Introduction to The Numbers

First, when we heard about the numbers. What image will appear in our mind?

Simply 1,2,3...so on.

Why do we think about numbers always from 1?

It's Natural because when we look around nature and starts amounting to something that always comes from one. For example, one moon, one sun, one God and one World...there doesn't occur the concept of zero in nature.

Then why zero came?

What're the numbers?

The first point, I want to mention to you is, Right from the beginning/learning about numbers from the first year when we come to the school started learning about the numbers and now we already know many things about them and use in the daily life and we are surrounded by numbers everywhere. Numbers help us count concrete objects they help us to say which collection of objects either bigger or smaller and arrange them as we want. However, A Number is a mathematical object manipulated to count, measure and label. Universally, the individual numbers can be represented by symbols called numerals.

The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called digits.

There are several types of numbers, but they fall into two main groups, counting numbers and scalars.

Counting Numbers=Natural Numbers

Scalars=Measurement of quantity

Types of Numbers or Set of Numbers

$$ \mathbb{N} \subseteq \mathbb{W} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C} $$

$$ Natural Numbers ( \mathbb{N} ) $$

Also called Positive Integers or Counting Numbers. 1,2,3,...

$$ Whole Numbers ( \mathbb{W} ) $$

Natural Numbers including zero called Whole Numbers. 0,1,2,3,...

$$ Integers ( \mathbb{Z} ) $$

Whole Numbers including negatives of Natural Numbers called Integers. ....-3,-2,-1,0,1,2,3,...

$$ Rational Numbers ( \mathbb{Q} )$$

It is all the fractions where numerator and denominator are integers that is, it is of the form: p/q; q is not equal to zero. $$ \frac {1} {2}, \frac {-5} {6}, \frac {3} {4} \dots $$

$$ Real Numbers ( \mathbb{R} ) $$

Real Numbers can be defined as the union of both rational and irrational numbers. $$ 2,34, \frac {5} {7}, \frac {-7} {9}, π, √3 \dots $$

$$ Complex Numbers ( \mathbb{C} ) $$

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and I represent the “imaginary unit”, satisfying the equation $$ i^2 = -1. $$ Because no real number satisfies this equation, i is called an imaginary number or iota. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols ℂ or C. It is also an uncountable number.

Discrete and Continuous Numbers

$$ \mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q} $$ are referred to as Discrete. Each of these sets is Countable. The set of Real Number can't be counted. It is an uncountable set. This is because they are Continuous.

Number Line

The numbers on the number line increase as one move from left to right and decreases on moving from right to left such that:

A number line can be extended infinitely in any direction and usually represented horizontally. It serves as an abstraction for real numbers denoted by $ \mathbb {R} $.

Prime Numbers

Prime numbers are the natural numbers or positive integers having only two factors 1 and the integer itself.

The few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41...

Composite Numbers

A Composite number is a natural number or positive integers having more than two factors.

The few composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16,....

Relative Prime Numbers

Two integers (a, b) are relatively prime (or coprime) if gcd (greatest common divisor) of two integers is 1 that is, $ gcd(a, b) = 1 $. $ For\ example, \ gcd(2,3) = 1 $.

G.C.D or H.C.F (Greatest common divisor or Highest common factor)

The G.C.D or H.C.F of two numbers is the largest positive integer that divides both the number means that we take the largest common factor of both numbers. Denoted as $ GCD(a, b)\ or\ HCF(a, b) $.

For example, $ H.C.F\ of\ 6\ and\ 8\ or\ G.C.D(6,8) =? $.

Factors of 6 = 1, 2, 3, 6 and Factors of 8 = 1, 2, 4, 8.

Common factor = 1, 2.

The highest common factor = 2 is the answer.

L.C.M or L.C.D (Least common multiple or least common divisor)

The LCM of two numbers is the smallest positive integer that divides both the number means that we take the smallest common multiple of both numbers. Denoted as LCM(a, b).

For example, LCM(4,8) =?

Factors of 4 = 1, 2, 4 and Factors of 8 = 1, 2, 4, 8

Smallest Common Multiple = 1 × 2 × 4 = 8 is the answer.

