Real Numbers – Every Number on the Number Line
The real numbers, written ℝ, are the union of all rational and all irrational numbers – in other words, every number that can be placed as a point somewhere on the continuous number line. Formally: ℝ = ℚ ∪ (irrationals). Naturals, wholes, integers, fractions, terminating and repeating decimals, and numbers like π and √2 are all real numbers.
Mathematicians used real numbers informally for over two hundred years – ever since Isaac Newton and Gottfried Leibniz built calculus on them in the late 17th century – without ever giving them a fully rigorous logical definition. That changed in the 1870s, when the German mathematicians Richard Dedekind and Georg Cantor independently developed precise constructions of the real numbers (Dedekind's approach, called “Dedekind cuts,” defines each real number by how it splits the rational numbers into two sets). This finally put two centuries of calculus on completely solid logical ground, closing what had been a surprisingly shaky gap at the foundation of some of mathematics' most powerful tools.
What Are Real Numbers?
A real number is any number that corresponds to a point on the number line – which includes every number covered so far in this section.
√2 is not a whole number and cannot be written as a fraction, so its most specific classification is Irrational (it is also, more broadly, a real number).
How the Number Sets Fit Together
Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real, and separately, Irrational ⊂ Real. Every number in a smaller set automatically belongs to every larger set that contains it.
Real-Life Application
- Measurement: any physical quantity (length, weight, time) is a real number.
- Science and engineering: virtually all formulas operate on real numbers.
- Graphs: every point's coordinates on a standard graph are real numbers.
Key Takeaways
- Real numbers are the union of all rational and irrational numbers.
- Dedekind and Cantor gave real numbers a rigorous definition in the 1870s, centuries after calculus began using them.
- Every natural, whole, integer, and rational number is also a real number.
Practice: Real Numbers
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