer (ℤ): ...−2,−1,0,1,2... Rational (ℚ): p/q form. Irrational: non-repeating decimals. Real (ℝ): all rational + irrational. Complex (ℂ): includes √(−1)=i.

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A number written as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Example: 0.00045 = 4.5 × 10⁻⁴. 3,200,000 = 3.2 × 10⁶.

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(New − Old) / Old × 100%. Positive result = increase. Negative result = decrease. Example: price rises from 50 to 65 → (65−50)/50 × 100 = 30% increase.

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∈: element of. ∉: not element of. ⊂: subset. ∪: union (in A or B or both). ∩: intersection (in both A and B). A': complement of A (not in A). ∅ or {}: empty set.

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To visually represent sets and their relationships — showing union, intersection, and complement. Two circles that overlap show which elements are in only A, only B, or both.

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Assume the opposite of what you want to prove, then show this leads to a contradiction (an impossibility), proving the original statement must be true. Example: proof that √2 is irrational.

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A proof technique for statements about all natural numbers. Step 1 (Base case): Prove true for n=1. Step 2 (Inductive step): Assume true for n=k, prove true for n=k+1. Therefore true for all n.

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Floor ⌊x⌋: The greatest integer ≤ x. Example: ⌊3.7⌋=3, ⌊−2.3⌋=−3. Ceiling ⌈x⌉: The smallest integer ≥ x. Example: ⌈3.2⌉=4, ⌈−2.3⌉=−2.

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a mod m gives the remainder when a is divided by m. Example: 17 mod 5 = 2 (because 17 = 3×5 + 2). Essential in number theory, cryptography, and computer science.

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eⁱᶿ = cos θ + i sin θ. The famous special case: eⁱᵖ + 1 = 0 (Euler's identity) — connects e, i, π, 1, and 0 in one equation.

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n! = n × (n−1) × (n−2) × ... × 2 × 1. Example: 5! = 120. Special cases: 0! = 1 and 1! = 1. Used in permutations, combinations, and the binomial theorem.

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Direct: y = kx — as x increases, y increases by the same factor. Inverse: y = k/x — as x increases, y decreases. k is the constant of proportionality in both cases.

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Significant figures (SF): Count from the first non-zero digit. 0.00420 has 3 SF. Decimal places (dp): Count digits after the decimal point. Round using the next digit — ≥5 round up, <5 round down.