Relationship between GCD and LCM

$$ | a.b | = gcd(a, b) × lcm(a, b) ; a \neq 0, b \neq 0 $$

$$ GCD\ of\ given\ fractions\ = \frac {GCD of Numerator}{LCM of Denominator} $$ $$ LCM\ of\ given\ fractions\ = \frac {LCM of Numerator}{GCD of Denominator} $$

Number Sequences

A sequence is a list of numbers that are in order.

Here are a few examples,

Fibonacci Sequence= $ 1, 1, 2, 3, 5, 8, 13,...\text {(formed by adding the two previous terms to get the next one).} $
Square Numbers = $ 1, 4, 9, 16, 25, 36,... $
Cubic Numbers = $ 1, 8, 27, 64, 125, 216,... $
Triangular Numbers = $ 1, 3, 6, 10, 15, 21, 28,... $

Order of Operations

BODMAS is used to explain the order of operation of a mathematical expression.

B =Bracket

O =Of

D =Division

M =Multiplication

A =Addition

S =Subtraction

For example, 4÷2×2+6-8 If we have this type of equation to solve then what will do first that's where the BODMAS rule will apply.

Let's solve it, first, we divide 4÷2=2 Then multiply by 2 i.e., 2×2=4 then add 4+6=10 then finally we subtract 10-8=2. 2 is the answer.

BODMAS is also known as PEDMAS which stands for

P =Parantheses

E =Exponents

D =Division

M=Multiplication

A =Addition

S =Subtraction

Roots and Powers

Powers are very useful in math and provide a convenient way of writing multiplication problems that have many repeated terms.

e.g., $$ 4×4×4=4^3 $$ we multiply 4 three times by itself get $$ 4^3=64 $$ Here, 4 is the base and 3 is the power or exponent.

Let any number be x.

If x is raised to the power 1 then always get the same number x i.e., $$ x^1 = x $$

If $$ x^0 = 1 $$

If x is raised to the power of negative numbers such as $$ x^{-1}= \frac {1} {x} $$

If x is raised to the power of fractions such as

$$ 2^ { \frac {1} {2} } = \sqrt {2}\text { say, 2 is raised to the power 1/2, that is called square root of 2.} $$

Now, the question arises that, How do find the square root of any number?

For example, you are going to find the square root of 64. First, we find prime factors of 64 such as $$ 64=2×2×2×2×2×2 = 2^6 $$ Now, $$ 64^{ \frac{1} {2} } = \sqrt {64} = {(2^6)}^{ \frac{1} {2}} = 2^3 =8 $$ Also, we know that 8×8=64 and this type of square root called the perfect square root.

Some Practice Questions $$ 8^{ \frac{-1} {3} } =? $$ $$ 625^{ \frac {1} {4}} =? $$ $$ \sqrt {1024} =? $$

$ Note: If\ we\ have\ x^{ \frac{1} {n} }\text { then, it is called as the nth roots of x.} $

$ If\ you\ have\ x^\infty \text { then what will be the answer?} $

$ If\ x<1\text { on the number line then}\ x^\infty = 0. $

$ If\ x≥1\text { on the number line then}\ x^\infty = \infty. $

This concept is usually used in higher mathematics, in case of limit finding, but here is just an idea that I am transmitting.

Multiply and Divide Exponents

$ For\ example, \ 5^4×5^2\ can\ be\ written\ as\ (5×5×5×5) × (5×5)= 625×25=15,625\ is\ the\ answer. $

$ Let's\ divide\ 4^3 /4^2= 4×4×4/4×4=4\text { we cancel the top two fours with the bottom two fours and get 4.} $

If we multiply the numbers with the same base and different exponents or powers then we add the powers with the same base. For the above example, we get the base 5 to the power 6;5^6 by adding powers 4+2.

If we divide the numbers with the same base, different exponents then we subtract the powers. For the above example, $ 4^3\ divided\ by\ 4^2\ then\ subtract\ the\ powers\ 3-2=1\ get\ 4^1=4\ is\ the\ answer. $

If we have a different base and the same power then what to do?

