Mathematics
Mathematics is a prerequisite for success in all science, technology and engineering fields. Having taken advanced placement math will be your stepping stone to success for both entering college and preparing for the job market. I have included links to several online resources to help you prepare for a career in mathematics and the sciences and to have fun doing it.
Perhaps one of the best resources on the web for learning mathematics as well as a variety of other subjects is the Kahn Academy. Here you will find tutorials that teach math and science at all skill level from K - 12. If you are interested in math and science, I think you will find this site extremely useful. Kahn Academy tracks your progress and recommends activities that will improve your skills in arears where you may need additional work.
Another great resource for math students is Mathigon an intelligent tutoring system. You will find a variety of interactive activities and exercises that help you learn math using visual examples and a virtual personal tutor. Your progress is tracked, and you are provided lessons designed to improve your performance in areas that you may be having difficulty with. For kindergarten level students this Sesame Street site will be of interest. There are a series of videos that teach basic counting and arithmetic.
The Society for Industrial and Applied Mathematics(SIAM) is the premiere professional societies for mathematicians. This section of the SIAM website for Students K-12 provides you with a variety of tools and instructions which will broaden your perspective on mathematics. If you are truly passionate about mathematics, you should check out information on internships and careers in mathematics.
Numbering, Counting and Measuring
Numbering systems are fundamental to mathematics and is the basis for counting and which date back to ancient Sumerians and Babylonians in about 3000 BC. The Babylonians used a base 60 numbering system called Sexagesimal. This is credited with being the basis of our system of time, 60 seconds per minute and 60 minutes per hour; and geographic measurements using that are also in degree, minutes and seconds. The Babylonians are also credited with the first use of positional notation. In 3100 BC Egypt the decimal system (base 10) is first used. These are the Arabic numbers 0 through 9, with the number zero being discovered much later by the Indian astronomer and mathematician Aryabhatta in about 500 AD. Today the binary (base 2) numbering system, which consisting of the two numbers 0 and 1 (off and on) are used universally in digital computers. There are a variety of other numbering systems used within computer such as octal (base 8) and hexadecimal (base 16) that are used for internal representation of data and for addressing.
The Systems of Measurement in the United States is based on the Imperial system and is called United States Customary Units (USCS). In 1975 the US entered into a treaty starting the process of voluntary converting to the metric system (called mtricifacitation). This conversion is well underway in the military and in the scientific community and in most of the world.
The metric system is called the International System of Units (SI) and is used by almost all countries outside the US for measurements. The US system based on the Imperial system use feet to measure length, gallons to measure volume and pounds to measure weight. The metric system uses meters for length, liters for volume and kilograms for weight. A useful characteristic of the metric system is that units are powers of 10 which makes it easy to perform calculations. The imperial system units are not and essentially must be memorized. For example, a 1 mile has 5280 feet, but a Kilometer has 1000 meters a simple power of 10. Learning and becoming comfortable using the metric system is essential for a career in STEM.
Arithmetic deals with basic counting and manipulating quantities using the basic operators of addition, subtraction, multiplication and division. Number Theory is the study of the natural numbers or set of positive whole numbers. Cryptography makes use of number theory for securing data. Algebra, Differential and Integral Calculus, Trigonometry, and Differential Equations are all sub-disciplines of mathematics. Mathematical knowledge like that of other disciplines is cumulative and the techniques can be applied to a variety of phenomena.
Data Analysis
The availability of high-speed computers has made Data Analysis (DA) also knows as Big Data, a widely used and highly profitable discipline. Statisticians are leading the way in data analysis, and make use of statistics and probability to predict behavior based on a large numbers of observations. Data analytics is at the forefront of computational science today with areas of applications in almost every discipline and form the basis of a lot of artificial intelligence techniques. Because lots of the techniques used for data analysis can be automated, some will be and made available for use by non-specialist. Just as spreadsheets were once the domain of accountants but are now usable by the general public so will some DA techniques.
