Working With Negative Numbers Revision Notes for Leaving Cert Mathematics

Working With Negative Numbers

What are integers?

Integers are the complete set of positive and negative whole numbers, including zero. This set is represented by the symbol Z. Understanding how to work with integers, especially negative numbers, forms the foundation of algebra and is essential for success in your Leaving Cert mathematics exam.

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The symbol Z comes from the German word "Zahlen" which means "numbers". This is the standard mathematical notation used worldwide to represent the set of integers.

The integers include numbers like: ..., -3, -2, -1, 0, 1, 2, 3, ...

Basic rules for working with negative numbers

When working with negative numbers, there are specific rules that you must follow to get the correct answer. These rules apply to all integer operations and are frequently tested in exams.

Adding integers with the same sign

When you add two integers that have the same sign (both positive or both negative), you simply add their numerical values together and keep the common sign.

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Adding Integers with Same Sign:

$8 + 4 = 12$ (both positive)
$(-3) + (-7) = -10$ (both negative)

This rule makes intuitive sense - if you're moving in the same direction on a number line, you continue further in that direction.

Adding integers with different signs

When adding integers with different signs (one positive and one negative), the process requires more careful attention. You must identify which number has the larger absolute value, keep its sign, then subtract the smaller absolute value from the larger one.

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Key Process for Different Signs:

Find the larger absolute value
Keep the sign of the number with the larger absolute value
Subtract the smaller absolute value from the larger one

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Adding Integers with Different Signs:

$(-3) + 8 = 5$ (8 is larger, so result is positive: $8 - 3 = 5$ )
$4 + (-10) = -6$ (10 is larger, so result is negative: $10 - 4 = 6$ , with negative sign)

Think of this as a tug-of-war between positive and negative forces - the stronger force determines the direction.

Multiplying and dividing integers

For multiplication and division with integers, there's a simple rule about signs that you must memorise:

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Sign Rules for Multiplication and Division:

Like signs give a positive answer: $(+) \times (+) = (+)$ and $(-) \times (-) = (+)$
Unlike signs give a negative answer: $(+) \times (-) = (-)$ and $(-) \times (+) = (-)$

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Multiplying Integers:

$(-5) \times (-4) = 20$ (like signs: negative × negative = positive)
$(-3) \times 6 = -18$ (unlike signs: negative × positive = negative)
$7 \times (-4) = -28$ (unlike signs: positive × negative = negative)

The same rules apply for division operations.

Removing brackets

When you encounter expressions with brackets, you must multiply each term inside the brackets by the number outside. This process is called expanding or removing brackets.

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The key principle is that every term inside the brackets gets multiplied by the term outside the brackets. This is also known as the distributive property.

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Worked Examples: Removing Brackets

$3(4 + 5) = 3 \times 4 + 3 \times 5 = 12 + 15 = 27$
$5(7 - 3) = 5 \times 7 + 5 \times (-3) = 35 - 15 = 20$
$-3(7 - 2) = -3 \times 7 + (-3) \times (-2) = -21 + 6 = -15$

Notice in the third example how the negative sign outside the brackets affects both terms inside. When you multiply by a negative number, the signs of all terms inside the brackets change.

Order of operations - BIMDAS

When solving complex expressions, you must follow the correct order of operations. This sequence is remembered using the acronym BIMDAS:

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BIMDAS Order:

Brackets
Indices (or powers)
Multiplication
Division
Addition
Subtraction

Always work through this order systematically - don't skip steps!

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Worked Examples: BIMDAS Order

$3 + (5 \times 4) = 3 + 20 = 23$ First multiply inside brackets, then add.
$5 + 4 + 3(10 - 4) = 9 + 3(6) = 9 + 18 = 27$ First solve brackets, then multiply, finally add.
$6^2 + (6 - 2)^2 \times 5 = 36 + (4)^2 \times 5 = 36 + 16 \times 5 = 36 + 80 = 116$ Brackets first, then indices, then multiplication, finally addition.

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Common Exam Tips:

Always work through BIMDAS systematically - don't skip steps
When removing brackets with negative signs, be extra careful with sign changes
Double-check your work by substituting simple values
Remember that $-(-5) = 5$ when dealing with double negatives
Practice identifying like and unlike signs in multiplication and division

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Key Points to Remember:

Integers include all positive and negative whole numbers plus zero
When adding integers with same signs, add the numbers and keep the sign
When adding integers with different signs, subtract and keep the sign of the larger number
For multiplication and division: like signs give positive, unlike signs give negative
Always follow BIMDAS order when solving complex expressions
When removing brackets, multiply the outside number by every term inside the brackets

Working With Negative Numbers (Leaving Cert Mathematics): Revision Notes