$ e.g., 2^2 × 3^2 =?\text { Then we can't add or subtract the powers in such type of problems.} $ $ We\ solve\ it\ as\ 2×2×3×3=4×9=36\ or\ (2×3)^2=6^2=6×6=36 $

How to raise an exponent to a power?

$ For\ example, \ (5^2)^3 = ?\text { In this case, we can write as, } $ $$ 5^2 × 5^2 × 5^2 = 25×25×25= 15,625 $$

$ First, \ We\ multiply\ the\ powers\ 2×3=6\ and\ write\ as, \ 5^6=\ the\ same\ answer. $

$ Also,\ 1/2^3=?\ solve\ as\ (1/2)^3=1^3/2^3=1/8= 0.125 $

$ Or\ first,\ we\ solve\ the\ value\ of\ 1/2=0.5\text { then write as }\ (0.5)^3 = 0.5×0.5×0.5=0.125. $

Scientific Notations and Approximations

We find always two types of numbers or solutions that are called Exact and Approximate numbers or solutions.

In higher mathematics, to solve engineering and scientific problems that can't be found by simple formulas where we find approximate solutions with errors.

However, the exact numbers arise from the counting, while the approximate numbers arise from the measurement or calculation. This is because we can never calculate or complete measure something accurately. There is always some inaccuracy or error involved.

How to add or subtract approximate numbers or decimal numbers?

When we add or subtract the approximate numbers, the results should have the precision of the least precise number.

For example, we have to add these numbers: $$ 1.2 + 0.325 + 0.24=? $$ We will write as, 1.200 + 0.325 + 0.240 = 1.765 is the answer. ( decimal point will lie in the same place.)

Also, will subtract in the same way.

When multiplying or dividing approximate numbers, the result should have the accuracy of the least accurate number.

$$ e.g., 5.625\times 3.87 =\ 21.76875 $$

But since we have only 3 significant digits the answer becomes ≈ 21.8

Scientific Notation is used to represent numbers that are either very large or very small.

$$ e.g., 990,000 = 9.9\times 100,000 = 9.9\times 10^5 $$

So, as you can see a number in scientific notation is expressed as, $$ B\times 10^k\ where B\ge 1\text { and}\ <10\text { and k is an integer.} $$

Significant Digits or Significant Figures

Zeroes to the left of the first non-zero digit are not significant digits. e.g., 0.000076 has only two significant digits '7' & '6'.
Zeroes between two non-zero numbers will be count as significant digits. e.g., 102 has three significant digits, 1.20 has three significant digits, 0.009900 has four significant as '9900'. Also, 160.000 has six significant.
All non-zero digits are significant. e.g., 12345.67890 has ten significant figures.

Round-Off the Numbers

Rounding off numbers is a mathematical technique of adjusting the digits of a number to make the number easier to use during calculations.

How to round off the whole numbers?

Identify the rounding digits.
If the number to the right is less than 5, round down.
If the number to the right is more than 5 or 5 then round up.
Zero out all digits to the right of the rounding digit.

For example,

Rounding the number to the nearest ten 842.

The rounding digit is the tenth place is 4.

The number to the right is less than 5.

Rounding down, 840 is the answer.
Rounding the number to the nearest hundred 1874.

The rounding digit is the hundredth place is 8.

The number to the right is greater than 5.

Round-Up, 1974.

So, zero out all numbers to the right of the rounding digit. 1900 is the answer.

Rounding the number to the nearest tenths place 584.21.

The rounding digit is the tenths place is 2.

The number to the right is less than 5.

Round-down, 584.20 is the answer.

If the digit to the right side of the rounding digit is 0, 1, 2, 3, or 4, then the rounding digit doesn’t change. All digits to the right of the rounding digit become zero.

If the digit to the right side of the rounding digit is 5, 6, 7, 8, or 9, the rounding digit increases by one digit. All the digits to the right are dropped to zero.

