What is a Prime numbers

Prime numbers are positive integers that are divisible only by themselves and 1. The first eleven are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31, but there are infinitely many. By convention, 1 is not considered prime, while 2 is the only even prime. A number that is neither 1 nor a prime is called a composite number. 

 Every composite number can be written uniquely as a product of prime factors multiplied together: for example, 12 = 2² × 3, 21 = 3 × 7, and 270 = 2 × 3³ × 5. Since prime numbers cannot be factorized themselves, they can be thought of as the fundamental building blocks of positive integers. However, determining whether a number is prime, and finding the prime factors if it is not, can be extremely difficult. This process is therefore an ideal basis for encryption systems. 

 There are many deep patterns to the primes, and one of the great outstanding hypotheses of mathematics, the Riemann hypothesis, is concerned with their distribution.

Prime Numbers Chart

Squares, square roots, and powers

The square of any number x is the product of the number times itself, denoted x². The term originates from the fact that the area of a square (with equal sides) is the length of a side times itself. The square of any nonzero number is positive, since the product of two negative numbers is positive, and the square of zero is zero. Conversely, any positive number must be the square of two numbers, x and −x. These are its square roots.

More generally, multiplying a number x by itself n times gives x to the power of n, written xn. Powers have their own combination rules, which arise from their meaning:

xn × xm = xn⁺m, (xn)m = xnm, x⁰ = 1, x¹ = x, and x⁻¹ = 1/X

It also follows from the formula (xn)m = xnm that the square root of a number can be thought of as that number raised to the power of one-half. 

                                      Example:

What is a Rational numbers

Rational numbers

Rational numbers are numbers that can be expressed by dividing one integer by another nonzero integer. Thus all rational numbers take the form of fractions or quotients. These are written as one number, the numerator, divided by a second, the denominator.

When expressed in decimal form, rational numbers either come to an end after a finite number of digits, or one or a number of digits are repeated forever. For instance, 0.3333333... is a rational number expressed in decimal form. In fraction form, the same number is 1/3. It is also true to say that any decimal number that comes to an end or repeats must be a rational number, expressible in fractional form.

Since there is an infinite number of integers, it is not surprising to find that there are an infinite number of ways of dividing one by another, but this does not mean there is a “greater infinity” of rational numbers than that of the integers.

What is a Combining Numbers

Combining Numbers

There are a number of different ways of combining any two given numbers. They can be added together to form their sum, subtracted to form their difference, multiplied together to form their product, and divided, provided the divisor is nonzero, to form their ratio. In fact, if we think of a − b as a + (−b) and a/b  as a × (1/b) , then the only operations really involved are addition and multiplication, together with taking the reciprocal to calculate 1/b.

Addition and multiplication are said to be commutative, in that the order of the numbers involved does not matter, but for more complicated sequences, the order in which operations are performed can make a difference. To aid clarity in these cases, certain conventions have been developed. Most important, operations to be performed first are written in brackets. Multiplication and addition also satisfy some other general rules about how brackets can be reinterpreted, known as associativity and distributivity, demonstrated opposite.

What is a Families of Numbers

Families of Numbers

Numbers can be classified into families of numbers that share certain properties. There are many ways of putting numbers into classes in this way. In fact, just as there is an infinity of numbers, there is an infinite variety of ways in which they can be subdivided and distinguished from one another. For example the natural numbers, whole numbers with which we count objects in the real world, are just such a family, as are the integers—whole numbers including those less than zero. The rational numbers form another family, and help to define an even larger family, the irrational numbers. The families of algebraic and transcendental numbers are defined by other behaviors while the members of all these different families are members of the real numbers, defined in opposition to the imaginary numbers.

Saying that a number is a member of a certain family is a shorthand way of describing its various properties, and therefore clarifying what sort of mathematical questions we can usefully ask about it. Often, families arise from the creation of functions that describe how to construct a sequence of numbers. Alternatively, we can construct a function or rule to describe families that we recognize intuitively.

