Order of Operations - BODMAS

As said here in this topic, when using Infix notation, we either need brackets or some rule to say the order in which we evaluate an expression. Say we have to evaluate 3 + 5 x 8, this thing possibly has two solutions depending on the priority we give to the addition and the multiplication operator, that is, 8 x8=65 or 3+40=43. When there isn't a particular standard followed, we will end up in wrong interpretation. Thus came the order or precedence of operators.

1) B or P - Brackets or Parenthesis
2) O or E or I - Orders or Exponents or Indices (square roots, powers, radix etc.)
3) D - Division
3) M - Multiplication
4) A - Addition
4) S - Subtraction

Since 5 x 8 / 4 will yield the same result when either one of multiplication or division is given the higher priority, either of it can be calculated first and thus they share a same level of precedence. Same is the case with the addition and the subtraction operators.

Let us have an example. (3 * (5² + 5) ) /5 + (3*15)/5

Here there are two (outer) brackets, hence we have to evaluate those two first. The expression in each bracket should be considered as a separate expression and should be evaluated separately by following BEDMAS separately. Two brackets that are not nested, can be evaluate them in parallel.

we have an exponent. 5² = 25. We don't have division. Then we have multiplication. 3* 25 = 75. Then we have addition 75+5=80. Now that the first bracket is evaluated. We can evaluate the other bracket in parallel to this. Here goes simple steps.

(3 * (5² + 5 )) /5 + (3*15)/5 - Outer Brackets - Ist and IInd
= (3 *( 25 + 5) )/5 + 15/5 - Inner bracket (power) of Ist, Multiplication of IInd
= (3 * 30)/5 + 15/5 - Addition in inner bracket of Ist - BEDMAS in Ist and IInd bracket over
= 90/5 + 15/5 - Multiplication in Ist outer bracket
= 18 + 3 - Division can be performed in parallel.
= 21 - BEDMAS for the expression is over - Final Result.

If the bracket operator is nested, like
(3 + (5*4)) , then the inner bracket should be evaluated first

When we come across something like this, 27/3/3, we need to know whether we evaluate from left to right, or right to left. Mostly we take it left to right. Hence we would evaluate this as (27/3)/3 = 9/3 = 3

Infix, Prefix and Postfix

An operator is a function that specify specific actions on objects. Those objects which acts as an input to the operator is called an operand. For eg., in 3+5 , the symbol "+" is a addition function (operator) which operates on the operands "3" and "5" to produce the result 8. Depending on where we place this operator and the operands, we have three different notations.

Infix: This is the common notation that we use in mathematics. Here the operators are put in between the two operands as in 3+5. Infix operations either require parenthesis or order of precedence to correctly evaluate an expression.

Prefix: Also called Polish notation due to the Polish logician Jan Łukasiewicz, who invented this notation. In this, the operators are put before the operands. Eg., +AB , where + is the operator. A,B are operands which has some value. It is mainly developed to avoid parenthesis in expressions.

For eg., let us take 2 + 6 / 8
Here we can use a bracket around 3+5 or we can give higher priority to division and evaluate the expression as (2+6)/8 = 1 or 2+6/8 = 2+0.75 = 2.75
So if we want the correct results we need to use brackets or precedence rule. The same expression, in prefix notation, can be written as / + 2 6 8
= / + 2 6 8
= / 8 8 (Note: / A B means "divide A by B" and not "divide B by A")
= 1
Here / operates on "the result of + 3 5" and 8. The result of + 3 5 is 8. This eliminates the ambiguity on what to evaluate first in the expression. If we note here, in prefix notation, the number of operators will always be one less than the number of operands. In our case, 2 operators and 3 operands.

Postfix: In postfix notation, also called reverse polish notation (RPN), we place the operators after the operands. This notation was proposed by F. L. Bauer and E. W. Dijkstra to reduce computer memory access and utilize the stack to evaluate expressions. eg., 5 3 -
Note: 5 3 - equals 5 - 3 = 2 and not 3 - 5 = -2
Thus - 5 3 in prefix is the same as 5 3 - in postfix

These prefix and postfix operations are often used in computer languages and in calculators for ease of calculation in the background. Since we are much familiar with the infix notation, we generally have a user interface which is in infix notation.

Let us take the discriminant of quadratic equations, B²-4AC
Infix : B^2-4*A*C
Prefix : -^B2**4AC
Postfix: B2^4A*C*-

Decimal or Base Ten or Denary Numeral System

Decimal is a base (or radix) 10 numeral system. Meaning, the system has ten symbols or numerals to represent any quantity. These symbols are called Digits. The Hindu-Arabic numeric system and Chinese counting rods system are the two systems that required no more than ten symbols. Decimal numeral system is the most commonly used numeral system perhaps because humans have ten digits over both hands.

