Ancient Mathematics VS Modern Mathematics

 

Ancient Mathematics VS Modern Mathematics

1.      

Numerical Systems

2.      Geometry

3.      Algebra

4.      Statistics

5.      Practical Applications

 

Numeral Systems

Ancient Mathematics	Modern Mathematics
1.     Roman Numerals 2.    Egyptian Numerals 3.    Babylonian Numerals	1.     Decimal System 2.     Binary System

 

1. Roman Numerals

Roman numerals use combinations of letters from the Latin alphabet to represent values. The basic symbols are:

- I = 1

- V = 5

- X = 10

- L = 50

- C = 100

- D = 500

- M = 1000

**Principle:** Roman numerals are formed by combining these letters and adding their values, with a few rules for subtraction (e.g., IV for 4).

 

**Example: Convert 1987 to Roman Numerals**

1. Break down the number into 1000, 900, 80, and 7.

2. Convert each part:

   - 1000 = M

- 900 = CM

   - 80 = LXXX

   - 7 = VII

3. Combine the parts: 1987 = MCMLXXXVII

 

**Another Example: Convert 2023 to Roman Numerals**

1. Break down the number into 2000, 20, and 3.

2. Convert each part:

   - 2000 = MM

   - 20 = XX

   - 3 = III

3. Combine the parts: 2023 = MMXXIII

 

2. Egyptian Numerals

Egyptian numerals used hieroglyphs to represent numbers. Different symbols represented different values, and numbers were formed by combining these symbols.

Symbols:

- 1 = single stroke (𓏺)

- 10 = heel bone (𓎆)

- 100 = coil of rope (𓍢)

- 1000 = lotus plant (𓆼)

- 10000 = finger (𓂭)

- 100000 = tadpole (𓆐)

- 1000000 = astonished man (𓁨)

**Example: Convert 2431 to Egyptian Numerals**

1. Break down the number into 2000, 400, 30, and 1.

2. Convert each part:

   - 2000 = two lotus plants (𓆼𓆼)

   - 400 = four coils of rope (𓍢𓍢𓍢𓍢)

   - 30 = three heel bones (𓎆𓎆𓎆)

   - 1 = one stroke (𓏺)

3. Combine the parts: 2431 = 𓆼𓆼𓍢𓍢𓍢𓍢𓎆𓎆𓎆𓏺

 

3. Babylonian Numerals

The Babylonian numeral system was a base-60 system, using a combination of symbols for 1 and 10.

 

Symbols:

- 1 = vertical wedge (𒐕)

- 10 = horizontal wedge (𒐖)

 

**Example: Convert 123 to Babylonian Numerals**

1. Break down the number using base-60:

   - 123 = 2 * 60 + 3

2. Convert each part:

   - 2 * 60 = two horizontal wedges (𒐖𒐖)

   - 3 = three vertical wedges (𒐕𒐕𒐕)

3. Combine the parts: 123 = 𒐖𒐖𒐕𒐕𒐕

 

**Another Example: Convert 70 to Babylonian Numerals**

1. Break down the number using base-60:

   - 70 = 1 * 60 + 10

2. Convert each part:

   - 1 * 60 = one horizontal wedge (𒐖)

   - 10 = one vertical wedge (𒐕)

3. Combine the parts: 70 = 𒐖𒐕

 

4. Modern Decimal System

The decimal (base-10) system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

 

Principle: Each digit's place value is a power of 10.

