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| Numbers 0 - 9. |
Numbers
A number is a concept from mathematics, used to count or
measure. Depending on the field of mathematics, where
numbers are used, there are different definitions: |
- People use symbols to represent
numbers; they call them numerals. Common places where
numerals are used are for labeling, as in telephone
numbers, for ordering, as in serial numbers, or to put a
unique identifier, as in an ISBN, a unique number that
can identify a book.
- Cardinal numbers are used to measure
how many items are in a set. {A,B,C} has size "3".
- Ordinal numbers are used to specify
a certain element in a set or sequence (first, second,
third).
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Numbers are also used for other things besides counting.
Numbers are used when things are measured. Numbers are used
to study how the world works. Mathematics is a way to use
numbers to learn about the world and make things. The study
of the rules of the natural world is called science. The
work that uses numbers to make things is called engineering. |
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Numbering methods
Numbers for people
There are different ways of giving symbols to numbers.
These methods are called number systems. The most common
number system that people use is the base ten number
system. The base ten number system is also called the
decimal number system. The base ten number system is
common because people have ten fingers and ten toes.
There are 10 different symbols {0, 1, 2, 3, 4, 5, 6, 7,
8, and 9} used in the base ten number system. These ten
symbols are called digits.
A symbol for a number is made up of these ten digits.
The position of the digits shows how big the number is.
For example, the number 23 in the decimal number system
really means (2 times 10) plus 3, and 101 means 1 times
a hundred (=100) plus 0 times 10 (=0) plus 1 times 1
(=1).
Numbers for machines
Another number system is more common for machines. The
machine number system is called the binary number
system. The binary number system is also called the base
two number system. There are two different symbols (0
and 1) used in the base two number system. These two
symbols are called bits.
A symbol for a binary number is made up of these two bit
symbols. The position of the bit symbols shows how big
the number is. For example, the number 10 in the binary
number system really means 1 times 2 plus 0, and 101
means 1 times four (=4) plus 0 times two (=0) plus 1
times 1 (=1). The binary number 10 is the same as the
decimal number 2. The binary number 101 is the same as
the decimal number 5. |
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Names of numbers
English has special names for some of the numbers in the
decimal number system that are "powers of ten". All of
these power of ten numbers in the decimal number system
use just the symbol "1" and the symbol "0". For example,
ten tens is the same as ten times ten, or one hundred.
In symbols, this is "10 × 10 = 100". Also, ten hundreds
is the same as ten times one hundred, or one thousand.
In symbols, this is "10 × 100 = 10 × 10 × 10 = 1000".
Some other power of ten numbers also have special names: |
- 1 – one
- 10 – ten
- 100 – one hundred
- 1000 – one thousand
- 1,000,000 – one million
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When dealing with larger numbers than this, there are two
different ways of naming the numbers in English. Under the
"long scale" a new name is given every time the number is a
million times larger than the last named number. It is also
called the "British Standard". This scale used to be common
in Britain but is not often used in English-speaking
countries today. It is still used in some other European
nations. Another scale is the "short scale" under which a
new name is given every time a number is a thousand times
larger than the last named number. This scale is a lot more
common in most English-speaking nations today. |
- 1,000,000,000 – one billion (short
scale), one milliard (long scale)
- 1,000,000,000,000 – one trillion
(short scale), one billion (long scale)
- 1,000,000,000,000,000 – one
quadrillion (short scale), one billiard (long scale)
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Types of numbers
Natural numbers
Natural numbers are the numbers which we normally use
for counting, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 etc. Some
people say that 0 is a natural number, too.
Another name for these numbers is positive numbers.
These numbers are sometimes written as +1 to show that
they are different from the negative numbers. But not
all positive numbers are natural.
If 0 is called a natural number, then the natural
numbers are the same as the whole numbers. If 0 is not
called a natural number, then the natural numbers are
the same as the counting numbers. So if the words
"natural numbers" are not used, then there will be less
confusion about whether zero is included or not. But
unfortunately, some say that zero is not a whole number,
either, and some say whole numbers can be negative.
"Positive integers" and "non-negative integers" are
another way to include zero or exclude zero, but only if
people know those words.
Negative numbers
Negative numbers are numbers less than zero.
One way to think of negative numbers is using a number
line. We call one point on this line zero. Then we will
label (write the name of) every position on the line by
how far to the right of the zero point it is, for
example the point one is one centimeter to the right,
the point two is two centimeters to the right.
Now think about a point which is one centimeter to the
left of the zero point. We cannot call this point one,
as there is already a point called one. We therefore
call this point minus 1 (−1) (as it is one centimeter
away, but in the opposite direction).
All the normal operations of mathematics can be done
with negative numbers:
If people add a negative number to another this is the
same as taking away the positive number with the same
numerals. For example, 5 + (−3) is the same as 5 − 3,
and equals 2.
If they take away a negative number from another this is
the same as adding the positive number with the same
numerals. For example, 5 − (−3) is the same as 5 + 3,
and equals 8.
