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As explained in a previous article, there is an infinite number of numbers. But what's more important is that there are different types of numbers with unique properties that makes them useful in one kind of mathematics while at the same time makes it seems like they are useless.
In this article, we will be addressing the different types of numbers. For their use on the other hand, we will be needing your help. |
"Number is the ruler of forms and ideas, and the cause of gods and daemons."
Pythagoras, as quoted in 'Life of Pythagoras' by Iamblichus of Chalcis
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 (for example 1/2 is positive, but not 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 possitve. We simbolise the square root of a negative number as i.
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.
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 (for example 1/2 is positive, but not 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 possitve. We simbolise the square root of a negative number as i.
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.
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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. |
√2 is irrational
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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.
Imaginary numbers were called imaginary because when they were first found, many mathematicians did not think they existed. The person who discovered imaginary numbers was Gerolamo Cardano in the 1500s. The first to use the words imaginary number was René Descartes. The first people to use these numbers were Leonard Euler and Carl Friedrich Gauss. Both lived in the 18th century.
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.
Imaginary numbers were called imaginary because when they were first found, many mathematicians did not think they existed. The person who discovered imaginary numbers was Gerolamo Cardano in the 1500s. The first to use the words imaginary number was René Descartes. The first people to use these numbers were Leonard Euler and Carl Friedrich Gauss. Both lived in the 18th century.
Subsets of the complex numbers
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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. To multiply two complex numbers is complicated. It is easiest to describe in general terms, with two complex numbers a + bi and c + di. 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.
Well-known transcendental numbers are: e, π, e^a for algebraic a ≠ 0
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.
Well-known transcendental numbers are: e, π, e^a for algebraic a ≠ 0
Ponder this
Are irrational, complex, and imaginary numbers merely derivatives or are they legitimately unique as natural numbers are?
Why do many of the irrational, complex, and imaginary numbers appear in nature?
Discuss
What are the applications of each form of numbers other than simple arithmetic? Show examples in different branches of science. For example, the golden ratio or phi in the arrangement of seeds on a flower and leaves on a plant stem. Try to figure out why these numbers are used, and in what relation they have with the nature of that field or subject.
Further readings
Number theory, is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions.
Patterns in nature, are visible regularities of form found in the natural world. They recur in different contexts and can be modeled mathematically.