We would look into the Hindu-Arabic system as it is commonly used all around the world. The ten symbols are 1,2,3,4,5,6,7,8,9 and 0. The two great Indian mathematicians could be given credit for developing the Hindu-Arabic system. They are Aryabhatta of Kusumapura who lived during the 5th century developed the place value notation and Brahmagupta a century later introduced the symbol zero. Though the concept of zero has been around for a long time, Indians gave the symbol for zero '0' which is used today.

Many system follow some way to represent the quantities by using finite number of symbols. That some way in this decimal number system is weightage. Each digit in a number is given weightage depending on its position. Eg on how a number is represented in a decimal number system is show below.
As we move towards left from the decimal point, the weight given to the digit increases by a factor of 10. As we towards right from the decimal point, the weight given to the digit decreases by a factor of 10.

So 17.591 can be expanded as 1x10¹ + 7 x 10⁰ + 5 x 10^-1 + 9 x 10^-2+ 1 x 10^-3 which is equal to 17.591

There are many systems similar to decimal system with different bases. Hence if two or more systems are used simultaneously, to distinguish what numeral system is used to represent a number we use the form something like this 17.591₁₀. This indicates that 17.591 is expressed in base 10.

Since weights are given to the digits in the number, it is one of the positional or place value numeral system.

Numbers, Number system and Numeral system

A Number is an abstract object, tokens of which are symbols, used for counting and measuring quantities. These symbols which represent the number are called numerals.

1,2,3 etc of the Arabic Numeral System,
I, II, III etc of the Roman Numeral System are examples of symbols alias numeral.

However in common usage the word number is used for both the abstract object and the symbol

The realization that two apples and two oranges have something in common was a breakthrough in human thought that lead to the discovery of numbers. In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time — days, seasons, years. Arithmetic calculations (addition, subtraction, multiplication and division), naturally followed them.

A Number system is a set of numbers, together with one or more operations, such as addition or multiplication. In other words, Numbers can be classified into sets, called number systems, like Natural numbers, Real numbers etc.

A Numeral system (or system of numeration) is a mathematical notation for representing numbers of a given set by symbols in a consistent manner. Arabic numeral system is the most commonly used numeral system. Other systems like Roman numeral system, Maya numeral system etc which are numeral systems by culture and Decimal, Binary, Octal numeral systems etc are positional notional or place value notation systems.

Numeral systems are sometimes called number systems, but that name is misleading, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers etc.

How do I learn Maths ?

Maths - one of the most important subject, that comes with you throughout your life. The only subject which is independent of other subjects. All other subjects receives help from this unique one. Yet most people think that it is the toughest of all subjects.

To me, the subject I like the most is Maths. Of course, most people score more marks in the subject they like and most like the subject in which they score more. Whatever you do with interest, you will succeed in it. So first thing is to develop interest towards it. It is another subject like Physics or Chemistry or English or whatever.

How to prepare for it ? Maths is not a subject to study like other subjects. Other subjects is more of theory whereas Maths is analytical. So you should not train by-hearting Maths. Practise makes a man perfect. So work out as many problems as you can. More important find out shortcuts to solve problems. This will help you save time, and will also make you stand unique among others. Following are some tips that you can follow to excel in Maths.

1) When there is some formula, learn to derive it first. Know what it is the purpose of the formula. When to apply it? What data you will be needing? This will help you to alter the formula here and there to match the question you have.

2) You don't need to keep too much formulas in mind. Most formulas are inter-related. With one or two steps, you can derive at the required formula if you know some basics formulas in each chapter.

3) There are some cases (especially in Choose and proofs) , where we can start from the solution rather than the question. Though it is not a good practice, it saves time for you.

4) When you prepare notes in your class room, don't write all the steps. When you look at your notes in the future, you will question what I have done here, and you will remember the step better. If you write all the steps in the notes, you will skid through your notes and there are chances that you forget things. When learning a problem, make note of important steps alone, eg I have to make a substitution at this step.

5) Do mind calculations for small steps and skip unnecessary steps. Like 5+(3*5)=5+15=20. Here second step is unnecessary.

Meaning, Etymology of Mathematics

Mathematics

1) is a science that deals with the study of measurement, properties, and relationships of quantities and sets, using numbers and symbols.

2) is the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically.

3) a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement

Origin: 1350-1400

From Middle English mathematik, from Old French mathematique, from Latin mathēmatica, from Greek mathēmatikē (téchné)

The word "mathematics" (Greek: μαθηματικά or mathēmatiká) comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art.

Latin mathematica was a plural noun, which is why mathematics has an -s at the end even though we use it as a singular noun. Thus in English, however, the noun mathematics takes singular verb forms. It is often shortened to math in English-speaking North America and maths elsewhere.