If they multiply two negative numbers together they get
a positive number. For example, −5 times −3 is 15.
If they multiply a negative number by a positive number,
or multiply a positive number by a negative number, they
get a negative result. For example, 5 times −3 is −15.
As finding the square root of a negative number is
impossible as negative times negative equals positive.
We symbolize the square root of a negative number as i.
Integers
Integers are all the natural numbers, all their
opposites, and the number zero. Decimal numbers and
fractions are not integers.
Rational numbers
Rational numbers are numbers which can be written as
fractions. This means that they can be written as a
divided by b, where the numbers a and b are integers,
and b is not equal to 0.
Some rational numbers, such as 1/10, need a finite
number of digits after the decimal point to write them
in decimal form. The number one tenth is written in
decimal form as 0.1. Numbers written with a finite
decimal form are rational. Some rational numbers, such
as 1/11, need an infinite number of digits after the
decimal point to write them in decimal form. There is a
repeating pattern to the digits following the decimal
point. The number one eleventh is written in decimal
form as 0.0909090909 ... .
A percentage could be called a rational number, because
a percentage like 7% can be written as the fraction
7/100. It can also be written as the decimal 0.07.
Sometimes, a ratio is considered as a rational number.
Irrational numbers
Irrational numbers are numbers which cannot be written
as a fraction, but do not have imaginary parts
(explained later).
Irrational numbers often occur in geometry. For instance
if we have a square which has sides of 1 meter, the
distance between opposite corners is the square root of
two, which equals 1.414213 ... . This is an irrational
number. Mathematicians have proved that the square root
of every natural number is either an integer or an
irrational number.
One well-known irrational number is pi. This is the
circumference (distance around) of a circle divided by
its diameter (distance across). This number is the same
for every circle. The number pi is approximately
3.1415926535 ... .
An irrational number cannot be fully written down in
decimal form. It would have an infinite number of digits
after the decimal point. Unlike 0.333333 ..., these
digits would not repeat forever.
Real numbers
Real numbers is a name for all the sets of numbers
listed above: |
- The rational numbers, including
integers
- The irrational numbers
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This is all numbers that do not involve imaginary
numbers.
Imaginary numbers
Imaginary numbers are formed by real numbers multiplied
by the number i. This number is the square root of minus
one (−1).
There is no number in the real numbers which when
squared makes the number −1. Therefore, mathematicians
invented a number. They called this number i, or the
imaginary unit.
Imaginary numbers operate under the same rules as real
numbers: |
- The sum of two imaginary numbers
is found by pulling out (factoring out) the i. For
example, 2i + 3i = (2 + 3)i = 5i.
- The difference of two imaginary
numbers is found similarly. For example, 5i − 3i =
(5 − 3)i = 2i.
- When multiplying two imaginary
numbers, remember that i × i (i2) is −1.
For example, 5i × 3i = ( 5 × 3 ) × ( i × i ) = 15 ×
(−1) = −15.
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Imaginary numbers were called imaginary because when
they were first found, many mathematicians did not think
they existed.
Complex numbers
Complex numbers are numbers which have two parts; a real
part and an imaginary part. Every type of number written
above is also a complex number.
Complex numbers are a more general form of numbers. The
complex numbers can be drawn on a number plane. This is
composed of a real number line, and an imaginary number
line.
All of normal mathematics can be done with complex
numbers: |
- To add two complex numbers, add
the real and imaginary parts separately. For
example, (2 + 3i) + (3 + 2i) = (2 + 3) + (3 + 2)i= 5
+ 5i.
- To subtract one complex number
from another, subtract the real and imaginary parts
separately. For example, (7 + 5i) − (3 + 3i) = (7 −
3) + (5 − 3)i = 4 + 2i.
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To multiply two complex numbers is complicated. It is
easiest to describe in general terms, with two complex
numbers a + bi and c + di.
( a + b \mathrm{i} ) \times ( c + d\mathrm{i} ) = a
\times c + a \times d\mathrm{i} + b\mathrm{i} \times c +
b\mathrm{i} \times d\mathrm{i} = ac + ad\mathrm{i} +
bc\mathrm{i} -bd = ( ac - bd ) + ( ad + bc )\mathrm{i}
For example, (4 + 5i) × (3 + 2i) = (4 × 3 − 5 × 2) + (4
× 2 + 5 × 3)i = (12 − 10) + (8 + 15)i = 2 + 23i.
Transcendental numbers
A real or complex number is called a transcendental
number if it can not be obtained as a result of an
algebraic equation with integer coefficients.
Proving that a certain number is transcendental can be
extremely difficult. Each transcendental number is also
an irrational number. The first people to see that there
were transcendental numbers were Gottfried Wilhelm
Leibniz and Leonhard Euler. The first to actually prove
there were transcendental numbers was Joseph Liouville.
He did this in 1844. |
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 Kiddle: Numbers
Wikipedia: Numbers |
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