Decimal Expansions

Decimal Representation of Real Numbers

Once the number line with 0 and 1 has been fixed, the location of any other real number is given by a precise address called the DECIMAL REPRESENTATION.

We start with 0 and 1 and begin counting by adding 1 each time and naming a new digit:

2 = 1 + 1
3 = 2 + 1
4 = 3 + 1
:
:
9 = 8 + 1

By continuing this process, we get the set of counting numbers, or the NATURAL NUMBERS which is denoted N.

The formal definition of Natural Numbers states:

N is the smallest subset of R with the following two properties:


(i)

1 in N


(ii)

if n is in N then n + 1 is in N

We can express any Natural Number using the exponential notation 10^j = (10)(10)(10)...(10) which is the product of 10 with itself j-times. Here are a few brief examples.

In general if a_j, a_j-1, a_j-2, ..., a₁, a₀ are digits from {0,1, 2, ... , 9}, then a_ja_j-1a_j-2...a₁a₀ represents the natural number a_j(10^j) + a_j-1(10^j-1) + a_j-2(10^j-2) + ... + a₁(10) + a₀ which is built up from the digits and powers of 10 using the basic operations of addition and multiplication.

We often write the set of Natural Numbers as {1, 2, 3, ...} where "..." is read as "and so on". If 0 is to be included then we denote N° as the set {0, 1, 2, 3, ...}.

How do we represent a positive number that is not a natural number? Again we use a string of digits but now a "." is inserted and the digits to the right of the "." corresponds to amounts (1/10)^k where k in N. Examples.

Some numbers have an infinite number of decimal digits, however it is possible for us to work with such numbers. Here is an example.

Vital Point:

Let x in R. If x 0, there is an n in N° and an infinite sequence of digits d₁, d₂, d₃, ... such that x = n.d₁d₂d₃... . This means that for any j in N, n.d₁d₂...d_j x < n.d₁d₂...d_j + (1/10)^j.

If x < 0 then -x > 0. So there is an n in N° and a sequence of digits d₁, d₂, d₃, ... such that for any j in N, n.d₁d₂...d_j -x < n.d₁d₂...d_j + (1/10)^j.
This implies that -n.d₁d₂...d_j - (1/10)^j < x -n.d₁d₂...d_j
We write x = -n.d₁d₂...d_j Click here to view an example.

Note that if we start with a decimal expression, then there is always a real number that is represented by that decimal expression.

Integers and Rational Numbers

A real number n is called an INTEGER if one of the following is true:

(i)

n = 0

(ii)

n in N

(iii)

-n in N

The set of integers, denoted Z, is given by Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

The addition, subtraction and multiplication of integers results in forming another integer. With division, however, we sometimes generate non-integers. For example, 1/2, 113/52, and -2/7. These numbers are called the RATIONAL NUMBERS, which is denoted Q. Then we say Q = {n/m such that n, m in Z, and m 0}. If m = 1 then we get the integers. We note that N is a subset of Z which is a subset of Q which is a subset of R, denoted
.

Every rational number has a decimal representation that has a repeating component. Examples.

Also, any decimal representation of a number with a repeating component can be changed to a rational number. Here are three examples

Irrational Numbers

The real numbers which are not rational are called irrational. For the irrationals, we can not determine their full decimal representation. We can get a close representation of the irrationals by doing the same kind of procedure that we did in the example on the cube root on 10 in the section Decimal Representations of Real Numbers.

Aside:

let n,j in N. We say j divides n if there exists a k in N such that n = jk. This is denoted j | n.
Examples: 2 | 6, 17 | 51, 10 | 320, but 5 does not divide 18.

But how do we know that the cube root of 10 is irrational? Try proving it yourself first... and if you get stuck have a peek at the proof.

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