## 1. Language & Concepts (sabrina)

### 1.1. Maths as a universal language

1.1.1. Symbolism and early counting in maths

1.1.1.1. Less effiecient methods for assigning symbols to numbers

1.1.1.1.1. Tallying

1.1.1.1.2. Subtracting and adding from a base value

1.1.1.2. More efficient method for assigning symbols to numbers

1.1.1.2.1. Positional system

1.1.2. Vocab of maths

1.1.2.1. Like any language has their verbs, adjectives and vowels, different symbols in math can be categorized to have different functions

1.1.2.1.1. Terms

1.1.2.1.2. Expressions

1.1.2.1.3. Equations

1.1.2.1.4. Operations

1.1.2.1.5. Variable

1.1.2.1.6. Constant

1.1.3. How do different languages interpret Maths

1.1.3.1. Since Maths is a universal language, can there be disputes/confusions when it is then translated again into the spoken languages we use everyday?

1.1.3.1.1. Eg "What is a billion?"

1.1.4. Using language to make sense of something infinite or intangible

1.1.4.1. The symbol infinity has vast, intangible connotations, however we can still understand it.

1.1.4.1.1. https://www.ancient-symbols.com/images/wp-image-library/fullsize/infinity.jpg

1.1.4.2. Are some concepts in Maths too transcendental to be explained in terms of language?

1.1.4.2.1. eg, Cantors conjecture of the continuum hypothesis was proved to be false AND proven to be true by later mathematicians.

1.1.4.3. Set theory

1.1.4.3.1. https://resourcekraft.com/wp-content/uploads/2017/05/4.png?x66911

1.1.4.3.2. The use of symbols to make concrete numbers that are not concrete.

1.1.4.3.3. For example, the symbol π represents a decimal which does not end - yet we can come to that understanding w/o even knowing the actual number

1.1.4.3.4. Maths also uses fractions to denote rational numbers. Hippasus claimed that all numbers that "we can see/make sense of" can be expressed as a ratio/fraction

1.1.4.3.5. Complex numbers are numbers that (based on widely accepted mathematical conjectures) cannot exist. However, with the use of i, we can now use them in problem solving.

1.1.5. Use of metaphors to paint a visual picture

1.1.5.1. eg, Rational numbers are stars in the galaxy, and irrational numbers are the darkness

## 2. Personal Knowledge (Prarthana)

### 2.1. What is the nature of the contribution of individuals you know personally to this area, in terms of your experience?

2.1.1. Pythagoras

2.1.1.1. Sir Isaac Newton,

2.1.1.1.1. Both Sir pythagoras and Isaac Newton are examples of how mathematics and help expand the realm of possibility

2.1.1.1.2. Sir Isaac Newton is a great example of someone who used their personal knowledge

2.1.1.2. Sir Pythagoras used what he saw (patterns with triangles) to help develop his theorem

### 2.2. What responsibilities rest upon YOU by virtue of YOUR knowledge in this area?

2.2.1. We have responsibility to explore the realm of possibility

2.2.1.1. Personal Knowledge is pivotal to understanding math

2.2.1.1.1. Whether it be finding out how to pay for groceries quickly and effectively; math surrounds us

2.2.1.1.2. We have a responsibility towards the advancement of science

### 2.3. What are the implications of this area of knowledge in terms of YOUR individual perspective?

2.3.1. Implications can vary from individual to individual

2.3.1.1. For some it may be the advancement of science and the betterment of the future

2.3.1.1.1. For others it may mean deep study that will yield no benefits

### 2.4. What assumptions underlie YOUR own approach to this knowledge?

2.4.1. How to assume margin of error?

2.4.1.1. How do we know if our margins for error have extremely negative implications?

2.4.1.1.1. Rockets:

2.4.1.1.2. Nuclear Bombs :

## 3. Scope & Application (Mo)

### 3.1. Social functions of mathematics

3.1.1. Importance of mathematics

3.1.1.1. deal with quantities and measurements

3.1.1.1.1. Calculating money (banking) and foods (baking)

3.1.1.1.2. use of numbers and statistics in an argument or as evidence are more convincing

3.1.1.2. way for interpreting the world

3.1.1.2.1. analysing survey and data

3.1.2. Aims of mathematics

3.1.2.1. Solving real life problems

3.1.2.1.1. algorithms can perform calculation, data processing, and automated reasoning tasks to explain difficult problems.

