“THE REAL NUMBERS”
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Real Number (Definition):
The real numbers are those numbers we normally use, such as :
1, 15.82, -0.1, 3/4, etc…
Positive or negative, large or small, whole numbers or decimal numbers are all Real
Numbers.
They are called "Real Numbers" because they are not Imaginary Numbers.
Real Numbers are just numbers like:
1 12.38 -0.8625 3/4 √2 1998
Real Numbers include:
Whole Numbers (like 1,2,3,4, etc)
Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc )
Irrational Numbers (like π, √3, etc )
Real Numbers can also be positive, negative or zero.
So ... what is NOT a Real Number?
√-1 (the square root of minus 1) is not a Real Number, it is
anImaginary Number
Infinity is not a Real Number
And there are also some special numbers that mathematicians play with
that are not Real Numbers
Why are they called "Real" Numbers?
Because they are not Imaginary Numbers.
The Real Numbers did not have a name before Imaginary Numbers were thought of. They
got called "Real" because they were not Imaginary. That is the actual answer!
The Real Number Line :
The Real Number Line is like an actual geometric line.
A point is chosen on the line to be the "origin", points to the right will be positive, and
points to the left will be negative.
A distance is chosen to be "1", and the whole numbers can then be marked off: {1,2,3,...),
and also in the negative direction: {-1,-2,-3, ...}
Any point on the line is a Real Number:
The numbers could be rational (like 20/9)
or irrational (like π)
But you won't find Infinity, or an Imaginary Number.
Real does not mean they are in the real world :
They are not called "Real" because they show the value of something real.
In mathematics we like our numbers pure, if we write 0.5 we
mean exactly half, but in the real world half may not be exact (try cutting an
apple exactly in half).
The set N of natural numbers :
N = {1, 2, 3, 4, ...}
These are numbers for counting things, and early man hundreds of thousands of years ago
probably used them. It wasn't until about 200 BC., however, that Greek mathematicians
made the jump from finite numbers to infinite numbers. This jump is suggested by the three
dots placed after the 4.
2. The set I of integers :
I = {…-3, -2, -1, 0, 1, 2, 3, ...}
The set of integers consists of the union of the natural numbers, their negatives, and the
number zero. Oddly enough, the incorporation of negative numbers was a long time coming.
Certainly the concept was understood with the use of black and red entries in the ledgers of
traders, but negative numbers were not fully incorporated into mathematics until the Italian
mathematician, Girolamo Cardano, used them in 1545 AD. The zero was introduced much
earlier, some say around 700 to 800 AD, by Hindu mathematicians.
3. The set Q of rational numbers :
This set includes fractions as well as integers. Fractional numbers are quite ancient. They
appear in the earliest mathematical writings, and were discussed at some length as early as
1550 BC. in the Rhind Papyrus of Egypt. All rational numbers are characterized by having a
repeating decimal form.
The Number Line :
Before continuing our study of the real number system, let us consider a geometric
interpretation of numbers called the number line. It allows us to set up a one-to-one
correspondence between numbers and points on the line.
At this point it appears that we can completely fill up our number line with rational
numbers. In fact, Greek mathematicians thought that this was the case, until in the 6th
century BC, the mathematical school of Pythagoreans encountered a number that could not
be written as a ratio of two integers: i.e. the number was not rational. The number was the
square root of two , the length of the diagonal of a square whose sides are one unit long.
Since the Greeks originally thought that all numbers were rational, this discovery was
tantamount to finding that the diagonal of a square did not have a mathematical length.
Most Greek mathematicians resolved this paradox by thinking of numbers as lengths; i.e. in
geometric terms, which inhibited the development of algebra which might have rivaled or
surpassed their work in geometry. It took centuries of development and sophistication in
mathematics to resolve this problem. What was done was to fill in the gaps of the number
line, which are not filled by rational numbers, with irrational numbers.
4. The set L of irrational numbers :
Note that the prefix 'ir' means not. Also, unlike;
rational numbers which contains a pattern of digits to the right of the
decimal point that repeats itself over and over again,
irrational numbers are non-repeating decimals; i.e. no pattern is ever
established. Two well known examples are -
We are now able to define a number in which addition, subtraction, multiplication, division,
and roots of positive numbers are closed operations.
5. The set R of real numbers :
This set includes all rational and irrational numbers. The set of real numbers is closed for
the operations of addition, subtraction, multiplication, division, and roots of positive
numbers. Odd roots of negative numbers exist, however, even roots of negative numbers
do not.
As you can see, we started with the most basic type of number, and through a series of
abstractions, reached the level of the real numbers. We can represent this schematically
with the following diagram.