The Real Numbers
How the real numbers split into rational and irrational, and how the natural, whole, integer and rational groups nest inside one another.
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The numbers used in ordinary mathematics all belong to one family: the real numbers. Within that family sit smaller groups, each contained in the next, and knowing which group a number falls into tells you a great deal about how it behaves.
Concept
The first division
Every real number is either rational or irrational. There is no overlap and nothing left over — each number belongs to exactly one of the two.
Rational — can be written as a fraction.
Irrational — cannot.
Concept
Rational numbers
A number is rational if it can be written as one integer over another, with the bottom number not zero. That definition catches more numbers than you might expect:
Ordinary fractions: 
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Terminating decimals: 
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Repeating decimals: 
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Whole numbers: 
.
Every integer is rational, since any integer can be placed over 1.
Concept
Irrational numbers
An irrational number cannot be written as an exact fraction. Its decimal expansion runs forever without ever repeating a pattern.
Approximations such as 3.14 are used in practice, but they are never the exact value. The distinction from a repeating decimal matters: also runs forever, yet its pattern repeats, which is what makes it rational.
Concept
Groups inside groups
Each group sits entirely inside the next. Moving outwards, every step admits new kinds of number while keeping everything that came before.
Natural — counting numbers: 1, 2, 3, …
Whole — the naturals plus 0.
Integers — the whole numbers plus the negatives.
Rational — the integers plus all other fractions.
Real — the rationals plus the irrationals.
Example
Placing a few numbers
Notice how membership accumulates. A number sits in its smallest group and in every group that contains it.
Example
Where each group is used
Natural — counting objects, numbering pages.
Whole — counts that may legitimately be zero.
Integers — temperatures, debts and profits.
Rational — precise measurements and financial ratios.
Irrational — geometry, physics and mathematical analysis.
Each group came into use because a real problem demanded it: zero to record nothing, negatives to record loss, fractions to record parts, irrationals to describe circles and diagonals exactly.
Note
Mistakes to avoid
Calling a repeating decimal irrational — a repeating pattern makes it rational.
Counting 0 as a natural number; it is whole but not natural.
Forgetting that every integer is also rational.
Treating 3.14 as the exact value of 
.
Assuming a number belongs to only one group rather than to every group containing it.
Summary
- Every real number is either rational or irrational, never both.
- Rational numbers can be written as a fraction of two integers.
- Irrational decimals run forever without repeating.
- Natural ⊂ whole ⊂ integers ⊂ rational ⊂ real.
- A number belongs to its own group and to every group containing it.