Build a solid mathematical foundation with this number theory and fundamentals flashcard deck. These core concepts underpin all of mathematics and are essential for GCSE, SAT, AMC (American Mathematics Competition), and everyday mathematical reasoning.
This d...
Build a solid mathematical foundation with this number theory and fundamentals flashcard deck. These core concepts underpin all of mathematics and are essential for GCSE, SAT, AMC (American Mathematics Competition), and everyday mathematical reasoning.
This deck covers prime and composite numbers, factors and multiples, HCF and LCM, divisibility rules, fractions and decimals, percentages and ratios, indices and surds, set theory and Venn diagrams, types of numbers (natural, integer, rational, irrational, real), and basic proof techniques including proof by contradiction and mathematical induction. Based on publicly available educational resources including OpenStax Prealgebra and CK-12 Number Theory.
Ready to test yourself?
Flip through all 24 cards in interactive study mode.
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.
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.
Every integer greater than 1 can be written as a unique product of prime numbers (unique prime factorization). Example: 60 = 2² × 3 × 5.
÷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.
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.
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.
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.
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.
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.
√(ab) = √a × √b. √(a/b) = √a / √b. √a × √a = a. (√a + √b)(√a − √b) = a − b (rationalizing the denominator).
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.
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.
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.
(New − Old) / Old × 100%. Positive result = increase. Negative result = decrease. Example: price rises from 50 to 65 → (65−50)/50 × 100 = 30% increase.
∈: 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
