Geometry is a branch of mathematics dealing with questions of shape, size, the relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

In two dimensions there are 3 geometries: Euclidean, spherical, and hyperbolic.

What about in three dimensions, which corresponds to space we actually live in? It has been shown that in three dimensions there are eight possible geometries.

There is a 3-dimensional version of Euclidean geometry, a 3-dimensional version of spherical geometry and a 3-dimensional version of Hyperbolic geometry. There is also a geometry which is a combination of spherical and Euclidean, and a geometry which is a combination of hyperbolic and Euclidean.

What is Euclidean Geometry?

The study of plane and solid figures on the basis of axioms and theorems captured by the Greek mathematician Euclid. In its irregular outline, Euclidean geometry is the plane and solid geometry often taught in secondary schools.

 In Euclid’s great work, the Elements, the only tools employed for geometrical constructions were the ruler and the compass—a reduction retained in fundamental Euclidean geometry to this day.

What is Non-Euclidean Geometry?

Non-Euclidean geometry is a type of geometry. This geometry only uses some “postulates” (assumptions) that Euclidean geometry is built on. In normal geometry, parallel lines can never meet. In non-Euclidean geometry they can meet, either boundless many times (elliptic geometry), or never (hyperbolic geometry).

The non-Euclidean geometries grew along two different historical threads. The first thread started with the search to understand the movement of stars and planets in the obviously hemispherical sky.

An example of Non-Euclidean geometry can be seen by drawing lines on a round object, straight lines that are parallel at the equator can meet at the poles.

What Is Spherical Geometry?

From early times, people noticed that the shortest distance between two points on Earth were great circle routes. For example,  in geography

the Greek astronomer Ptolemy wrote:

“It has been demonstrated by mathematics that the surface of the land and water is in its totality a sphere…and that any plane which passes through the centre makes at its surface, that is, at the surface of the Earth and of the sky, great circles”

Great circles are the “straight lines” of spherical geometry. This is an outcome of the properties of a sphere, in which the shortest distances on the surface are great circle routes.

There are many ways of projecting a portion of a sphere, such as the surface of the Earth, onto a plane. These are known as maps or charts and they must certainly buckled distances and either area or angles.

For example, Euclid wrote about spherical geometry in his astronomical work circumstances. In addition to looking to the heavens, the earliest attempted to know the shape of the Earth and to use this understanding to solve problems in navigation over long distances. These activities are features of spherical geometry.

What Is Hyperbolic Geometry?

Hyperbolic geometry, also called Lobachevskian Geometry, a non-Euclidean geometry that turned down the soundness of Euclid’s fifth, the “parallel,” postulate. Clearly declared, this Euclidean postulate is: through a point not on a given line there is exactly one line parallel to the given line. In this geometry, through a point not on a given line there are at least two lines parallel to the specified line. The doctrine of hyperbolic geometry, however, confess the other four Euclidean postulates.

Although many of the theorems of hyperbolic geometry are uniform to those of Euclidean, others differ. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In hyperbolic geometry, two parallel lines are taken to converge in one direction and split in the other. In Euclidean, the sum of the angles in a triangle is equal to two right angles; in hyperbolic, the sum is less than two right angles. In Euclidean, polygons of different areas can be similar; and in hyperbolic, similar polygons of differing areas do not exist.

Applications

Geometry applied to many fields, including art, architecture, as well as to other branches of mathematics.

Art

Mathematics and art are related in a variation. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of panorama geometry.

Architecture

Mathematics and architecture are associated, since, as with other arts, architects use mathematics for multiple reasons. Apart from the mathematics required geometry when engineers start constructing buildings, architects  to create forms considered melodious, and thus to lay out buildings and their surroundings according to mathematical, decorative and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to keep down wind speeds around the bases of tall buildings.

An algebraic expression is an expression is made up of integer constants, variables, and algebraic operations (addition, subtraction, multiplication, division, and exponent. For example, 8x² + 4xy – z is an algebraic expression.

We are now leading you from basic arithmetic to algebra.

In algebra, letters are used to represent numbers.

