Parentheses

Parentheses are used in mathematics to determine the order of calculations. They can both change which terms must be calculated first and how an expression can be removed or expanded. This makes them an important tool in almost all types of calculations.

 

Rules

When solving an expression with several terms, you must always multiply and divide before adding and subtracting. (The order of operations)

 

If the calculation requires adding something before multiplying/dividing, you put parentheses around what must be calculated first.

 

Examples:

 

$$ \large 1 + 2 \cdot 3 $$

 

Here you must multiply first (2 · 3) and then add 1 afterwards:

 

$$ \large 1 + 6 = 7 $$

 

In contrast to:

 

$$ \large (1 + 2) \cdot 3 $$

 

Here you must add first (1 + 2) and then multiply by 3:

 

$$ \large 3 \cdot 3 = 9 $$

 

So parentheses can be used to change the order of calculation and thus give a different result.

 

 

Signs

The sign in front of the parentheses (the symbol immediately before it) determines what we can and must do.

 

A parenthesis with a plus sign can be removed without doing anything else. The parentheses have no effect on how the calculation is carried out:

 

$$ \large a+(b-c+d) = a+b-c+d $$

 

Number example: \( \large 5+(3-2) = 5+3-2 = 6\).

 

A parenthesis with a minus sign can be removed if you change the sign of all the terms:

 

$$ \large a-(b-c+d) = a-b+c-d $$

 

Number example: \( \large 5-(3-2) = 5-3+2 = 4\).

 

Multiplying into parentheses

You multiply a multi-term expression by a number by multiplying each term by that number.

This is called "multiplying into the parentheses". For example, if it says: \( \large 2 \cdot (a + b)\)

You must multiply both \( \large a\) and \( \large b\) by \( \large 2\), and then add them together:

 

$$ \large 2 \cdot (a+b) = 2a + 2b $$

 

More examples:

 

$$ \large (a+b)(c+d) = ac + ad + bc + bd $$

$$ \large (a+b)(c-d) = ac - ad + bc - bd $$

$$ \large (a+b)^2 = a^2 + b^2 + 2ab $$

$$ \large (a-b)^2 = a^2 + b^2 - 2ab $$

$$ \large (a+b)(a-b) = a^2 - b^2 $$

 

 

Summary

Parentheses are used in mathematics for three main purposes:

 

• To determine the order of calculation.
• To change the signs of the terms inside the parentheses.
• To make it possible to multiply into or expand an expression.

 

Remember that parentheses are not just "extra decoration" but an important part of how to calculate correctly.

 

A typical mistake is to ignore the order of operations. For example, there is a big difference between:

 

$$ \large 1+2\cdot3 = 7 $$

$$ \large (1+2)\cdot3 = 9 $$