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A question that stumped the ancient Greeks

• Is every number a fraction? The Greeks thought yes — until someone proved $\sqrt{2}$ cannot be written as $\dfrac{a}{b}$.
• Legend says the discoverer was thrown overboard for revealing this uncomfortable truth.
• Today we sort numbers into families, and that split — rational 有理数 vs irrational 无理数 — is still the big one.

The number families

• Natural numbers 自然数 $\mathbb{N} = \{1, 2, 3, \dots\}$ — the counting numbers (some definitions include $0$).
• Integers 整数 $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ — whole numbers, positive, negative and zero.
• Rational numbers $\mathbb{Q}$ — anything that can be written $\dfrac{a}{b}$ where $a, b$ are integers and $b \neq 0$.
• Irrational numbers — everything else ($\pi$, $\sqrt{2}$, $e$): they cannot be expressed as a fraction.

Natural → Integer → Rational, each a bigger family. Irrational numbers live outside.

Factors 因数, multiples 倍数 and primes 质数

• A factor of $n$ divides into $n$ exactly. Factors of $12$: $1, 2, 3, 4, 6, 12$.
• A multiple of $n$ is $n \times$ an integer. Multiples of $5$: $5, 10, 15, 20, \dots$.
• A prime has exactly two factors: $1$ and itself.
• $2, 3, 5, 7, 11, 13, \dots$ are prime. $1$ is not prime (only one factor).

Every natural number is an integer, every integer is rational, and every rational is real; irrational numbers like $\pi$ and $\sqrt{2}$ are real but not rational

Prime factorisation 质因数分解, HCF 最大公因数 and LCM 最小公倍数

• Write any number as a product of primes: $72 = 2^3 \times 3^2$, $120 = 2^3 \times 3 \times 5$.
• HCF (highest common factor): take the lowest power of each shared prime.
• HCF of $72$ and $120$ $= 2^3 \times 3 = 24$.
• LCM (lowest common multiple): take the highest power of every prime that appears.
• LCM of $72$ and $120$ $= 2^3 \times 3^2 \times 5 = 360$.

Keep splitting until every branch ends on a prime (circled); collecting them gives $72 = 2^3 \times 3^2$

Fun fact

• There are infinitely many primes (Euclid proved this around 300 BC).
• The largest known prime (as of 2024) has over 41 million digits — it's $2^{136{,}279{,}841} - 1$, a Mersenne prime.

Reciprocals and place value

• The reciprocal 倒数 of a nonzero number multiplies it to 1: for $3/7$ it is $7/3$, since $(3/7)(7/3)=1$. Zero has no reciprocal.
• Group large numbers into thousands: 6000042 is six million and forty-two. $6\,000\,042=6(1\,000\,000)+42$; the zeros hold the missing place values.

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