Types of Algebraic expressions:

There are 3 main branches of algebraic expressions which include:

• Monomial Expression
• Binomial Expression
• Polynomial Expression

Monomial Expression:

An algebraic expression which has only one term is known as a monomial.

Examples of monomial expression include 2×3, 4xy, etc.

Binomial Expression:

A binomial expression is an algebraic expression which has two terms.

Example 1:

Since 8x and 2x are like terms then,

8x+2x =10x

Since 8 and 2 are the multiples of x, therefore 8 and 2 are the coefficients.

Example:2

Add 8x and 2

Here both the terms are unlike so, the sum will be, 8x+2

Polynomial Expression:

Polynomial expression having more than one terms with non-negative integral exponents of a variable. Example 2x²+5x+9-4x²+2x -6

Further branches of Expression:

Including monomial, binomial and polynomial branches of expressions, an algebraic expression can also be categorized into two additional branches which are:

• Numeric Expression
• Variable Expression

Numeric Expression:

In numeric expression, never include any variable. These only consist of numbers and operations. Some of the examples of numeric expressions are 20+5, 15÷3, etc.

Variable Expression:

A variable expression is an expression which occupies variables along with numbers and operation to define an expression. A few examples of a variable expression include 4x+y, 6ab + 25, etc.

An algebraic expression is a mathematical phrase that depends on numbers and/or variables. Though it cannot be solved because it does not contain an equals sign (=), it can be simplified. Algebraic equations, however, solve, which contain algebraic expressions separated by an equals sign.

Following are the fundamental steps to simplify an algebraic expression:

You need to simplify an algebraic expression before you evaluate it. This will make all your calculations much easier. Here are the main steps to follow for simplifying an algebraic expression:

1. Remove parentheses by multiplying factors
2. In terms with exponents. Use exponent rules to remove parentheses 
3. Merge like terms by adding coefficients
4. Merge the constants.

Let’s work through an example.

In this expression, we can use the distributive property to get rid of the sets of parentheses. When simplifying an expression, the first thing you need to look for clear parentheses. We multiply the factors to the terms inside the parentheses.

= 3(3+x)+4(2x+2)+(x²)²

We have to get rid of the parentheses in the term with the exponents by using the exponent rules. When a term with an exponent is raised to a power, we multiply the exponents, so (x²)² becomes x⁴.

= 9+3x+8x+8+x²

The next step in simplifying we have to simplify the like terms and combine them. Here 3x and 8x are like terms, because they have the same variable with common powers, the first power since the exponent is understood to be 1. We can combine these two terms to get 11x.

= 9+11x+8+x⁴

= 17+11x+x⁴

Finally, our expression is simplified. Keep in mind one more thing that algebraic expression is usually written in a certain order. We start with the terms that have the largest exponents to the constants. Using the commutative property of addition, we rearrange the terms and put this expression in correct order, like this.

= x⁴+11x+17

1. Difference between an algebraic expression and an algebraic equation:

An algebraic equation can be solved and does include a series of algebraic expressions separated by an equals sign.

An algebraic expression is a mathematical phrase that can contain numbers and/or variables. It does not contain an equals sign and cannot be solved.

Expression:

 3X + 9  or 4x-8

Equation:

4x +2=18 or 9x-2=14

2. Solve an algebraic equation with exponents.

If the equation has exponents, then you have to isolate the exponent on one side of the equation and then to solve by “removing” the exponent by finding the root of both the exponent and the constant on the other side. Let’s know how you do it:

• 2x² + 6 = 14

First, subtract 12 from both sides.

• 2x² + 6 -6 = 14 -6 = 0
• 2x²/2 = 8/2 = 0
• x² = 4

Solved by taking the square root of both sides, since that will turn x² into x.

