# The Most Useful Curve in Mathematics [Logarithms]

TLDRThis video explores the significance of logarithms as the most useful mathematical curve, highlighting their historical impact on simplifying complex calculations. From the early navigational challenges to the advent of electronic calculators, logarithms have revolutionized computation. The video delves into the invention by John Napier, the refinements by Henry Briggs, and the practical applications of logarithms in slide rules and modern AI, emphasizing their enduring relevance in various scientific fields.

### Takeaways

- ๐ The logarithm is considered the most useful curve in mathematics due to its ability to simplify complex problems.
- ๐ Logarithms transform multiplication and division problems into addition and subtraction, respectively, by mapping mathematical operations from one axis to another.
- ๐ Logarithms were revolutionary for navigation and mathematical computation before the advent of electronic calculators, simplifying tasks like calculating angles for navigation.
- ๐ Scottish mathematician John Napier introduced logarithms in 1614 with his book, which contained detailed tables of logarithms to the eighth decimal place.
- ๐ข Logarithms, initially called 'artificial numbers' by Napier, were later named 'logarithms' meaning 'ratio numbers', reflecting their computation method.
- ๐ Henry Briggs and John Napier collaborated to simplify logarithms by basing them on the more user-friendly base of 10, making them easier to compute and use.
- ๐ Briggs computed logarithms to high precision using a method of repeated square roots to zoom in on the curve and linearize it for values close to one.
- ๐ The slide rule was a mechanical device that utilized logarithms for rapid calculations, embodying the principle that multiplication and division could be converted into addition and subtraction.
- ๐ Logarithms are crucial in modern applications such as AI and data visualization, playing a key role in the loss functions of neural networks.
- ๐ซ Despite the decline in use of logarithm tables and slide rules, the underlying concept of logarithms remains essential in various scientific and mathematical fields.
- ๐ก Logarithms are fundamental to understanding growth, scaling, and proportional relationships in a wide range of mathematical and scientific contexts.

### Q & A

### What unique property of logarithms allows complex problems to be transformed into simpler ones?

-Logarithms have the unique property of mapping mathematical operations from one axis to another, effectively transforming multiplication problems into addition problems and division problems into subtraction problems on a different axis.

### How do logarithms simplify the process of multiplication and division?

-Logarithms simplify multiplication by converting it into addition of exponents, and division by converting it into subtraction of exponents. This is because the logarithm of a product is the sum of the logarithms, and the logarithm of a quotient is the difference of the logarithms.

### Why were logarithms so revolutionary for navigation in the 1600s?

-Logarithms revolutionized navigation by simplifying the complex calculations involved in determining angles for course settings. Instead of performing long division, navigators could use logarithm tables to quickly find the angle by converting division into subtraction.

### Who is credited with the invention of logarithms, and how did they initially call their invention?

-Scottish mathematician John Napier is credited with the invention of logarithms. Initially, he called his invention 'artificial numbers,' but ultimately decided to call them 'logarithms,' meaning ratio numbers, due to how they are computed.

### How did the introduction of logarithms by John Napier change the way mathematicians computed values?

-The introduction of logarithms by John Napier allowed mathematicians to perform complex computations much more quickly and with fewer errors. Instead of using large numbers and cumbersome division, they could use logarithm tables to simplify calculations.

### What was the significance of the collaboration between John Napier and Henry Briggs?

-The collaboration between John Napier and Henry Briggs led to the reconceptualization of logarithms with a base of 10, making them more practical and easier to use. This collaboration also resulted in the creation of more accurate and comprehensive logarithm tables.

### How did the slide rule work, and how was it related to logarithms?

-The slide rule worked by using logarithmically spaced tick marks that allowed for the direct conversion of multiplication and division into addition and subtraction, respectively. This was achieved by aligning the scales and reading the result from the position of the logarithmic tick marks.

### What was the impact of the electronic calculator on the use of slide rules and logarithm tables?

-The advent of the electronic calculator, such as the HP-35, rendered slide rules and logarithm tables obsolete for everyday use. The calculators could perform the same functions as slide rules and tables but with greater speed, accuracy, and convenience.

### Why are logarithms still important today despite the decline in the use of logarithm tables and slide rules?

-Logarithms remain important today because they play a crucial role in various fields such as science, mathematics, engineering, and machine learning. They are used in the loss functions of neural networks and are key to understanding data visualization and large language models.

### How did the method of 'zooming in' on the logarithm curve by Henry Briggs contribute to the development of logarithms?

-Henry Briggs' method of 'zooming in' on the logarithm curve allowed for the computation of logarithms for values close to one by linearizing the curve in that region. This approach greatly simplified the calculation of logarithms and contributed to the precision of the logarithm tables.