For instance, we instinctively recognize even numbers, but what are they? Mathematically, we could define them as all natural numbers of the form 2 × n where n is itself a natural number. Similarly, odd numbers are natural numbers of the form 2n + 1, while prime numbers are numbers greater than 1, whose only divisors are 1 and themselves.

Other families arise naturally in mathematics—for example in the Fibonacci numbers (1, 2, 3, 5, 8, 13, 21, 34, . . .), each number is the sum of the previous two. This pattern arises naturally in both biology and mathematics. Fibonacci numbers are also closely connected to the golden ratio.

Other examples include the multiplication tables, which are formed by multiplying the positive integers by a particular number, and the squares, where each number is the product of a natural number with itself: n times n, or n², or n squared.

What is a number line?

Number Line

The number line is a the concept about the meaning of mathematical operations. It is either horizontal or vertical line, with major divisions marked by the positive (+) and negative (-) whole numbers in a opposite direction. The entire range of whole numbers covered by the number line are also known as the integers.

In basic mathematics, it is a picture of horizontal straight line that divided into two symmetric half from its point of origin, zero (0) and stuff with points in which each points correspond to a whole number or real number.  

In between the whole number integers shown, there are other numbers, such as half, thirds, and quarters. These are ratios formed by dividing any integer by a nonzero integer. Together with the natural numbers—zero and the positive whole numbers, which are effectively ratios divided by 1—they form the rational numbers. These are marked by finer and finer subdivisions of the number line.

But do the rational numbers complete the number line? It turns out that almost all the numbers between zero and one cannot be written as ratios. These are known as irrational numbers, numbers whose decimal representations never stop and are not eventually repeating. The complete set of rationals and irrationals together are known as the real numbers.

What is a Natural numbers?

Natural Numbers

Natural numbers are the simple counting numbers (0, 1, 2, 3, 4, . . .). The skill of counting is intimately linked to the development of complex societies through trade, technology, and documentation. Counting requires more than numbers, though. It involves addition, and hence subtraction too.

As soon as counting is introduced, operations on numbers also become part of the lexicon—numbers stop being simple descriptors, and become objects that can transform each other. Once addition is understood, multiplication follows as a way of looking at sums of sums—how many objects are in five groups of six?—while division offers a way of describing the opposite operation to multiplication—if thirty objects are divided into five equal groups, how many objects are in each?

But there are problems. What does it mean to divide 31 into 5 equal groups? What is 1 take away 10? To make sense of these questions we need to go beyond the natural numbers.

What is a number System?

 Number System

A number system is a way of writing down numbers. In our everyday decimal system, we represent numbers in the form 434.15, for example. Digits within the number indicate units, tens, hundreds, tenths, hundredths, thousandths and so on, and are called coefficients. So 434.15 = (4 × 100) + (3 × 10) + (4 × 1) + (1/10) + (5/100) . This is simply a shorthand description of a sum of powers of ten, and any real number can be written in this way.

But there is nothing special about this “base 10” system. The same number can be written in any positive whole-number base n, using coefficients ranging from 0 up to n - 1. For example, in base two or binary, the number 8^5/16 can be written as 1000.0101. The coefficients to the left of the decimal point show units, twos, fours, and eights—powers of 2. Those to the right show halves, quarters, eighths, and sixteenths. Most computers use the binary system, since two coefficients (0 and 1) are easier to work with electronically.

What is an Infinity

Infinity Symbol

Infinity (represented mathematically as 8) is simply the concept of endlessness: an infinite object is one that is unbounded. It is hard to do mathematics without encountering infinity in one form or another. Many mathematical arguments and techniques involve either choosing something from an infinite list, or looking at what happens if some process is allowed to tend to infinity, continuing toward its infinite limit.

Infinite collections of numbers or other objects, called infinite sets, are a key part of mathematics. The mathematical description of such sets leads to the beautiful conclusion that there is more than one sort of infinite set, and hence there are several different types of infinity.

In fact there are infinitely many, bigger and bigger, kinds of infinite set, and while this may seem counter intuitive, it follows from the logic of mathematical definitions.