Please enable JavaScript.
Coggle requires JavaScript to display documents.
Math IA: RIEMANN HYPOTHESIS, Fret placement and accurate frequencies in…
Math IA: RIEMANN HYPOTHESIS
About the Riemann Hypothesis
a conjecture about the Riemann zeta Function and its non-trivial zeros.
States that all non-trivial zeroes of the Riemann zeta function have a real part equal to 1/2
Directly linked to te distribution of prime numbers
One of the 7 Millennium Prize Problems
how to approach in ia
explain hypothesis in simple terms with examples
show how it connects to the prime number distribution
The Riemann Zeta Function
Formula zeta s equals summation from n equals 1 to infinity to infinity of one divided by n to the power of s.
S is a complex number (s= a+bi)
Converges when s > 1. But diverges at s = 1
Extended into the complex plane using analytic continuation
How to approach in ia
write and break down the formula
show convergence and divergence through calculations
Types of Zeros
trivial zeros; where at negative even integers like -2, -4, -6, -8,....
Non-trivial zeros; occur in the
critical strip
where the [0 < Re(s) < 1]
The Riemann Hypothesis states that all non-trivial zeros lie on the critical line where Re(s) = 1/2
how to approach in ia
define trivial and non trivial zeros with examples
graph the critical strip and locate zeros using an online computational tool [most likely]
Mathematical Application in real world
Prime Number Theorem connects the zeta function to the distribution of primes
The no. of primes less than a given number is closely related to the zeros of the zeta function.
Used in cryptography as prime distribution impact encryption security
Application in quantum physics and random matriz theory
how to approach in ia
research and explain the prime number theorem
show calculations that connect primes and the zeta functions
Attempts to proving up until now
trillions of non-trivial zeros have been tested numerically and lie on the critical line
Prtial results show some zeros lie on the line but no proof for ALL zeros
unsolved since 1859
how to appoach in ia
explain numerical verifications of zeros on the critical line
discuss major mathematical approaches attempted up until now
Why the hypothesis matters today
Prime numbers are fundamental to number theory and crytography
If proven true, i would confirm that primes follows a highly structured pattern
If proven false it would reshape entire fields of pure and applied mathematics
how to approach in ia
show real world applicationss in crytography and physics
discuss how solving or disproving it would impact math
Aim and Topic Question
Aim: The ultimate aim of this IA is to explore the relationship between the non-trivial zeros of the Riemann zeta function and the distribution of prime numbers, using mathematical analysis and numerical approximations.
By investigating the behavior of these zeros and testing computationally generated values, the IA aims to provide insight into why the Riemann Hypothesis—the conjecture that all non-trivial zeros lie on the critical line Re(s) = 1/2—is believed to be true.
Topic Question How does the distribution of non-trivial zeros of the Riemann zeta function relate to the distribution of prime numbers, and can numerical approximations provide insights into the Riemann Hypothesis?
Fret placement and accurate frequencies in guitars
Purpose
To understand how the positions of the frets on a guitar neck are decided so that the notes produced are accurate according to musical standards.
I want to explore the math behind fret placement and see how it relates to the sound the guitar makes.
Aim/RQ
How can we use mathematics to find the optimal placement of frets on a guitar neck that produces accurate musical notes?
BG Info
The equal temperament tuning system divides the octave into 12 equal steps, so each fret on the guitar raises the note by one equal musical step.
A guitar’s strings produce different notes by shortening the vibrating length of the string, which is done by pressing the string against frets.
Frets are placed to divide the string length into specific ratios so that each fret corresponds to a specific musical note.
The fret positions follow a mathematical pattern based on the
equal temperament tuning system
.
The standard musical scale divides an octave into 12 equal parts (semitones).
frequency and string length are inversely proportional
Math Involved
Functions: Modeling the fret position as a function of fret number.
Exponential and Logarithmic functions: The distance from the bridge to each fret follows a geometric sequence.
Inverse functions: To understand how fret position relates to note frequency.
Differentiation (Calculus): explore optimisation, like minimising the error between ideal and actual fret placements.
frets get closer and closer together as you move up the neck
plot a graph of
fret number
on the x-axis and
distance from nut
on the y-axis.
not a straight line because the change in distance between frets decreases.
If you differentiate this graph, it tells you how fast the fret distances are shrinking
Sequences and Series: the spacing pattern of frets.