These systems will be accessible as cloud base applications where in addition to the algorithms needed for processing, curated data sets will be available. The Census Bureau and the Bureau of Labor Statistics are the two of leading government agencies where statisticians apply their trade analyzing large volumes of data.
GOOGLE (Now Alphabet) developed MapReduce to manage the large volume of data captured for their search engines by indexing the web. Hadoop is an Apache open-source software framework written in the Java programming language for processing and storage of very large distributed data sets on computer clusters built using commodity hardware. This software is designed to run on a variety of inexpensive hardware and different types of operating systems. A fundamental assumption is that hardware failures are common and should be automatically handled by the framework. Hadoop has become the de-facto tool used for Distributed computing.
Database technology is another area that makes use of mathematical disciplines such as logic and set theory. E.F. Codd while working at IBM established the basic theory supporting Relational Databases and The Structured Query Language (SQL). In addition to relational databases, hierarchical databases consisting of partially and fully inverted file structures are used. The first commercial implementation of SQL by Chamberlin and Raymond Boyce occurred in 1970 and they are credited as the developers for the SQL Standard. SQL is now both an International Standards Organization (ISO) and American National Standards Institute (ANSI) standard. Anyone interested in gaining hands on experience with SQL can use Microsoft Office Professional software that includes both Excel and Access, which is a relational database. Microsoft Office software also has its own object orient programming language called Visual Basic for Applications (VBA) that provides a good programming platform for writing macros and it is included with the MS Office software.
Cryptography
Cryptography is the basis of almost all of secure transactions today and provides a method for encoding data so that it can only be easily decoded and read by a recipient with a decryption key. Of course, brute force methods of decrypting data exist but these require extensive computations on high speed computers and depending on the strength of the encryption may take days or even weeks to ‘crack’.
Cryptography depends on factoring prime numbers and becomes extremely difficult for large primes. Prime numbers are numbers that are evenly divisible by itself and 1, such as 1, 3, 5, 7, 11, 13 and 17. As computers have become faster NIST recommended prime key lengths has increased to 2048 bits which is stated to be secure until around 2030. On the horizon is a new kind of computing called quantum computing which is capable of breaking current encryption quite easily. Once perfected quantum computing will present a threat to any of today’s encryption algorithms.
Digital or Crypto Currency
New digital currencies called Crypto Currencies are becoming quite popular and provides a method for exchanging digital cash between users. There are now over 20 of these digital cash systems. Bitcoin is one of the oldest and most widely known of these crypto currencies. Bitcoin is described as a currency built using mathematics. It is designed to provide a decentralized, digital electronic system capable of replacing cash and that will allow ‘money’ to be exchanged anonymously over the Internet. The Bitcoin digital currency depends on and underlying technology called Blockchain which acts a global distributed ledger that keeps track of who own what in the system.
Blockchain is a distributed ledger system used to keep track of transactions and to document who owns the digital currency and how much in the Bitcoin network. Blockchain is now being used independently of bitcoin in a number of applications. IBM offers a service based on blockchain to provide a variety of secure services
Wall Street Mathematics
Wall Street has become one of the biggest employers of mathematicians. Statisticians using Applied Mathematics develop models that allow them to not only predict some trends in market behavior, but to hedge their bets and protect against losses in both bull and bear markets. Listen to this interview with Ed Thorp who was one of the original Wall Street Quants as he discusses how he used his math skills to build a Hedge Fund and beat the casinos at blackjack. The math wizzes among you should not be sanguine however because Artificial Intelligence is being developed and will play a prominent role in future markets. If you like puzzles you will find a collection from the Wall Street Journal called Mo Math.
Math Fun Facts has several recreational math problems for those who enjoy a challenge. Having a good background in mathematics is often a determining factor in college acceptance for those interested in the sciences or engineering. Advance placement calculus is usually a prerequisite for most of these programs.