maths > algebra

Arithmetic Operations And Precedence

what you'll learn...

Overview

Arithmetic Operations :

»  Fundamental Arithmetic Operations
→  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

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÷2×2$$4 \div 2 \times 2$ equals $\left(4÷2\right)×2=4$$\left(4 \div 2\right) \times 2 = 4$ and not $4÷\left(2×2\right)=1$$4 \div \left(2 \times 2\right) = 1$.
With the above definition of PEMA, this rule is not required.
eg: $4÷2×2=4×\frac{1}{2}×2$$4 \div 2 \times 2 = 4 \times \frac{1}{2} \times 2$ and simplify either way
$\left(4×\frac{1}{2}\right)×2=4$$\left(4 \times \frac{1}{2}\right) \times 2 = 4$ or
$4×\left(\frac{1}{2}×2\right)=4$$4 \times \left(\frac{1}{2} \times 2\right) = 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$

$2×4×3$$2 \times 4 \times 3$ 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$$2 - 4 - 3$ $=2+\left(-4\right)+\left(-3\right)$$= 2 + \left(- 4\right) + \left(- 3\right)$.

In this case both students will get the correct answer.

•  $2+\left(-4\right)+\left(-3\right)$$2 + \left(- 4\right) + \left(- 3\right)$ $=-2-3$$= - 2 - 3$ $=-5$$= - 5$

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

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

•  $2÷4÷3$$2 \div 4 \div 3$ $=\frac{2}{4}÷3$$= \frac{2}{4} \div 3$ $=\frac{2}{12}$$= \frac{2}{12}$

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

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

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

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

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

which one first

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

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

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

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

Which one of the following is correct?

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

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

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

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

In a numerical expression, the precedence order is:
•  exponents
•  multiplication

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+4×3$$2 + 4 \times 3$ as the result of this expression does not equal the example.

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

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÷2×2$$4 \div 2 \times 2$ equals $\left(4÷2\right)×2=4$$\left(4 \div 2\right) \times 2 = 4$ and not $4÷\left(2×2\right)=1$$4 \div \left(2 \times 2\right) = 1$.

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

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.

→   Numerical Arithmetics

→   Arithmetic Operations and Precedence

→   Properties of Comparison

→   Properties of Multiplication

→   Properties of Exponents

→   Algebraic Expressions

→   Algebraic Equations

→   Algebraic Identities

→   Algebraic Inequations