Methodology
Measure Your Guitar’s Open String Length
Find the Frequency of the Open String
Measure the Distance of Each Fret from the Nut
Play Each Note and Measure Its Frequency (and organise)
Research How Guitar Frets Are Traditionally Placed
Analyze the Relationship (frequency and fret/distance)
plotting on graph and use Calculus Concepts to Understand the Changes
Compare Your Guitar’s Fret Placement with Theoretical Models
Reflect on Your Findings, and present conclusion of how fret placement and frequency are related (an if my findings are matching the traditional methods/research)
write up the report
materials list
Ruler or tape measure
Guitar
Frequency measuring app
Calculator or software (Excel/Google Sheets) to calculate values and plot graphs
Research resources to confirm the maths and theory behind fret placement
outcome / what to expect by the end
mathematical model linking fret position, string length, and frequency.
Real-world data to compare against the model.
Use calculus (derivatives) to analyze fret spacing and frequency changes.
reflectino of math in music (specifically an acoustic guitar)
Strong personal engagement through determining my own data of measurements.
Math IA: How to Approach the Investigations
Research and understanding
study the hypothesis and its properties
learn how complex numbers and analytical continuation work
explore past attempts to prove the hypothesis and their mathematical techniques
Mathematical understanding
Calculate andvisualise the zeta function in real and complez planes
use graphin tools to analyse zeros in the critical strip
test numerical approximation of non-trivial zeros
Application of mathematics
use prime no. theorem to connect primes and the zeta function
analyse how prime gaps are influenced by the distribution of zeros
apply HL maths like series summation and limits
How will i link thi to the IA criteriaa
A - Prseentation [4]
Clearly state the research question in the introduction and explain its significance
Structure the IA into introduction, body, and conclusion with clear section headings
Include logically connected sections with smooth transitions between ideas
Use labeled graphs, tables, and diagrams where necessary and place large data in appendices
Explain how technology such as graphing software or programming tools is used for calculations and visualizations
B- matheatical Communication[4]
Use correct mathematical notation consistently for the zeta function, summations, and complex numbers
Define key terms like zeta function, complex plane, and prime number theorem upon introduction
Use multiple mathematical representations, including equations, tables, graphs, and models
Clearly state rounding precision and approximation used in numerical calculations
Provide step-by-step explanations for calculations, ensuring all data presentations are well-described
C - Personal engagement[3]
Explain my personal interest in the Riemann Hypothesis and why I chose this topic
Formulate predictions about prime number patterns and test them using my own approach
Explore the topic from multiple perspectives, such as pure mathematics, cryptography, and number theory
D Reflection [3]
Assess the strengths and limitations of my investigation and mathematical approaches
Suggest improvements and further extensions to explore beyond the current research
Evaluate different methods used in my exploration and discuss their effectiveness
Relate my conclusions back to my original research question and objectives
E - use of mathematics [6]
Use mathematical concepts from the HL syllabus such as complex numbers, limits, and series summations
Extend beyond the syllabus by analyzing zeta function properties and connections to prime number distribution
Ensure all calculations are accurate, properly justified, and use appropriate approximations
Demonstrate sophistication by linking different mathematical areas such as calculus, number theory, and complex analysis
Avoid unnecessary complexity while maintaining a high level of rigor
Justify all mathematical claims with logical arguments or numerical verifications
Modeling the burning of a candle using exponential decay
about:
objective: Predict how the volume of a candle decreases over time based on a constant burn rate.
Useful for understanding burn time predictions, designing candles, and studying heat transfer.
The burning of a candle can be modeled as a continuous process where the volume decreases over time.
This situation fits the exponential decay model, where the rate of change depends on the current volume.
Possible RQ: How can the volume of a candle be modelled over time using differential equations, and how does the rate of decay change with different sizes and burn rates of candles?
BG info
Exponential Decay: The volume of the candle decreases over time at a rate proportional to the current volume.
Real-World Example: A candle burns, and its volume decreases based on how much wax is used up over time.
Key Variables: Volume of the candle (V), time (t), decay constant (k).
differential equation: A mathematical equation that relates a function with its derivatives (rate of change). It shows how a quantity changes over time, like volume, temperature, or speed.
exponential decay: A process where a quantity decreases at a rate proportional to its current value. In candle burning, the volume of the candle decreases over time, and the rate of decrease is proportional to the volume.