We have defined the natural numbers via the Peano Postulates, and now would like to turn our attention to other famous sets of numbers, namely the integers, rationals, and real numbers.
Natural numbers, integers, and rational numbers all turn out to be special kinds of real numbers so it suffices to define the real numbers first, and then specify which real numbers are considered to be natural, integers, and rational numbers.
In addition to defining the real numbers themselves, we would like our real numbers to be ordered so that we can tell when one real number is less than another. If $x,y$ are real numbers, we will write $x<y$ as usual to indicate that $x$ is less than $y$.
We would also like to be able to add and multiply any two real numbers. As usual, if $x,y$ are real numbers, we will write $x+y$ for their sum and $x\cdot y$ as their product. With this notation in hand we can now define the real numbers.
In the following axioms, 0 and 1 are constants, the variables $x,y,z,m,M$ are real numbers, and $a$ is an infinite sequence of real numbers.
The Axioms for Real Numbers
| Axiom Name | Definition |
|---|---|
| closure of $+$ | $x+y$ is a real number |
| identity of $+$ | $0$ is a real number and $x+0=0+x=x$ |
| inverse for $+$ | $\negative{x}$ is a real number and $x+(\negative{x})=\negative{x}+x=0$ |
| commutativity of $+$ | $x+y=y+x$ |
| associativity of $+$ | $(x+y)+z=x+(y+z)$ |
| closure of $\cdot$ | $x\cdot y$ is a real number |
| identity of $\cdot$ | $1$ is a real number, $1\neq 0$, and $1\cdot x=x\cdot 1=x$ |
| inverse for $\cdot$ | if $x\neq 0$ then $x^-$ is a real number and $x\cdot x^-=x^-\cdot x=1$ |
| commutativity of $\cdot$ | $x\cdot y=y\cdot x$ |
| associativity of $\cdot$ | $(x\cdot y)\cdot z=x\cdot(y\cdot z)$ |
| irreflexive | $\neg(x<x)$ |
| transitive | $x<y \xand y<z \ximplies x<z$ |
| trichotomy | $x=y$ or $x<y$ or $y<x$ |
| distributivity | $x\cdot (y+z)=x\cdot y + x\cdot z$ and $(y+z)\cdot x=y\cdot x + z\cdot x$ |
| translation | $x<y \ximplies x+z<y+z$ |
| product of positives | $0<x \xand 0<y \ximplies 0<x\cdot y$ |
| complete | Every bounded above set of real numbers has a least upper bound |
Any set that has addition, multiplication, and order defined in such a way that satisfies these axioms will be called the set of real numbers, $\mathbb{R}$, i.e., $\mathbb{R} = \Set{x : x \text{ is a real number}}$.
It is also helpful to make some convenient definitions. In the following definitions, let $x,y,z$ be real numbers.
Definitions for Real Numbers
| Name | Definition |
|---|---|
| $\leq$ | $x\leq y \,\iff\, x<y \xor x=y$ |
| quotient | if $y\neq 0$ then $\frac{x}{y}=x/y=x\cdot y^-$ |
| subtraction | $x-y=x+(\negative{y})$ |
| positive | $x$ is positive $\,\iff~\, 0<x$ |
| negative | $x$ is negative $\,\iff~\, x<0$ |
| nonnegative | $x$ is nonnegative $\,\iff~\, 0\leq x$ |
| nonzero | $x$ is nonzero $\,\iff~\, x\neq 0$ |
| distance | the distance from $x$ to $y$ is $\lvert x-y\rvert $ |
| increasing | $\langle a _ n\rangle _ {n=0}^\infty$ is increasing $\iff~ \forall k.a _ k<a _ {k+1}$ |
| nondecreasing | $\langle a _ n\rangle _ {n=0}^\infty$ is nondecreasing $\iff~ \forall k.a _ k\leq a _ {k+1}$ |
| decreasing | $\langle a _ n\rangle _ {n=0}^\infty$ is decreasing $\iff~ \forall k.a _ {k+1}<a _ k$ |
| nonincreasing | $\langle a _ n\rangle _ {n=0}^\infty$ is nonincreasing $\iff~ \forall k.a _ {k+1}\leq a _ k$ |
| absolute value | $\lvert x\rvert =\begin{cases} x & \text{if }0\leq x \\ \negative{x} & \text{otherwise} \\ \end{cases}$ |
We also interchangeably use the abbreviations $x\nless y$, $x\nleq y$, $x\ngtr y$, and $x \ngeqslant y$ to say that $x$ is not $y$, not less than or equal to $y$, not greater than $y$, and not greater than or equal to $y$ respectively. As before, we will also use the shortcut of identifying the statements $x\geq y$ with $y\leq x$, and $x > y$ with $y < x$, without the need to justify it as a separate step.