√x² = √4

State both answers:x = 2, -2

Solve an Algebraic Expression with subtraction

Subtract (3x+2y) from (6x+7y)

= (6x+7y) – (3x+2y)

Now, we multiply negative sign to (3x + 2y):

= 6x+7y-3x-2y

Now, we arrange the expression as:

= 6x-3x+7y-2y

So, we got the answer 3x+5y

Example: 2 

Subtract (4X² – 2X + 6) from (2x² + 5x + 9)

= (2x² + 5x + 9) – (4X² – 2X + 6)

= 2x² + 5x + 9 – 4X² + 2X – 6

Now, arrange the expression:

= 2x² – 4X² + 5x + 2X + 9 – 6

= -2x²+7x+3

Solve an Algebraic Expression with Multiplication:

Example # 1

We have the algebraic expression (2x²)(3)

We multiply (2x²) by 3 

So we got the answer 6×2

Example # 2

Multiply (x + 2) by (3x – 5)

= x (3x – 5) + 2 (3x – 5)

= 3x² – 5x + 6x – 10

= 3x² + x – 10

Solve an Algebraic expression with division:

Example # 1

= -25×3 by 5x

= -25x³ / 5x= -5x

Example # 2

Divide (12x² + 3x + 9) by 3

=12x²+3x +9 / 3

=12x² + 3x³ + 9 / 3

=4x² + x + 3

Solve an algebraic expression with fractions. 

If you want to solve an algebraic expression with fractions, then you have to cross multiply the fractions, combining like terms, and then isolate the variable. Here’s an example of how you would do it:

• (x + 3)/6 = 2/3

First, cross multiply to remove the fraction. You have to multiply the numerator of one fraction by the denominator of the other.

• (x + 3) x 3 = 2 x 6 =
• 3x + 9 = 12

Now, combine like terms. Combine the constant terms, 9 and 12, by subtracting 9 from both sides.

• 3x + 9 – 9 = 12 – 9
• 3x = 3

Isolate the variable, x, by dividing both sides by 3 and you’ve got your answer.

• 3x / 3 = 3 / 3
• x =1

Solve an algebraic expression that contains an absolute value.

The absolute value is always positive.

For example, the absolute value of -7 (also known as |7|), is simply 7. To find the absolute value, you have to separate the absolute value and then solve for x twice, solving both for x with simply removed the absolute value, and for x when the terms on the other side of the equals sign have changed their signs from positive to negative and vice versa. Here’s how to do it.

• |6x +2| – 4 = 8 =0
• |6x +2| = 8 + 4 =0
• |6x +2| = 12 =0
• 6x + 2 = 12 =0
• 6x = 12
• x = 2

Now, solve again by flipping the sign of the term on the other side of the equation after you’ve isolated the absolute value:

• |6x +2| = 12 =0
• 6x + 2 = -12
• 6x = -12 -2
• 6x = -14
• 6x/2 = -14/2 =0
• x = -7

Now, just state both answers: x = 3, -7

As we have learned conversion from other systems to the decimal system in our article “Understanding the ‘inside out’ of number system: (Part 1 )”. Now we are going to understand the “ conversion between:

1. Binary and octal: In this conversion, we are going to focus on how can we convert binary to octal numbers and how to convert octal to binary digits.
2. Binary and hexadecimal: In this conversion, we will understand in detail how to convert binary to hexadecimal and how to convert hexadecimal to binary digits.

Conversion between binary and octal:

Each digit of an octal number is equivalent to three binary digits because any octal number ( 0 to 7) can be represented by three binary digits as shown in the given table.

Conversion from Binary to Octal number system:

The binary digits are split into two groups of three digits starting from right to left. Zeros can be added at the left end to make a complete group of three digits if required. Each group represents an equivalent octal digit.

Octal number system provides an appropriate way of converting large binary numbers into more dense and smaller groups.

There are multiple ways to convert a binary number into an octal number. You can transform using direct methods or indirect methods. First, you need to convert a binary into other base systems. 

Since there are only 8 digits (from 0 to 7) in the octal number system, so we can represent any digit of octal number system using only 3 bit as following below.

Octal Symbol	Binary equivalent
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

So, if you make each group of 3-bit binary input number, then replace each group of binary number from its equivalent octal digits. That will be the octal number of a given number. Note that you can add any number of 0’s in leftmost bit (or in most significant bit) for integer part and add any number of 0’s in rightmost bit (or in the least significant bit) for fraction part for completing the group of 3 bit, this does not change the value of input binary number.

