Math 2.0

statement theorem

The rationals form a field

Statement

The rational numbers Q\mathbb{Q}, with their addition and multiplication, form a field. That is, Q\mathbb{Q} is a commutative ring with 0≠10 \neq 1, and each rational number q≠0q \neq 0 has a multiplicative inverse q−1q^{-1} with q⋅q−1=1q \cdot q^{-1} = 1.

Remarks

The inverse of a/ba / b with a≠0a \neq 0 is b/ab / a. The field Q\mathbb{Q} is the smallest subfield of the real numbers. Each field of characteristic 00 contains a copy of Q\mathbb{Q}. The integers are not a field, and Q\mathbb{Q} is the field of fractions of Z\mathbb{Z}.

Explanations

L1 human

In the rational numbers, you can add, subtract, multiply, and divide by any number that is not zero. The result is always a rational number again.

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Sources

  1. Web https://en.wikipedia.org/wiki/Rational_number, Properties

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