I'm just giving some practice questions to solve and write answers in the comment section given below:

Round off the following numbers to the nearest tens.

$$ 28, 34, 56, 78, 99, 789, 1999, 99999. $$
Round off the following numbers to the nearest hundred.

$$ 2029, 765, 987, 2212121. $$

How to round off the decimal numbers?

Decimal numbers can be rounded off to the nearest integer or whole number, tenths, hundredths, thousandths etc.

Round off to the nearest integer or whole number

The rounding rule is simple. Look at the digit in the tenths place (the first digit to the right of the decimal point).

If the digit in the tenths places less than 5 then round down.

If the digit in the tenths places greater than or equal to 5 then round up.

$$ e.g., 4.3\ round\ down\ to\ 4, 4.9\ round-up\ to\ 5,\ -2.9\ round-down\ to\ -3, 7.49\ or\ 7.49999\ or\ 7.4999999999\ and\ so\ on,\ all\ these\ will\ be\ round-down\ to\ 7. $$

Round off to the nearest tenth

$$ e.g., 845.324 $$ Look at the hundredths digit is 2 which is less than 5. Drop all the digits after hundredths place. 845.30 is the answer.

Round Off to the nearest hundredth

$$ e.g., 745.3286 $$ Look at the hundredths digit is 8 which is greater than 5.

So, Round-up, 745.33 is the answer.

Absolute Value or Modulus Value

Absolute value refers to the distance of a point from the origin on the number line. The absolute value of a number is always positive. Represented as, $ | x | $.

$ | x | = x\ or\ | -x | = x $.

$$ e.g., | -8 | = 8\ means\ that, \ the\ distance\ from\ origin\ is\ 8\ units. $$

In general, defined as,

Algebraic Numbers and Transcendental Numbers

An Algebraic numbers is any complex number including real numbers which are the roots of any non-zero polynomials in one variable with rational coefficients or integer coefficients. The number which is not algebraic called Transcendental Numbers.

All integers and rational numbers are Algebraic. Real and complex numbers are not algebraic as π and e are the Transcendental Numbers.

Algebraic Integers

An Algebraic Integer is an algebraic number that is the root of a polynomial with integer coefficients with leading coefficients 1 that is, a monic polynomial. $$ e.g., 2+3\sqrt2, 4-9i, 25+i\sqrt36, \frac {1}{2} {(1+2i)} $$

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a Ring.

Countable and Uncountable set of numbers

A set A is countable if it has a one-to-one correspondence with the natural number set $ \mathbb{N} $.

In other words, a set A is countable, if there exists an injective function f from A to $ \mathbb{N} $.

Also, we can say that a set A is countable if, cardinality (number of elements) of A is equal to the cardinality of the set of a natural number denoted as, $ |A| = |\mathbb{N}| $.

$$ |\mathbb{N}| = \aleph = alephnot $$

If sets are finite then it is called a finite countable set.

If sets are infinite then it is called countably infinite set. For example, $ \mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q} $ are the countably infinite set as their cardinality equal to the cardinality of $ \mathbb{N} $.

$ \mathbb{R}\ and\ \mathbb{C} $ are not countable sets as there doesn't exist one-to-one correspondence with the $ \mathbb{N} $. These are uncountable sets.

The cardinality of an Uncountable set are a continuum number, denoted as $ C $.

The cardinality of a power set of a finite set is $ 2^n $.

The cardinality of a power set of countably infinite set is $ 2^{ \aleph} = C $.

The cardinality of a power set of an uncountable set is $ 2^C > C $.

Roman Number

Roman Numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet.

The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced by Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some applications to this day.

Urdu Numbers

Numbers in Urdu from 1 to 10 are represented as,

1 = ۱, 2 = ۲, 3 = ۳, 4 = ۴, 5 = ۵, 6 = ٦, 7 = ٧, 8 = ٨, 9 = ٩, 10 = ١٠

Urdu Number Chart from 1 to 100

Hindi Numbers

Numbers in Hindi from 1 to 10 are represented as,

1 = १, 2 = २, 3 = ३, 4 = ४, 5 = ५, 6 = ६, 7 = ७, 8 = ८, 9 = ९, 10 = १०