Step By Step Calculus » 2.2 - Number Operations

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Synopsis
Addition and multiplication of real numbers satisfy certain algebraic assumptions, called the field axioms.
For all a,b,c\in \mathbb{R} a,b,c\in \mathbb{R} :
  • commutativity: a+b=b+aa+b=b+a, and ab=baab=ba.
  • associativity: (a+b)+c=a+(b+c)(a+b)+c=a+(b+c), and (ab)c=a(bc)(ab)c=a(bc).
  • distributivity: a(b+c)=ab+aca(b+c)=ab+ac.
  • additive/multiplicative identities: a+0=aa+0=a, and a\cdot 1=aa\cdot 1=a.
  • additive/multiplicative inverses: a + (-a )\equiv a - a=0, \qquad b\cdot \dfrac{1}{b} \equiv b \cdot b^{-1} =1\; \mathrm{for} \; b\neq 0 \, . a + (-a )\equiv a - a=0, \qquad b\cdot \dfrac{1}{b} \equiv b \cdot b^{-1} =1\; \mathrm{for} \; b\neq 0 \, .
In that last axiom we use \equiv\equiv to indicate that the two expressions are equivalent ways to say the same thing with slightly different notation.
An integer d\neq 0d\neq 0 is a divisor of an integer nn if n=dqn=dq for some integer qq; dd is then also called a factor of nn. Notice that an integer nn always has the divisors (factors) nn and 11 since n=1\cdot nn=1\cdot n. A natural number pp is said to be a prime number iff it has exactly two positive divisors, namely itself and 11. For example, 22 is a prime number since it has exactly two positive divisors \{1,2\}\{1,2\} among its divisors \{-2, -1, 1, 2\}\{-2, -1, 1, 2\}. An integer dd is a common divisor of two nonzero integers, nn and mm, if dd is a divisor of both nn and m.m. And, the Greatest Common Divisor (GCD) of two nonzero integers nn and mm is the largest integer by which nn and mm can both be divided by with no remainder. The GCD is often called the greatest common factor of nn and mm. This definition implies that GCD(n,mn,m)==GCD(|n|,|m||n|,|m|).
An integer nn is a multiple of an integer mm if n=pmn=pm for some integer p.p. The Least Common Multiple (LCM) of two integers n\neq0, m\neq0n\neq0, m\neq0 is the smallest positive nonzero integer that is a multiple of both nn and mm. It is useful when adding two fractions; one then needs to find the LCM of denominators. It is related to the GCD by the formula:
\displaystyle \mathop{\mathrm{LCM}}(m,n)=\dfrac{\vert m \cdot n\vert}{\mathop{\mathrm{GCD}}(m,n)} \; .
\displaystyle \mathop{\mathrm{LCM}}(m,n)=\dfrac{\vert m \cdot n\vert}{\mathop{\mathrm{GCD}}(m,n)} \; .