Overview

Arithmetic Operations :

» Fundamental Arithmetic Operations

→ Addition

→ Multiplication

→ Comparison

→ Exponent

» Derived Arithmetic Operations

→ Subtraction : Inverse of Addition

→ Division : Inverse of Multiplication

→ Root : Inverse of Exponent to find the index

→ Logarithm : Inverse of Exponent operation to find the base

» Precedence : PEMA

* PEMDAS or BODMAS *

→ Parenthesis

→ Exponents

→ Multiplication

→ Addition

*When a number of operations of same precedence is encountered, it is prescribed that the operations be carried out from left to right in sequence. eg: $4\xf72\times 2$ equals $(4\xf72)\times 2=4$ and not $4\xf7(2\times 2)=1$. With the above definition of PEMA, this rule is not required. eg: $4\xf72\times 2=4\times \frac{1}{2}\times 2$ and simplify either way $(4\times \frac{1}{2})\times 2=4$ or $4\times (\frac{1}{2}\times 2)=4$.*

to express = to say

*$2+4+3$ is an example of a numerical expression.*
The expression can be evaluated as $2+4+3=9$

*$2\times 4\times 3$ is another example of a numerical expression.*

one or other

What is the value of $2-4-3$?

• One student did $2-4-3$ $=-2-3$ $=-5$

• Another student did $2-4-3$ -- $2-1$ =$1$.

Which one is correct?

Subtraction is to be handled as inverse of addition $2-4-3$ $=2+(-4)+(-3)$.

In this case both students will get the correct answer.

• $2+(-4)+(-3)$ $=-2-3$ $=-5$

• $2+(-4)+(-3)$ $=2-7$ =$-5$.

Simplify $2\xf74\xf73$. Which one of the following is correct?

• $2\xf74\xf73$ $=\frac{2}{4}\xf73$ $=\frac{2}{12}$

• $2\xf74\xf73$ -- $2\xf7\frac{4}{3}$ $=\frac{6}{4}$

Division is to be handled as inverse of multiplication $2\xf74\xf73$ $=2\times \frac{1}{4}\times \frac{1}{3}$.

In this case, both methods will give the correct answer.

• $2\times \frac{1}{4}\times \frac{1}{3}$ $=\frac{2}{4}\times \frac{1}{3}$ $=\frac{2}{12}$

• $2\times \frac{1}{4}\times \frac{1}{3}$ $=2\times \frac{1}{12}$ $=\frac{2}{12}$.

which one first

Simplify $2+4\times 3$. Which one of the following is correct?

• $2+4\times 3$ -- $6\times 3$ $=18$

• $2+4\times 3$ $=2+12$ $=14$

*Multiplication has higher precedence to addition.* In $2+4\times 3$, the multiplication is to be done ahead of addition and so $2+4\times 3$ $=2+12$ $=14$

Which one of the following is correct?

• $2\times {4}^{3}$ $=2\times 64$ $=128$

• $2\times {4}^{3}$ -- $8}^{3$ $=512$

*Exponent has higher precedence to multiplication.* In $2\times {4}^{3}$, the exponent is to be done ahead of multiplication. So $2\times {4}^{3}$ $=2\times 64$ $=128$

The word "precedence" means: priority over another; order to be observed.

In a numerical expression, the precedence order is:

• exponents

• multiplication

• addition.

out of order

Multiplication has higher precedence to addition. In some expressions, addition has to be carried out before multiplication.

For example: Result of $2+4$ has to be multiplied by $3$. This cannot be given as $2+4\times 3$ as the result of this expression does not equal the example.

Parenthesis or brackets help to define such expressions. $(2+4)\times 3=\left(6\right)\times 3=18$

*Parenthesis or brackets have higher precedence.*

PEMA / BOMA

Precedence order is "PEMA" or "BOMA" is also known as PEMDAS / BODMAS

**LPA - Precedence ** : Precedence Order in arithmetics is PEMA or BOMA

* PEMDAS or BODMAS *

→ Parenthesis

→ Exponents

→ Multiplication

→ Addition

Note 1: Subtraction is handled as inverse of addition.

Note 2: Division is handled as inverse of multiplication

Note 3: Roots and Logarithm are handled as inverse of exponents

why pema

For a number of operations of same precedence, it is prescribed that the operations be carried out from left to right in sequence.

eg: $4\xf72\times 2$ equals $(4\xf72)\times 2=4$ and not $4\xf7(2\times 2)=1$.

With the definition of PEMA, the rule of "left-to-right-sequence" is not required.

eg: $4\xf72\times 2=4\times \frac{1}{2}\times 2$ and simplify either way and both result in the same correct answer.

$(4\times \frac{1}{2})\times 2=4$ or

$4\times (\frac{1}{2}\times 2)=4$.

*This is very important in the context of Algebra, as variables or terms may require to be handled in different order than the prescribed left to right order.*

summary

**LPA - Precedence ** : Precedence Order in arithmetics is PEMA or BOMA

→ Parenthesis

→ Exponents

→ Multiplication

→ Addition

Note 1: Subtraction is handled as inverse of addition.

Note 2: Division is handled as inverse of multiplication

Note 3: Roots and Logarithm are handled as inverse of exponents

Outline

The outline of material to learn "Algebra Foundation" is as follows.

Note: *click here for detailed outline of Foundation of Algebra*

→ __Numerical Arithmetics__

→ __Arithmetic Operations and Precedence__

→ __Properties of Comparison__

→ __Properties of Addition__

→ __Properties of Multiplication__

→ __Properties of Exponents__

→ __Algebraic Expressions__

→ __Algebraic Equations__

→ __Algebraic Identities__

→ __Algebraic Inequations__

→ __Brief about Algebra__