Here we have the binary numbers which we are going to convert to octal numbers: (10111010001111)

Binary:    010  111  010  001  111

Octal:      2 7   2 1 7

So, we got the octal numbers (27217).

Conversion from Octal to Binary number system:

An octal number can be easily converted to a binary number by replacing each octal digit with the corresponding three binary digits.

There are various direct or indirect methods to convert an octal number into a binary number. In an indirect method, you need to convert an octal number into other number systems (e.g., decimal or hexadecimal), then you can convert into binary number by converting each digit into binary numbers from the hexadecimal system and using conversion system from decimal to binary number.

Here we have an octal number which we are going to convert in binary digits: ( 42153 )₈

Octal: 4   2   1   5 3

Binary: 100 010 101  011

So, we got the answer ( 100010001101011 )₂

Conversion between Binary and Hexadecimal:

Like the octal system, the hexadecimal system can be easily derived from the binary system. Each Hexadecimal digits (0-9 and A- F) is equivalent to four binary digits as shown in the table given under:

Decimal	Hex	Binary
0	0	0
1	1	1
2	2	10
3	3	11
4	4	100
5	5	101
6	6	110
7	7	111
8	8	1000
9	9	1001
10	A	1010
11	B	1011
12	C	1100
13	D	1101
14	E	1110
15	F	1111
16	10	10000
17	11	10001
18	12	10010
19	13	10011
20	14	10100
…	…	…
25	19	11001
26	1A	11010
27	1B	11011
28	1C	11100
29	1D	11101
30	1E	11110
31	1F	11111
32	20	100000
33	21	100001
34	22	100010

Binary to hexadecimal Conversion:

Conversion between binary and hexadecimal is simply accomplished by grouping the binary numbers into four digits replacing each group with a hexadecimal equivalent digit.

Hexadecimal number system provides an appropriate way of converting large binary numbers into more compact and smaller groups. There are various ways to convert a binary number into a hexadecimal number. You can convert using direct methods or indirect methods. First, you need to convert a binary into other base systems (e.g., into a decimal, or into octal). Then you need to convert it into a hexadecimal number.

Since there are only 16 digits (from 0 to 7 and A to F) in the hexadecimal number system, so we can represent any digit of hexadecimal number system using only 4 bit as following below.

Hexa	0	1	2	3	4	5	6	7
Binary	0000	0001	0010	0011	0100	0101	0110	0111

Hexa	8	9	A=10	B=11	C=12	D=13	E=14	F=15
Binary	1000	1001	1010	1100	1101	1110	1111

Here we have the binary numbers to convert in hexadecimal number system (111010000110111000)₂

In this conversion we make the pair of four digits, so we have to add two leading zeros to make perfect pairs like:

Binary numbers :  0011  1010  0001  1011  1000

Hexadecimal:        3 A 1 B 8

We got the answer (3A1B8)₁₆. So, if you make each group of 4-bit binary input number, then replace each group of binary number from its equivalent hexadecimal digits. That will be a hexadecimal number of given numbers. Note that you can add any number of 0’s in leftmost bit (or in most significant bit) for integer part and add any number of 0’s in rightmost bit (or in the least significant bit) for fraction part for completing the group of 4 bit, this does not change the value of input binary number.

Conversion from Hexadecimal to Binary number system:

Conversely, a hexadecimal number can be converted into binary by replacing each digit by the equivalent four binary digits.

Here we have the hexadecimal number system to change in binary digits: (5DC7).

Hexadecimal number: 5 D C 7 

Binary Numbers: 0101 1101 1100 0111

So, we got the answer 0101110111000111

This method of using the binary equivalent of digits of numbers in Interbase conversion is called “Direct Method”.

Number System Conversions:

The number system, which we have discussed in the previous article named “let’s operate the number system”. In which we talk about the relationship of (Decimal, Binary, Octal, and Hexadecimal). We can convert a number of a particular system to its equivalent number in other systems.

Conversion from Decimal system to other systems:

The method of converting from decimal to another system is by successive division by the base of the objective system. The remainders of these divisions are arranged in reverse order.

Decimal to Binary Conversions:

To convert from decimal to binary, divide the decimal number by 2, which is the base of the binary system.

