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.
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 powers of ten also have special names:
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.
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. The set of all natural numbers is written as formula_1.
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, formula_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, while others say that 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 are numbers less than zero.
One way to think of negative numbers is to use 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 is. For example, the point one is one centimeter to the right, and the point two is two centimeters to the right.
However, the point one centimeter to the left of the zero point cannot be point one, since there is already a point called one. We therefore call this point minus one (−1, as it is one centimeter away but in the opposite direction).
A drawing of a number line is below.
All the normal operations of mathematics can be done with negative numbers:
Since finding the square root of a negative number is impossible for real numbers (as negative times negative equals positive for real numbers), the square root of -1 is given a special name: "i". This is also called the imaginary unit.
Integers are all the natural numbers, all their opposites, and the number zero. Decimal numbers and fractions are not integers.
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 zero.
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 are numbers which cannot be written as a fraction, but do not have imaginary parts (explained later).
Irrational numbers often occur in geometry. For example, 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, and unlike 0.333333 ..., these digits would not repeat forever.
Real numbers is a name for all the sets of numbers listed above:
The real numbers form the real line. This is all the numbers that do not involve 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:
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.
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 multiply two complex numbers is more complicated. It is easiest to describe in general terms, with two complex numbers a + bi and c + di.
formula_3
For example, (4 + 5i) × (3 + 2i) = (4 × 3 − 5 × 2) + (4 × 2 + 5 × 3)i = (12 − 10) + (8 + 15)i = 2 + 23i.
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.
formula_4
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.
Some well-known transcendental numbers include:
November
November (Nov.) is the eleventh and penultimate (second-last) month of the year in the Gregorian calendar, coming between October and December. It has 30 days. Its name is from the Latin word "novem", which meant "nine". It was the ninth month of the year before January and February were added to the Roman Calendar.