Mathematics

Real Numbers

Number sets, generating fractions, representation on the real number line, absolute value, intervals, measurement errors, and scientific notation.

Nested number sets diagram: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ with imaginary numbers outside ℝ
  • ℕ: Natural numbers
  • ℤ: Integers
  • ℚ: Rational numbers
  • 𝕀: Irrational numbers
  • ℝ: Real numbers
  • i: Imaginary numbers
  • ℂ: Complex numbers
Terminating decimal: Divide the number without the decimal point by a 1 followed by as many zeros as there are decimal places:
2,25=225100=942{,}25 = \dfrac{225}{100} = \dfrac{9}{4}
Pure repeating decimal: Subtract the non-repeating part from the entire number without the decimal point. Divide this difference by as many 9s as there are digits in the repeating part (the period). Ex.:
2,45^=245−299=24399=27112{,}\widehat{45} = \dfrac{245-2}{99} = \dfrac{243}{99} = \dfrac{27}{11}
Mixed repeating decimal: Subtract the non-repeating part from the entire number without the decimal point. Divide this difference by as many 9s as there are repeating digits, followed by as many zeros as there are non-repeating decimal digits. Ex.:
6,345^=6345−63990=6282990=349556{,}3\widehat{45} = \dfrac{6345-63}{990} = \dfrac{6282}{990} = \dfrac{349}{55}
Proper fraction (numerator < denominator). The interval from 0 to 1 is divided into as many equal parts as the denominator indicates, using Thales' intercept theorem. Ex.:
35⇒\dfrac{3}{5} \Rightarrow
Representation of 3/5 on the real number line using Thales theorem
Improper fraction (numerator > denominator). It is expressed as a mixed number: integer part + fractional part (as a proper fraction). We draw the fraction, as in the previous case, starting from the unit indicated by the integer part. Ex.:
−83=− ⁣(2+23)=−2−23⇒-\dfrac{8}{3} = -\!\left(2+\dfrac{2}{3}\right) = -2-\dfrac{2}{3} \Rightarrow
Representation of -8/3 on the real number line as a mixed number
Square roots: The root we want to represent will be the hypotenuse of a right triangle, h=a2+b2h = \sqrt{a^2+b^2} with legs aa and bb. We decompose the radicand into the sum of two squares, and map the hypotenuse onto the real line using a compass. Ex.:
5=22+12⇒\sqrt{5} = \sqrt{2^2+1^2} \Rightarrow
Representation of square root of 5 on the real line using a right triangle and compass
It is the distance of a number from zero on the number line, disregarding its sign. Ex.:
∣−4∣=4  ;∣4∣=4|-4|=4 \;; \quad |4|=4
∣x∣=2⇒x=−2; x=2|x|=2 \Rightarrow x=-2;\, x=2
∣x∣<2⇒−2<x<2x∈(−2,2)|x|<2 \Rightarrow -2<x<2 \quad x\in(-2,2)
∣x∣>2⇒(x<−2)∨(x>2)x∈(−∞,−2)∪(2,∞)|x|>2 \Rightarrow (x<-2)\vee(x>2) \quad x\in(-\infty,-2)\cup(2,\infty)
∣x−2∣<5⇒−5<x−2<5x∈(−3,7)|x-2|<5 \Rightarrow -5<x-2<5 \quad x\in(-3,7)
Absolute value functions: They can be rewritten as piecewise-defined functions: we find the roots of the expression inside the absolute value and analyze their signs. The intervals where the expression is negative are multiplied by -1. Ex.:
f(x)=3x+∣x2−4∣⇒x2−4=0⇒x=−2; x=2f(x)=3x+|x^2-4| \Rightarrow x^2-4=0 \Rightarrow x=-2;\, x=2
Sign chart of x²−4 with roots x=−2 and x=2
f(x)={3x+(x2−4)ifx≤−23x−(x2−4)if−2<x<23x+(x2−4)ifx≥2f(x)=\begin{cases}3x+(x^2-4) & \text{if}\quad x\leq -2\\3x-(x^2-4) & \text{if}\quad -2<x<2\\3x+(x^2-4) & \text{if}\quad x\geq 2\end{cases}
Open interval: the endpoints are not included in the set
(1,3)⇒1<x<3⇒(1,3)\Rightarrow 1<x<3\Rightarrow
Open interval (1,3) on the real line
Closed interval: the endpoints are included in the set
[1,3]⇒1≤x≤3⇒[1,3]\Rightarrow 1\leq x\leq 3\Rightarrow
Closed interval [1,3] on the real line
Half-open (or half-closed) interval:
(1,3]⇒1<x≤3⇒(1,3]\Rightarrow 1<x\leq 3\Rightarrow
Half-open interval (1,3] on the real line
Rays (Infinite intervals): an interval where one of the endpoints is unbounded (goes to infinity)
(1,∞)⇒x>1⇒(1,\infty)\Rightarrow x>1\Rightarrow
Open ray (1,+∞) on the real line
[1,∞)⇒x≥1⇒[1,\infty)\Rightarrow x\geq 1\Rightarrow
Closed ray [1,+∞) on the real line
Absolute Error (EaE_a): The absolute value of the difference between the true value (VrV_r) and the measured or approximate value (VmV_m).
Ea=∣Vr−Vm∣E_a = |V_r - V_m|
Relative Error (ErE_r): The ratio of the absolute error to the true value. It has no units and is often expressed as a percentage (%).
Er=Ea∣Vr∣  ;%(Er)=Er⋅100E_r = \dfrac{E_a}{|V_r|} \;; \quad \%(E_r) = E_r \cdot 100
Write the number with a single non-zero digit to the left of the decimal point, followed by the remaining significant figures as decimals, multiplied by a power of 10. Ex.: 125000=1,25⋅105125000 = 1{,}25 \cdot 10^5
Addition in scientific notation: The exponents of 10 must be made equal. Ex.:
1,25⋅105+2,5⋅104=1,25⋅105+0,25⋅105=1,5⋅1051{,}25\cdot 10^5 + 2{,}5\cdot 10^4 = 1{,}25\cdot 10^5 + 0{,}25\cdot 10^5 = 1{,}5\cdot 10^5
Multiplication and division:
1,25⋅105⋅2,5⋅104=3,125⋅1091{,}25\cdot 10^5 \cdot 2{,}5\cdot 10^4 = 3{,}125\cdot 10^9
1,25⋅1072,5⋅104=0,5⋅103=5⋅102\dfrac{1{,}25\cdot 10^7}{2{,}5\cdot 10^4} = 0{,}5\cdot 10^3 = 5\cdot 10^2