arithmetic.rst

Arithmetic

Subsets of non-complex numbers

• Natural numbers ℕ:

  1, 2, 3, 4, 5, ...

• Prime numbers ℙ:

  A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.¹

  1, 2, 3, 5, 7, 11, ...

• Integers ℤ:

  ...,  − 2,  − 1, 0, 1, 2, ...

• Rational numers ℚ:

  $\frac{x}{y} | x, y \in \mathbb{Z}$

• Irrational numbers 𝕀:

  non-rational real numbers, e.g. $\sqrt{2}, e, \pi, ...$

• Real numbers ℝ:

  ℚ ∪ 𝕀

Order of operations

1. exponents and roots
2. multiplication and division
3. addition and subtraction

Powers or Integer exponents²

Base definitions

b¹ = b

b^n + 1 = bⁿ * b

b⁰ = 1

$b^{-n} = \displaystyle\frac{1}{b^n}$

$b^{n} = \displaystyle\frac{b^{n+1}}{b}, n \geq 1$

Properties

b^m + n = b^m * bⁿ

(b^m)ⁿ = b^m * n

(b ⋅ c)ⁿ = bⁿ ⋅ cⁿ

Roots or Rational exponents³

$\displaystyle b^{\frac{u}{v}} = (b^u)^{\frac{1}{v}} = \sqrt[v]{b^u}$

$\displaystyle \sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}$

$\displaystyle \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

Logarithms⁴

log_bx = y, ifb^y = x

log_b(xy) = log_bx + log_by

$\displaystyle \log_b \frac{x}{y} = \log_b x - \log_b y$

log_b(xⁿ) = nlog_bx

$\displaystyle \log_b \sqrt[n]{x} = \frac{\log_b x}{n}$

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1. https://en.wikipedia.org/wiki/Prime_number↩

2. https://en.wikipedia.org/wiki/Exponentiation#Integer_exponents↩

3. https://en.wikipedia.org/wiki/Nth_root#Identities_and_properties↩

4. https://en.wikipedia.org/wiki/Logarithm#Logarithmic_identities↩