Here we have decimal 57 and going to convert into its binary equivalent.

The binary number is obtained by arranging the remainders from bottom to top. Thus, the equivalent binary number is : (111001)

Decimal to Octal Conversion:

To convert a decimal number to its octal equivalent, divide the decimal number successively by 8 in a similar manner as was done with decimal to binary conversions. The remainders are read in reverse order to obtain the required octal number.

Here we have decimal digits 1769 and going to convert into its octal to equivalent:

Thus, we got octal number which is (3351).

Decimal to Hexadecimal conversions:

Decimal to Hexadecimal conversions can be carried out using the same technique as shown above binary and octal conversion with the division being done by 16.

Here we have decimal digits 6958 and going to convert into its hexadecimal equivalent.

Here, you need to know the in hexadecimal number system that alphabet

• A denotes 10
• B denotes 11
• C denotes 12
• D denotes 13
• E denotes 14
• F denotes 15
• G denotes 16

Therefore the equivalent of this decimal number in hexadecimal is (1B2E).

Let’s learn Conversion from other systems to the Decimal system:

Converting from other systems to decimal system involves multiplication.

Multiply each digit of the given number by its positional weight. The base of the given number multiplies each positional weight.

Final value is obtained by adding each positional value.

We can perform this process through these steps.

Step 1: Multiply each digit of the given  number by its base raised to positional power.

Step 2: Add all results of multiplication of each position.

Binary to Decimal Conversion:

We convert binary to decimal by finding the decimal equivalent of the binary array of digits (10011)₂ and expanding the binary digits into a series with a base of 2 giving an equivalent of (19)₁₀ in decimal or denary.

Note that in number conversion systems “subscripts” are used to indicate the relevant base numbering system, 1001₂ = 9₁₀. If no subscript is used after a number, then it is generally assumed to be decimal.

Let’s convert decimal number to binary (10011)²:

= 1 x 2⁴ + 0 x 2³ + 0 x 2² + 1 x 2¹ + 1 + 2⁰

= 1 x 16 + 0 x 8 + 0 x 4 + 1 x 2 + 1 x 1

= 16 + 0 + 0 + 2 + 1

= ( 19 )₁₀

The equivalent decimal number is 19 for binary number 10011.

When converting from Binary to Decimal or even from Decimal to Binary, we need to be careful that we do not mix up the two sets of numbers. 

For example, if we write the digits 10 on the page it could mean the number “ten” if we assume it to be a decimal number, or it could equally be a “1” and a “0” together in binary, which is equal to the number two in the weighted decimal format from above. for example, if we were using a binary number string we would add the subscript “2” to denote a base-2 number so the number would be written as 10₂. Likewise, if it was a standard decimal number we would add the subscript “10” to denote a base-10 number so the number would be written as 10₁₀.

Octal to Decimal Conversion:

Octal numbers, therefore, have a range of just “8” digits, (0, 1, 2, 3, 4, 5, 6, 7) making them a Base-8 numbering system and therefore, q is equal to “8”.

Convert the octal number (546)₈ to its decimal number equivalent, (base-8 to base-10).

Step 1: = 5 x 8² + 4 x 8¹ + 6 x 8⁰

= 5 x 64 + 4 x 8 + 6 x 1

Step 2: = 320 + 32 + 6

= (358)₁₀

Converting octal to decimal shows that (546)₈ in its Octal form is equivalent to (358)₁₀ in its Decimal form.

While Octal is another type of digital numbering system, it is little used these days instead of the more commonly used Hexadecimal Numbering System is used as it is more flexible.

Hexadecimal to Decimal Conversions:

The “Hexadecimal” or simply “Hex” numbering system uses the Base of 16 system and is a popular choice for representing long binary values because their format is quite compact and much easier to understand.

Then in the Hexadecimal Numbering System, we use the numbers from 0 to 9 and the capital letters A to F to represent it’s Binary or Decimal number equivalent, starting with the least significant digit at the right-hand side.

Convert the following Hexadecimal number (3AC8)₁₆ into Decimal or Denary equivalent using subscripts to identify each numbering system.

