Arithmetic¶
Subsets of non-complex numbers¶
Natural numbers \(\mathbb{N}\):
\(1, 2, 3, 4, 5, ...\)
Prime numbers \(\mathbb{P}\):
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 \(\mathbb{Z}\):
\(..., -2, -1, 0, 1, 2, ...\)
Rational numers \(\mathbb{Q}\):
\(\frac{x}{y} | x, y \in \mathbb{Z}\)
Irrational numbers \(\mathbb{I}\):
non-rational real numbers, e.g. \(\sqrt{2}, e, \pi, ...\)
Real numbers \(\mathbb{R}\):
\(\mathbb{Q}\cup\mathbb{I}\)
Order of operations¶
- exponents and roots
- multiplication and division
- addition and subtraction
Powers or Integer exponents [2]¶
Base definitions¶
\(b^1 = b\)
\(b^{n+1} = b^n*b\)
\(b^0 = 1\)
\(b^{-n} = \displaystyle\frac{1}{b^n}\)
\(b^{n} = \displaystyle\frac{b^{n+1}}{b}, n \geq 1\)
Roots or Rational exponents [3]¶
\(\displaystyle b^{\frac{u}{v}} = (b^u)^{\frac{1}{v}} = \sqrt[v]{b^u}\)
\(\displaystyle \sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}\)
\(\displaystyle \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\)
Logarithms [4]¶
\(\log_b x = y, if b^y = x\)
\(\log_b(xy) = \log_b x + \log_b y\)
\(\displaystyle \log_b \frac{x}{y} = \log_b x - \log_b y\)
\(\log_b(x^n) = n \log_b x\)
\(\displaystyle \log_b \sqrt[n]{x} = \frac{\log_b x}{n}\)
| [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 |