As usual, we can derive useful rules from these definitions.
Axioms for Real Numbers
Declare -, positive, negative, nonnegative, nonzero and abs to be constants.
Axioms of Addition
Additive identity
$x+0=x$ and
$0+x=x$
Additive inverse
$x+(−x)=0$ and
$−x+x=0$
Commutativity of addition
$x+y=y+x$
Associativity of addition
$(x+y)+z=x+(y+z)$
Axioms of Multiplication
Multiplicative identity
$1 \neq 0$ and
$1\cdot x=x$ and
$x\cdot 1=x$
Multiplicative inverse
If $x\neq 0$ then
$x\cdot x^-=1$ and
$x^-\cdot x=1$
Commutativity of multiplication
$x\cdot y=y\cdot x$
Associativity of multiplication
$(x\cdot y)\cdot z=x\cdot (y\cdot z)$
Axioms of Order
Less-than is irreflexive
$\xnot (x < x )$
Less-than is transitive
If $x<y$ and
$y<z$ then
$x<z$
Trichotomy
$x=y \xor x<y \xor y<x$
Axioms Relating $+$, $\cdot$, and $<$
Distributivity
$x\cdot (y+z)=x\cdot y+x\cdot z$
Translation
If $x<y$ then
$x+z<y+z$
Product of positives
If $0<x$ and
$0<y$ then
$0 < x\cdot y$
Definitions for Real Numbers
less than or equal to $x\leq y \equiv x<y \xor x=y$
Subtraction $x - y = x+(-y)$
Quotient If $y\neq 0$ then $\frac{x}{y} = x\cdot y^-$
Absolute value If $0\leq x$ then $\left|x\right|=x$
Absolute value If $x<0$ then $\left|x\right|=-x$
Positive $x$ is positive $\equiv 0<x$
Negative $x$ is negative $\equiv x<0$
Nonnegative $x$ is nonnegative $\equiv 0\leq x$
Nonzero $x$ is nonzero $\equiv x\neq 0$
Armed with the axioms and definitions above, we can now prove a lot of properties of real numbers.
Prove each of the following facts about the real numbers.
Product with zero $x\cdot 0=0\cdot x=0$
Warm up facts We know from the axioms that $0\neq 1$. Now prove the following.
Cancellation for addition If $x+z=y+z$ then $x=y$.
Cancellation for multiplication If $z\neq 0$ and $z\cdot x=z\cdot y$ then $x=y$.
Additive inverses are unique If $z+x=0$ and $z+y=0$ then $x=y$.
Multiplicative inverses are unique If $z\cdot x=1$ and $z\cdot y=1$ then $x=y$.
Inverse of additive inverse If $x$ is a real number then $\vphantom{x}^-(\negative{x})=x$.
Reciprocals are nonzero If $x\neq 0$ then $x^-\neq 0$.
Inverse of multiplicative inverse If $x$ is a real number and $x\neq 0$ then $(x^-)\vphantom{x}^-=x$.
Alternate additive inverse If $x$ is a real number then $\negative{x}=\negative{1}\cdot x$.
Alternate multiplicative inverse If $x$ is a real number and $x\neq 0$ then $(x^-)=\frac{1}{x}$.
Generalized additive inverse If $x,y$ are any real numbers then there exists a unique real number $z$ such that $x+z=y$.
Generalized multiplicative inverse If $x,y$ are any nonzero real numbers then there exists a unique nonzero real number $z$ such that $x\cdot z=y$.
