Sponsored Ads
A number is a mathematical object used to count, measure, and label. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs).
Sponsored Ads
Classification of Number System
Numbers can be classified into Natural Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers, Complex Numbers etc.
In the decimal number system or the base 10 number system which is the most universally used today for mathematical operations 10 digits are used to write the symbols of natural numbers. They are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right.
Read More: Number System
The Natural Numbers
The natural numbers or sometimes called as the counting numbers are the most common among numbers. There are infinite number of natural numbers starting from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and so on….
The set of natural numbers, {1,2,3,4,5,...} is sometimes written N for short.
The sum and product of any two natural numbers is also a natural number
eg: 2 + 5 = 7 and
eg: 3 × 5 = 15
This is not true for subtraction and division, though.
Whole numbers
Whole numbers are the numbers with natural numbers and a 0. i.e. 0,1,2,3,4,5,6,7,8,9,10 and so on
The Integers
Integers are the set of natural numbers, their additive inverse and 0.
{….., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ……}
So here comes the concept of negative numbers. The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign).
For example the negative number of 2 is -2 and 2+(-2) = 0.
The symbol for integers is Z.
The sum, product, and difference of any two integers is also an integer. But this is not true for division.
Eg: 1÷2 = 0.5 or 1/2
The Rational Numbers
A rational number is a number that can be expressed as ratio between tow numbers or a fraction with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator.
Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction m/n represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number;
for example 1/2 and 2/4 are equal,
All the integers are included in the rational numbers, since any integer z can be written as the ratio z/1
The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don't divide by 0).
The symbol for rational number is Q (for quotient).
Sponsored Ads
The Irrational Numbers
An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.
The first such equation to be studied was 2=x2. What number times itself equals 2?
2√2 is about 1.4141.414, because 1.4142=1.999396, which is close to 2. But you'll never hit exactly by squaring a fraction (or terminating decimal). The square root of 2 is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern:
2√=1.41421356237309
Other famous irrational numbers are the golden ratio, a number with great importance to biology:
1+5√2=1.61803398874989
π (pi), the ratio of the circumference of a circle to its diameter:
π=3.14159265358979...
and e, the most important number in calculus:
e=2.71828182845904
Irrational numbers can be further subdivided into algebraic numbers, which are the solutions of some polynomial equation (like 2√2 and the golden ratio), and transcendental numbers, which are not the solutions of any polynomial equation. π and e are both transcendental.
The Real Numbers
The real numbers include all the measuring numbers. Real numbers are usually represented by using decimal numerals, in which a decimal point is placed to the right of the digit with place value 1. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left.
A finite decimal representation allows us to represent exactly only the integers and those rational numbers whose denominators have only prime factors which are factors of ten. Thus one half is 0.5, one fifth is 0.2, one tenth is 0.1, and one fiftieth is 0.02. To represent the rest of the real numbers requires an infinite sequence of digits after the decimal point. Since it impossible to write infinitely many digits, real numbers are commonly represented by rounding or truncating this sequence, or by establishing a pattern, such as 0.333..., with an ellipsis to indicate that the pattern continues. Thus 123.456 is an approximation of any real number between
1234555/10000 and 1234565/10000 (rounding) or any real number between 123456/1000 and 123457/1000 (truncation). Negative real numbers are written with a preceding minus sign: -123.456.
Every real number is either rational or irrational. Every real number corresponds to a point on the number line.
The symbol for the real numbers is R.
The Complex Numbers
The complex number is formed by two parts. The real part and the complex part. It is written as (a+bi). In the expression (a + bi), a is called the real part and b is called the imaginary part, where i is the imaginary unit, √−1.(root of -1).
If the real part of a complex number is 0, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer.
Sponsored Ads
The symbol for the complex numbers is C
Sponsored Ads