Algebra : Foundation with Numerical Arithmetics : Arithmetic Operations And Precedence

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Arithmetic Operations And Precedence

what you'll learn...

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: $4 \div 2 \times 2$ $4-:2xx2$ equals $(4 \div 2) \times 2 = 4$ $(4-:2)xx2=4$ and not $4 \div (2 \times 2) = 1$ $4-:(2xx2)=1$ .
With the above definition of PEMA, this rule is not required.
eg: $4 \div 2 \times 2 = 4 \times \frac{1}{2} \times 2$ $4-:2xx2 = 4xx 1/2 xx 2$ and simplify either way
$(4 \times \frac{1}{2}) \times 2 = 4$ $(4xx1/2)xx2 = 4$ or
$4 \times (\frac{1}{2} \times 2) = 4$ $4xx (1/2 xx 2)= 4$ .

to express = to say

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

$2 \times 4 \times 3$ $2xx4xx3$ is another example of a numerical expression.

one or other

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

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

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

Which one is correct?

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

In this case both students will get the correct answer.

• $2 + (- 4) + (- 3)$ $2+(-4)+(-3)$ $= - 2 - 3$ $= -2-3$ $= - 5$ $=-5$

• $2 + (- 4) + (- 3)$ $2+(-4)+(-3)$ $= 2 - 7$ $=2-7$ = $- 5$ $-5$ .

Simplify $2 \div 4 \div 3$ $2-:4-:3$ . Which one of the following is correct?

• $2 \div 4 \div 3$ $2-:4-:3$ $= \frac{2}{4} \div 3$ $=2/4 -:3$ $= \frac{2}{12}$ $= 2/12$

• $2 \div 4 \div 3$ $2-:4-:3$ -- $2 \div \frac{4}{3}$ $2-: 4/3$ $= \frac{6}{4}$ $=6/4$

Division is to be handled as inverse of multiplication $2 \div 4 \div 3$ $2-:4-:3$ $= 2 \times \frac{1}{4} \times \frac{1}{3}$ $=2xx 1/4 xx 1/3$ .

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

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

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

which one first

Simplify $2 + 4 \times 3$ $2+4xx3$ . Which one of the following is correct?

• $2 + 4 \times 3$ $2+4xx3$ -- $6 \times 3$ $6 xx 3$ $= 18$ $= 18$

• $2 + 4 \times 3$ $2+4xx3$ $= 2 + 12$ $=2 + 12$ $= 14$ $=14$

Multiplication has higher precedence to addition. In $2 + 4 \times 3$ $2+4xx3$ , the multiplication is to be done ahead of addition and so $2 + 4 \times 3$ $2+4xx3$ $= 2 + 12$ $=2 + 12$ $= 14$ $=14$

Which one of the following is correct?

• $2 \times 4^{3}$ $2xx4^3$ $= 2 \times 64$ $=2 xx 64$ $= 128$ $= 128$

• $2 \times 4^{3}$ $2xx4^3$ -- $8^{3}$ $8^3$ $= 512$ $=512$

Exponent has higher precedence to multiplication. In $2 \times 4^{3}$ $2xx4^3$ , the exponent is to be done ahead of multiplication. So $2 \times 4^{3}$ $2xx4^3$ $= 2 \times 64$ $=2 xx 64$ $= 128$ $= 128$

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 $2 + 4$ $2+4$ has to be multiplied by $3$ $3$ . This cannot be given as $2 + 4 \times 3$ $2+4xx3$ as the result of this expression does not equal the example.

Parenthesis or brackets help to define such expressions. $(2 + 4) \times 3 = (6) \times 3 = 18$ $(2+4)xx3 = (6)xx3 = 18$

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: $4 \div 2 \times 2$ $4-:2xx2$ equals $(4 \div 2) \times 2 = 4$ $(4-:2)xx2=4$ and not $4 \div (2 \times 2) = 1$ $4-:(2xx2)=1$ .

With the definition of PEMA, the rule of "left-to-right-sequence" is not required.
eg: $4 \div 2 \times 2 = 4 \times \frac{1}{2} \times 2$ $4-:2xx2 = 4xx 1/2 xx 2$ and simplify either way and both result in the same correct answer.
$(4 \times \frac{1}{2}) \times 2 = 4$ $(4xx1/2)xx2 = 4$ or
$4 \times (\frac{1}{2} \times 2) = 4$ $4xx (1/2 xx 2)= 4$ .

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