math-algebra-1.md

Algebra 1

Source: https://www.khanacademy.org/math/algebra

Irrational Numbers

• rational: any number that can be expressed as the ratio (!) or fraction p/q of two integers (https://en.wikipedia.org/wiki/Rational_number)

• irrational: every other number

• multiplying two rational numbers: rational

  • (a / b) * (m / n) = am / bn
  • integer * integer => integer
  • integer / integer => rational

• adding two rational numbers: rational

  • (a / b) + (m / n) = (an + bm) / bn
  • integer * integer => integer
  • integer + integer => integer
  • integer / integer => rational

• multiplying rational & irrational number: irrational

  • proof by contradiction: rational * irrational => rational
  • (a / b) * x = m / n
  • x = (mb / na) => x = integer / integer => x = rational => x => rational, but originally irrational

• adding rational & irrational number: irrational

  • proof by contradiction: rational + irrational => rational
  • (a / b) + x = (m / n)
  • x = (m / n) - (a / b)
  • x = ((mb - na) / nb) => x = (integer - integer) / integer => x = integer / integer => x = rational => x => rational, but originally irrational

• adding two irrational numbers: depends on the number

  • Pi + (1 - Pi) = 1

  • irrational + irrational = rational

  • Pi + Pi = 2Pi

  • irrational + irrational = irrational

• multiplying two irrational numbers: depends on the number

  • sqrt2 * sqrt2 = 2

  • irrational * irrational = rational

  • Pi * Pi = Pi^2

  • irrational * irrational = irrational