3.1.2.2. Discover theories to explain patterns and the relationships between the pattern, the nature and the universe

3.1.2.2.1. Axioms are not ‘self-evident’ truths, but assumptions premises, definitions, or givens at the base of a mathematical system

3.1.2.2.2. Theorems are concepts and “rules” that arise when one applies the axioms of a formal system onto examples. They arise through a deductive reasoning.

3.1.2.2.3. Conjectures are statements that are believed to be true, but has not been proven.

### 3.2. Different forms of mathematics

3.2.1. Pure mathematics

3.2.1.1. concerned with number, quantity, and space, either as abstract ideas with no intention of use.

3.2.1.1.1. Calculus

3.2.1.1.2. Algebra

3.2.1.1.3. Trigonometry

3.2.1.1.4. Statistics

3.2.1.1.5. Probability

3.2.2. Applied mathematics

3.2.2.1. applied to physics, engineering, and other subjects

3.2.2.1.1. Algorithm is a process with unambiguous steps that has a beginning and an end, and does something useful to achieve predictability.

### 3.3. Mathematics influenced by culture and society.

3.3.1. Mathematics are interpreted in the same way around the world but a small part of it may be influenced

3.3.1.1. Cultural influence

3.3.1.1.1. Symbolic and religious properties of geometric figures

3.3.1.2. Shared knowledge

3.3.1.2.1. Difference of contexts

### 3.4. What is mathematics

3.4.1. Mathematical formalism

3.4.1.1. an abstract game played according to invented rules

3.4.1.1.1. Mathematic is captured in a formal system, or common enough within something formalisable with claims.

3.4.2. Mathematical humanism

3.4.2.1. the nature of Mathematics is artificial and abstract

3.4.2.1.1. explore the human side of mathematical thought and encourage the imagination to discover this beauty of mathematics.

3.4.3. Mathematical platonism

3.4.3.1. Mathematics seen as a code to understand the world around us.

3.4.3.1.1. Platonists believe mathematics is discovered; it exists in a reality we cannot fully comprehend

## 4. Methodology (max)

### 4.1. Proof by substitution

4.1.1. Assigning values and variables to other variables

4.1.1.1. If a cat is an animal and an animal follows the principles of MRS NERG a cat also follows those principles.

### 4.2. Proof by contradiction

4.2.1. The proof that something is untrue, through assumption

### 4.3. Theorems

4.3.1. Definition: "A true statement based on other accepted truths"

4.3.1.1. Pythagoras' Theorem

4.3.1.2. Pigeon-hole Theorem

4.3.1.2.1. An analogy relating to pigeon holes. If there are more pigeons than holes, some holes will have at least 2 pigeons.

## 5. Historical Development (Hubert)

### 5.1. Paradigm Shifts

5.1.1. TS Kuhn

5.1.1.1. Internal Viewpoint

5.1.1.2. External Viewpoint

### 5.2. History

5.2.1. Development of Zero

5.2.1.1. zero was thought of as a placeholder in positional systems

5.2.2. Axiomatization

5.2.2.1. Euclid proving which truths are actually true

5.2.3. Fibonacci, F.Sequence and Phi

5.2.3.1. brought Arabic numerals to Europe

5.2.3.2. PHI can found in the Equiangular Spiral, also called Bernoulli's Logarithmic Spiral

5.2.3.3. Fibonacci observes that when one divides a number with the one before it

5.2.4. Al-Khwarizmi & The Invention of Algebra

5.2.4.1. solves equations related to shapes

5.2.4.2. one of the first methods figuring out which numbers the letters actually represent

5.2.4.3. central part of computer programming à actual use of words

5.2.5. Rene Descartes and Analytic Geometry

5.2.5.1. combination of algebra and geometry

5.2.5.2. essential tool for science and engineering

5.2.6. Fermat’s Last Theorem

5.2.6.1. Solved in 1994

5.2.6.2. 150 pages of proof

5.2.7. Pascal's Triangle

5.2.7.1. exploration of number theory and triangular numbers

5.2.7.2. Enables the expansion of binomials

5.2.8. Counting in Binary and Computers

5.2.8.1. using bits to count; bits are represented in Arabic numerals 0 and 1

5.2.8.2. first investigated by Gottfried Leibnitz

5.2.8.3. 1 standing for God and 0 for Void

5.2.9. Babbage Difference Engine

5.2.9.1. developed this first automatic machine producing more accurate tables using 2 orders of difference