Step 1: = 3 x 16³ + A x 16² + C x 16¹ + 8 x 16⁰

= 3 x 4096 + 10 x 256 + 12 x 16 + 8 x 1

Step 2 = 12288 + 2560 + 192 + 8

= (15048)₁₀

Then, the Decimal number of 15048 can be represented as 3AC8 in Hexadecimal.

The number system is the system of counting and calculations. Number system is formed of  characters called “Digits”. Each number is made up of these characters. The number of digits a system uses is called its base or Radix. For example the number system we use in our habitual life is called Decimal System.

There are four branches of the number system.

1) Decimal    2) Binary   3) Octal   4) Hexadecimal

Decimal system:

The number system which we use in our habitual life is based on ten digits. 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any numbers can be expressed as a combination of these ten digits. For example, 25 is the combination of two digits 2 and 5, 125 is the combination of three digits and so on. In this system, every digit of a number is a multiple of some power of 10.

Any decimal integer can be expressed as the sum of each of its digits times power of 10.

Since this number system has ten digits, this system is also called “Decimal Number System” Or “ Denary System” or  “ Base ten system”.

For example, 8326 can be expressed as:

= 8 x 10³ + 2 x 10² + 3 x 10¹ + 6 x 10⁰

= 8 x 1000 + 2 x 100 + 3 x 10 + 6 x 1

= 8000 + 200 + 30 + 6

Position Weight

1st 10⁰= 1
2nd 10¹= 10
3rd 10²= 100
4th 10³= 1000
5th 10⁴ =10000

Binary System:

The binary number system is ideal for internal networking of electronic computers.

The Binary System is especially useful in the construction of computers because the digits can be represented by switches in “ON” or “OFF” positions.Computer circuitry represents data in a pattern of ON or OFF state of electric current. Because there are only two states, they are represented by 1 (for ON) and 0( for OFF). Binary system is based on two fundamental digits 0 and 1 to represent its numeric values there the base of this system is two.So, it is also called “Base two number system”.

The positional weight for the binary system are based on the powers of two.To visualized the positional values of the binary system the weights for the first five positions are given as under:

1st 2⁰= 1
2nd 2¹= 2
3rd 2²= 4
4th 2³= 8
5th 2⁴ =16

Here, we have an example 100112 can be represented in decimal as:

=1 x 2⁴ + 0 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2⁰

= 1 x 16 + 0 x 8 + 0 * 4 + 1 x 2 + 1 x 1

= 16 + 0 + 0 + 2 + 1

= 19

The octal system:

Octal is latin word for eight (8). Octal has eight fundamental digits as, 0, 1, 2, 3, 4, 5, 6, and 7. So, the base of this system is eight. The weights of digits position are successive powers of eight as under :

Position Weight

1st 8⁰= 1
2nd 8¹= 8
3rd 8²= 64
4th 8³= 512
5th 8⁴ = 4096

Here, we have an example 352₈ can be expressed in decimal as:

= 3 x 8 ² + 5 x 8 ¹ + 2 x 8 ⁰

= 3 x 64 + 5 x 8 + 2 x 1

= 192 +40 + 2

= 234 ₁₀

Hexadecimal system:

Hexadecimal means 16, therefore this is a base -16 number system.It has 16 basic digits. The first ten digits are same as decimal (0 to 9) and the rest of the six digits are the first six letters of english alphabet (a, b, c, d, e, f).The letters A through F represent the decimal numbers 10 through 15.This system is often used in programming as a shortcut to the binary number system. The weight used in the hexadecimal system are the successive power of 16.

The weight of the first five digits positions are as under.

Position Weight

1st 16⁰ = 1
2nd 16¹ = 16
3rd 16² = 256
4th 16³ = 4096
5th 16⁴ = 65536

Here, we have an example 3AC8 can be expressed in decimal as:

= 3 x 16 ³ + A x 16 ² + C x 16 ¹ + 8 x 16 ⁰

= 3 x 4096 + 10 x 256 + 12 x 16 + 8 x 1

= 12288 + 2560 + 192 + 8

= 15048 ₁₀