Signed products For any real numbers $x,y$, $$\negative{x}\cdot y=x\cdot \negative{y}=\negative{\left(x\cdot y\right)}$$ and $$\negative{x}\cdot \negative{y}=x\cdot y$$
Signed quotients For any real numbers $x,y,z$ $$\frac{\negative{x}}{y}=\frac{x}{\negative{y}}=\negative{\left(\frac{x}{y}\right)}$$ and $$\frac{\negative{x}}{\negative{y}}=\frac{x}{y}$$
Trichotomy Law For all real numbers $x,y$ exactly one of the following is true:
Asymmetry If $x<y$ then $y\nless x$.
Generalized translation If $w<x$ and $y<z$ then $w+y<x+z$.
Product of signs For all nonzero real numbers $x,y$
Inverses are opposite $ (\negative{x}<0\iff 0<x)\xand (x<0\iff 0<\negative{x})$
Negating an inequality $x<y$ if and only if $\negative{y}<\negative{x}$.
Scaling by a positive If $0<z$ and $x<y$ then $z\cdot x < z\cdot y$.
Scaling by a negative If $z<0$ and $x<y$ then $z\cdot y < z\cdot x$.
Properties of $\leq$ For all real numbers $x, y, z,$
Squares are nonnegative If $x$ is a real number then $0\leq x^2$.
Zero divisors If $x\cdot y=0$ then $x=0$ or $y=0$.
Difference of two squares Let $x,y$ be real numbers. Then
$$x^2-y^2=(x+y)\cdot(x-y)$$
Every natural number can be considered to be a real number. To see this, let’s define the following set of real numbers.
Definition . We say a real number $n$ is a natural number if and only if $n\in\NN$ where $\NN$ is the set defined as follows.
If we define the successor $\sigma(x)$ of a real number $x$ to be the real number $x+1$ the elements of $\NN$, $\sigma$, $0$, $+$, $\cdot$, and $\leq$ satisfy the Peano Axioms (see problem 10.32 below). Notice that parts $1$ thru $4$ of the definition above correspond to Peano Axioms N0, N1, N3, and N4. Peano Axiom N2 holds for real numbers in general (see problem 10.3 above), as do axioms A0, A1, M0, and M1. Peano Axiom I is not true about every real number but can be proven for those in $\NN$. Therefore, anything we can prove about the natural numbers in the formal system defined by the Peano Axioms is true about the numbers in $\NN$.
Notice that the natural number we called $1$ is $\sigma(0)=0+1=1$ in the reals. So both $0$ and $1$ in the reals correspond to what we refer to as $0$ and $1$ in natural numbers. Similarly, we can refer to the rest of the natural numbers in the reals as we did before, namely $0=0$, $\sigma(0)=1$, $\sigma(1)=2$, $\sigma(2)=3$, $\sigma(3)=4$, and so on.
There are many more real numbers than just the natural numbers. For example, there is no natural number that when added to $2$ gives $1$. But there is such a real number. This allows us to define the integers.
Definition . A real number is an integer if and only if it is either (a) a natural number or (b) the additive inverse of a natural number.
These are the ordinary integers that we have all come to know and love. We can extend many of the definitions about natural numbers to integers. In particular, the definitions of summation and powers extend to sums and powers of integers and real numbers. We can also extend the definition of powers to include negative integer exponents.
Definition (negative integer exponents). If $z$ is a real number and $n$ is a natural number then $$z^{-n}=(z^n)^-$$
But there are many more real numbers besides the integers. For example, there is no integer that you can multiply by $5$ to get $3$. But there is such a real number. This allows us to define the rational numbers.
Definition . A real number is rational if and only if some integer multiple of it is an integer, i.e., a real number $r$ is rational if and only if there is some nonzero integer $m$ such that $r\cdot m$ is an integer.
Indeed, in this definition if we define $k=r\cdot m$ then we can show that $r=\frac{k}{m}$, i.e., the quotient of two integers. Thus, once again, the rational numbers are the familiar quotients of integers we all know and love.
We can now define constants to represent sets containing all numbers of a certain type.