 

**Example: Convert 1987 to Base-10 Representation**

1. Expand the number using place values:

   - 1987 = 1 * 10^3 + 9 * 10^2 + 8 * 10^1 + 7 * 10^0

2. Calculate each part:

   - 1 * 1000 = 1000

   - 9 * 100 = 900

   - 8 * 10 = 80

   - 7 * 1 = 7

3. Sum the parts: 1000 + 900 + 80 + 7 = 1987

 

**Another Example: Convert 2023 to Base-10 Representation**

1. Expand the number using place values:

   - 2023 = 2 * 10^3 + 0 * 10^2 + 2 * 10^1 + 3 * 10^0

2. Calculate each part:

   - 2 * 1000 = 2000

   - 0 * 100 = 0

   - 2 * 10 = 20

   - 3 * 1 = 3

3. Sum the parts: 2000 + 0 + 20 + 3 = 2023

 

5. Binary System

The binary (base-2) system uses only two digits: 0 and 1.

 

**Principle:** Each digit's place value is a power of 2.

 

**Example: Convert 13 to Binary**

1. Find the powers of 2 that sum up to 13:

   - \( 13_{10} = 1 * 2^3 + 1 * 2^2 + 0 * 2^1 + 1 * 2^0 \)

2. Write the binary number:

   - \( 13_{10} = 1101_2 \)

 

**Another Example: Convert 7 to Binary**

1. Find the powers of 2 that sum up to 7:

   - \( 7_{10} = 1 * 2^2 + 1 * 2^1 + 1 * 2^0 \)

2. Write the binary number:

   - \( 7_{10} = 111_2 \)

Geometry Systems

Ancient Geometry	Modern Geometry
1.    Euclidean Geometry 2.    Constructions with Compass and Straightedge	1.      Coordinate Geometry 2.      Transformations

1. Euclidean Geometry

Principle: Euclidean geometry is based on the postulates and axioms described by the Greek mathematician Euclid. It deals with the properties and relationships of points, lines, surfaces, and solids in a flat, two-dimensional plane.

Key Concepts:

• Point: A location with no size or dimension.
• Line: A straight one-dimensional figure that extends infinitely in both directions.
• Plane: A flat two-dimensional surface that extends infinitely.

Example: Pythagorean Theorem Formula:

Simple Numerical Example:

• Triangle with sides 3 and 4: $$a = 3, \; b = 4 \implies c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
• So, the hypotenuse $c$ is 5.

Explanation: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

2. Constructions with Compass and Straight edge

Principle: Ancient Greeks used a compass and straightedge for constructing geometric figures.

Example: Constructing a Perpendicular Bisector

1. Draw a line segment AB.
2. With the compass, draw arcs from points AAA and BBB with the same radius, creating two intersection points above and below the segment.
3. Draw a line through the intersection points. This line is the perpendicular bisector of AB.

 

Modern Geometry

1.    Coordinate Geometry

Coordinate geometry (or analytic geometry) uses a coordinate system to describe geometric figures.

Key Concepts:

• Coordinate Plane: Consists of an x-axis (horizontal) and a y-axis (vertical).
• Points: Represented as (x, y) coordinates.
• Distance Formula: Calculates the distance between two points (x1,y1) and (x2,y2) :

Example: Distance Between Points

• Given points: A(1,2) and B(4,6)
• Calculate the distance:

Distance=(x2​−x1​)2+(y2​−y1​)2​=(4−1)2+(6−2)2​=32+42​=9+16​=25​=5

Visual Aid: Plot the points on a coordinate plane and show the distance calculation.

 

2.    Transformation

Transformations change the position or size of shapes on the coordinate plane.

Key Transformations:

• Translation: Slides a shape without rotating or flipping it.
• Reflection: Flips a shape over a line (mirror image).
• Rotation: Turns a shape around a point.
• Dilation: Resizes a shape proportionally.

Example: Reflecting a Point

• Given point $A(3, 2)$
  
• Reflect $A$ over the y-axis
• A′=(−3,2)

Visual Aid: Show the original point and its reflection on a coordinate plane.

 

Ancient Applications

1. Architecture: Egyptians used geometry for constructing pyramids.
2. Astronomy: Greeks used geometry to study stars and planets.

Modern Applications

1. Computer Graphics: Geometry helps create images and animations.
2. Engineering: Geometry is used in designing buildings and machines.

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