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What are the areas of Mathematics ?

1. Algebra

Algebra is one of the most important areas of mathematics and one usually taught in its most basic form, to elementary school children. Despite this, it can get complex. It deals with properties and their operations, and also their structures to solve often quite simple problems.

Many people use it as a way of solving simple or slightly more complex arithmetic, although that only becomes algebra with the introduction of “variables” - usually represented by letters or other symbols that do not have set, defined values.

They vary because they may depend on the variable nature of other items in the equation. It's divided into many subareas with elementary algebra being the most commonly used in everyday use. As discussed above, it is named after the Persian thinker who came up with several concepts around the idea.

2. Calculus

It is another area of math that many people believe is more complicated than it actually is. It's the study of change including distance and time. It differs from arithmetic specified above in that it examines the changes rather than looking for an answer based on the integers, by breaking things down into infinitesimals.

There are two branches of calculus: differential calculus which is the study of instantaneous rates such as curves and slopes, and integral calculus, which studies quantities and accumulations and the areas around and underneath curves.

Although proto calculus may have existed before, it is believed that both Isaac Newton and Gottfried Wilhelm Leibniz invented it. It has many uses including and especially in science, design, engineering, and in economics.


To many people, arithmetic is the ability to - as most people would call it - do sums in one's head without needing paper and pen, calculator or any other device to work it out. But it's actually much simpler than that. Arithmetic concerns numbers, their values and relationships, and operations - addition, subtraction, multiplication and division.

It is one of the basics of number theory, dealing with decimals, and forms the basis of any economy, budgeting, bookkeeping (even household finance).

Arguably, it is the one area of math that we all need every minute, every day of our lives to work out times, measurements, distances and so on.

3. Foundational Mathematics

Many areas of science had a “foundational” aspect, that is, the study of the core principles. Mathematics is no different.

The field is dedicated to examining the algorithmic, logical, and the philosophical basis of math.

It is simply going back to the core of a subject and in this instance, means the study of the mathematics' most basic concept: numbering, sets, functions, geometrical figures, and the framework of the language of math. It has undergone several fundamental shifts with each now problem and addition of theories to solve them, most notably in Euclid's era, Newton's, and then Einstein's.

4. Geometry

The study of the shapes of objects - lines, points, circles, angles, circumference and volumes in both two and three-dimensional. This means it is integral to a number of areas including (and especially) cartography, route planning and navigation, civil engineering particularly in the built environment, computer-aided design, MRI and other medical imaging and many other areas.

5. Topology

This is the subarea of geometry within mathematics concerned with the properties of space when subject to pressure and deformations such as crumpling, bending, stretching, compression, and tearing. It is particularly useful for landscape analysis in geology.

6. Mechanics

Mechanics examines, through mathematics, behavior of physical bodies under certain pressures, forces, and displacements. It also examines the permanent effects of the changes to those bodies and the environment changes.

It began with Aristotle and Archimedes and developed greatly in the Renaissance and pre-enlightenment era. Kepler, Newton and Galileo are largely credited with its application in physics and why the two disciplines are so intertwined today, but that is not the limits of its influence. More recently, it has moved to incorporate motion and force on objects.

7. Operation Research

Related to analysis, this is the application of advanced methods in decision-making using a wide range of techniques. It is a method of applied math, using many principles and ideas to solve real-world problems.

It sometimes goes by the name of “management science” or “decision science” and uses computational science, modelling, statistical analysis, especially using powerful technologies. It has uses in complex engineering, environmental science, industry, operations management, and can use both quantitative and qualitative data depending on the area of application.

It was once the exclusive preserve of applying statistics and seeking efficiency and proficiency of military applications during World War II but has grown, along with computing power, and outstripped that.

8. Probability

Devised by Blaise Pascal, probability is the mathematics of likelihood of an outcome. Although largely perceived as an issue with gambling, it also has applications in risk management for investment and business, insurance premium calculation and a variety of other financial services such as the stock market. But probability also has some higher uses. Often expressed as a percentage (with 0% being impossible and 100% being certain), this is not always the case. In mathematical probability, many calculations are expressed in a range of 0-1 instead.

9. Combinatorics

Lesser known to many outside of mathematics and science, combinatorics is the science of things with finite combinations or structures.

For example, it looks to solving simply questions with difficult answers such as the number of possible combinations for giving cash change from all available coins when presented with a $1 bill or working out the total possible number of Sudoku puzzles that we can devise.

There is no set standard definition, but it is largely described as the being concerned with the enumeration theory, and with combinations and permutations, existing in a finite set of possible answers to a puzzle. This typically includes graph theory, amongst other things.

10. Computational Mathematics

For reasons that are understandable, this is a relatively new part of mathematics although it did exist before the age of information technology. How we use the term varies depending on whichever aspect of the relationship between computing and mathematics is under discussion.

It could refer to applied mathematics and the use of math principles in driving computing or it may refer to the use of computing power to solve or drive complex mathematical equations. Applied computational mathematics can be used in any area of science and has particular uses in data modeling and analysis in such things as atmospheric carbon measurement over time or geographical area.

11. Statistics

Statistics is about the collection and processing, and the interpretation and presentation of hard data. It is fundamental to any scientific research such as measuring demographics, population change, health issues and resource allocation, percentage changes.

Today it includes the methods of data collection as well as the process of collection in order to get a proper and fully accurate sample size. For example, standing outside a church on a Sunday and asking people who come out about their church attendance and whether they are a Christian might lead us to a conclusion that 100% of the population goes to church and 100% go to church every Sunday.

Therefore, statistics is as much a philosophy about the collection of data as a tool of collection.


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