Famous Sets of Numbers
| Name | Notation |
|---|---|
| The Natural Numbers | $\bbN =\Set{ 0,1,2,3,4,\ldots } $ |
| The Integers | $\bbZ =\Set{ \ldots,-3,-2,-1,0,1,2,3,\ldots } $ |
| The Rational Numbers | $\bbQ = \Set{ \frac{a}{b}:a,b\in\bbZ\xand b\neq 0 } $ |
| The Real Numbers | $\bbR = \Set{ x : x\text{ can be expressed as a decimal number} } $ |
| The Complex Numbers | $\bbC = \Set{ x+yi : x,y\in\bbR } $ where $i^{2}=-1$ |
| The positive real numbers | $\bbR^{+}=\Set{ x : x\in\bbR\xand x>0 } $ |
| The negative real numbers | $\bbR^{-}=\Set{ x : x\in\bbR\xand x<0 } $ |
| The positive reals in a set $A$ | $A^{+}=A\cap\bbR^{+}$ |
| The negative reals in a set $A$ | $A^{-}=A\cap\bbR^{-}$ |
| The first $n$ positive integers | $\mathbb{I} _ {n}=\Set{ 1,2,\ldots,n } $ |
| The first $n+1$ natural numbers | $\mathbb{O} _ {n}=\Set{ 0,1,2,\ldots,n } $ |
As usual, we can derive rules from some of these definitions.
Famous sets of numbers
Declare natural number, integer and rational to be constants. Note that $\NN,\ZZ,\QQ$ are declared to be constants by default.
Set names
Natural $x$ is a natural number $\equiv$ $x\in\NN$
Integer $x$ is an integer $\equiv$ $x\in\ZZ$
Rational $x$ is rational $\equiv$ $x\in\QQ$
Closure
Closure of $\NN$ If $n\in\NN$ and $m\in\NN$ then $n+m\in\NN$ and $n\cdot m\in\NN$.
Closure of $\ZZ$ If $n\in\ZZ$ and $m\in\ZZ$ then $n+m\in\ZZ$ and $n\cdot m\in\ZZ$
Closure of $\QQ$ If $n\in\QQ$ and $m\in\QQ$ then $n+m\in\QQ$ and $n\cdot m\in\QQ$
Relationships between the sets
Natural Numbers
Integers $n\in\ZZ \equiv n\in\NN \xor −n\in\NN$
Rational Numbers If $x=\frac{a}{b}, a\in\ZZ$ and $b\in\ZZ$ then $x\in\QQ$.
Rational Numbers If $r\in\QQ$ then $r=\frac{a}{b}\xand a\in\ZZ\xand b\in\ZZ$ for some $a$ and $b$.
Positives $x\in A^+ \equiv x\in A, 0<x$
Extended Definitions
Integer Exponents If $z\neq 0$ and $n\in\NN$ then $z^0=1$, $z^{n+1}=z\cdot z^n$ and $z^{-n}=\frac{1}{z^n}$.
Theorems
Inclusions If $-n\in\NN$ then $n\in\ZZ$.
Inclusions. $\NN\subseteq\ZZ$, $\ZZ\subseteq\QQ$ and $\NN\subseteq\QQ$.
Prove each of the following facts about the real numbers.
Unnatural integers There exist integers which are not natural numbers.
Additive inverses in the integers For every integer $m$ there exists a unique integer $n$ such that $m+n=n+m=0$.
Closure in the integers For any integers $m,n$ both $m+n$, $m-n$, and $m\cdot n$ are integers.
Closure in the rationals For any rational numbers $m,n$, both $m+n$ and $m\cdot n$ are also rational number. Furthermore, if $n\neq 0$ then $\frac{m}{n}$ is also a rational number.
Every natural is real The terms of the sequence $N$, $0$, $+$, $\cdot$, $<$ and $\sigma$ together satisfy the Peano Axioms.
Every natural is an integer Every natural number is also an integer.
Every integer is rational Every integer is also a rational number.
Alternate definition of rational A real number $r$ is rational if and only if for some integer $m$ and some nonzero integer $n$, $$r=\frac{m}{n}$$
Equivalent fractions If $k,m,n$ are nonzero integers, the $$\frac{k\cdot m}{k\cdot n}=\frac{m}{n}$$
No gaps There is a rational number strictly between any two distinct rational numbers, i.e., if $s,t$ are rational number with $s<t$ then there exists a rational number $r$ such that $$s<r<t$$
Completeness for lower bounds Every infinite sequence of real numbers that has a lower bound has a greatest lower bound.