### What is the significance of the logarithmic curve in the context of the video script?

-The logarithmic curve is significant as it is the basis for logarithm tables and slide rules, which were the most powerful computing instruments for over three centuries. The curve's properties allowed for the simplification of complex mathematical operations, making it the most useful curve in mathematics.

### Outlines

### ๐ The Power of Logarithms in Simplifying Mathematics

This paragraph introduces the concept of logarithms as a transformative tool in mathematics, which simplifies complex problems by mapping mathematical operations from multiplication on the x-axis to addition on the y-axis. It uses the historical context of navigation to illustrate how logarithms were used to simplify long division and multiplication in the 16th century, before the advent of electronic calculators. The paragraph also discusses the invention of logarithms by John Napier and how they were used to revolutionize computation, including the use of logarithmic tables to convert division into subtraction and the transformation of other mathematical operations.

### ๐ The Evolution of Logarithmic Computation

This section delves into the historical development of logarithmic computation, starting with Napier's initial logarithmic tables and the subsequent collaboration with Henry Briggs to simplify and improve them. It explains how Briggs computed logarithms using a base of 10, making them more practical for everyday use. The paragraph also describes the painstaking process Briggs undertook to compute the logarithms of numbers between 1 and 20,000 with high precision, and how these tables became the backbone of human calculation for centuries, influencing figures such as Isaac Newton and Albert Einstein.

### ๐ The Mathematical Foundation of Logarithms

This paragraph explains the mathematical principles behind logarithms, focusing on how they transform multiplication into addition and division into subtraction. It describes the geometric sequence and the logarithmic scale, illustrating how logarithms are used to count the number of times a base number is multiplied to reach a given number. The explanation includes the construction of a miniature version of Napier's table and the use of logarithms to simplify multiplication and division, highlighting the logarithmic identities that underpin these operations.

### ๐ The Practical Application of Logarithms in Slide Rules

The paragraph discusses the practical application of logarithms in the form of slide rules, which were used for computation for over 350 years until the advent of electronic calculators. It explains how the logarithmic spacing of tick marks on slide rules allowed for rapid and precise calculations, including multiplication, division, and finding square roots. The paragraph also describes how the slide rule embodies the logarithmic identities, converting physical movement into mathematical operations.

### ๐ฑ The End of an Era: Logarithms and the Advent of Electronic Calculators

This final paragraph marks the transition from slide rules and logarithmic tables to electronic calculators, specifically the HP-35, which effectively replaced the slide rule. It explains how the engineers behind the HP-35 used Briggs' method to linearize logarithms for values close to one, and how this development led to the decline of slide rule usage. The paragraph also reflects on the enduring importance of logarithms in modern science, mathematics, and engineering, and how they are integral to AI and data visualization.

### Mindmap

### Keywords

### ๐กLogarithms

### ๐กMathematical Wormhole

### ๐กJohn Napier

### ๐กSlide Rule

### ๐กHenry Briggs

### ๐กBase of Logarithms

### ๐กGeometric Sequence

### ๐กExponents

### ๐กNavigation Problem

### ๐กElectronic Calculator

### ๐กBriggs and Black

### Highlights

Logarithms transform complex mathematical problems into simple ones by mapping operations from one axis to another.

Multiplication on the x-axis corresponds to addition on the y-axis in logarithmic scale.

Division problems are transformed into subtraction problems using logarithms.

Historically, logarithms simplified navigational calculations by converting division into subtraction.

John Napier's invention of logarithms revolutionized mathematical computations.

Napier's logarithm values are effectively the Y-values for points on a logarithmic curve.

Logarithms can be used to quickly solve problems that were previously time-consuming and error-prone.

The East India Company adopted Napier's logarithms for its sailors due to their efficiency.

Logarithms transform powers into multiplication and roots into division.

Edmund Gunter and William Oughtred invented the slide rule based on logarithmic principles.

Logarithmic scale tick marks allow the slide rule to perform computations efficiently.

Napier's logarithms are computed based on the motion of two points in a geometric sequence.

Logarithms count the number of times a base number is multiplied to reach a given number.

Henry Briggs and John Napier collaborated to recreate logarithms in base 10 for easier use.

Briggs computed logarithms using a method of zooming in on the curve for high precision.

The slide rule was replaced by electronic calculators in the 1970s, ending its 350-year reign as a primary computation tool.

Logarithms continue to be crucial in modern fields such as AI and data visualization.

Brilliant.org offers courses on logarithms and their applications in various mathematical fields.