Existence of floor and ceiling For every real number $x$ there are unique integers $m$ and $M$ such that $m\leq x \leq M$.
ninety one The natural number $91$ is not prime.
perfect squares mod $8$ Any integer that is a perfect square has a remainder of $0$, $1$, or $4$ when divided by $8$. Note that an integer is a perfect square if and only if it is equal to the square of some integer.
same gcd If $a$ and $b$ are integers then $\gcd(a,b)=\gcd(a,a+b)$.
consecutives are relatively prime The greatest common divisor of two consecutive integers is $1$.
binary fun Define a sequence of natural numbers by $a _ 0=1$ and for all natural numbers $n$, $$a _ {n+1}=2\cdot a _ n+1$$ Then for all natural numbers $n$, we have $a _ n=2^{n+1}-1$.
somewhat odd Let $a$ be a sequence of natural numbers such that $a _ 0=1$, $a _ 1=3$, and for all natural numbers $n$, $$a _ {n+2}=2\cdot a _ {n+1}-a _ n$$ Then for all natural numbers $n$, $$a _ n=2\cdot n+1$$
interesting multiples of $25$ For all $n\in\bbN^+$, the number $6^n-5\cdot n-1$ is divisible by $25$.
As we have seen above, natural numbers can be expressed in the form $0,1,2,3,\ldots$. Similarly, integers can be expressed as $\ldots,\negative{3},\negative{2},\negative{1},0,1,2,3,\ldots$. As seen above, the rational numbers can all be represented as quotients of integers. But what about arbitrary real numbers?
For these we have their decimal or base ten representations. These can be defined in terms as a special kind of infinite series.
Definition . Let $\langle a\rangle _ {k=0}^\infty$ be a sequence of nonnegative real numbers such that the sequence of partial sums $$\left\langle\sum _ {k=0}^n a _ k \right\rangle _ {n=0}^\infty =a _ 0,\, a _ 0+a _ 1,,a _ 0+a _ 1+a _ 2,,\ldots$$ is bounded above. For such sequences we define the infinite series $$\sum _ {n=0}^\infty a _ k$$ to be the least upper bound of the sequence of partial sums.
This allows us to define the decimal representation of any real number.
Definition . Let $x$ be a nonnegative real number. A digit is one of the numbers $0,1,2,\ldots,9$. If $\left\langle d _ {n-k}\right\rangle _ {k=0}^\infty$ is a sequence of digits such that $$x=\Sigma _ {k=0}^\infty d _ {n-k}\cdot 10^{n-k}$$ then the expression $$d _ nd _ {n-1}\cdots d _ 0.d _ {\negative{1}}d _ {\negative{2}}d _ {\negative{3}}\cdots$$ is called a decimal representation of $x$. In this situation a decimal representation of $\negative{x}$ is just $$\negative{d _ n}d _ {n-1}\cdots d _ 0.d _ {\negative{1}}d _ {\negative{2}}d _ {\negative{3}}\cdots$$
Prove the following theorems about the real numbers.
Finite Geometric Series For every natural number $n$ and every real number $r$ other than $1$, $$\sum _ {k=0}^n r^k = \frac{1-r^{n+1}}{1-r}$$
Geometric Series For every every real number $r$, if $\negative{1}<r<1$ then $$\sum _ {k=0}^\infty r^k = \frac{1}{1-r}$$
One third The decimal representation of $\frac{1}{3}$ is $0.333333…$, i.e., $d _ 0=0$ and $d _ k=3$ for all negative integers $k$.
Decimal representation of rationals We say a decimal representation $$d _ nd _ {n-1}\cdots d _ 0.d _ {\negative{1}}d _ {\negative{2}}d _ {\negative{3}}\cdots$$ (or its additive inverse) is eventually repeating if there exists positive integers $k$ and $p$ such that for all $j$ greater than $k$, $d _ {\negative{j}}=d _ {\negative{(j+p)}}$. The decimal representation of a real number $x$ is eventually repeating if and only if $x$ is a rational number.
