Calculating with Positive and Negative Numbers

What are Directed Numbers?

Every time you check a bank balance that has dipped into an overdraft, you are working with negative numbers.

Directed numbers are numbers that have both a size and a specific direction (positive or negative) from zero.
They are represented on a number line, where zero is the origin. Zero is a unique integer because it is neither positive nor negative.
The size of the number, regardless of its sign, is called its magnitude or absolute value. For example, the magnitude of both $-5$ and $+5$ is $5$ .

Adding and Subtracting Directed Numbers

Taking away a penalty actually helps your score. In maths, subtracting a negative number works exactly the same way—it increases your total!

When adding and subtracting, think about directional movement on the number line. Adding moves you right, while subtracting moves you left.
If two signs appear directly next to each other in a calculation, you must simplify them using the "Two Sign" rule.
Same signs directly next to each other ( $+(+)$ or $-(-)$ ) become a plus ( $+$ ).
Different signs directly next to each other ( $+(-)$ or $-(+)$ ) become a minus ( $-$ ).
You can check your result using an inverse operation (an operation that "undoes" another, like using addition to check subtraction).

Worked Example: Addition (Different Signs)

Calculate $-2 + (-3)$

Step 1: Simplify adjacent signs. The $+$ and $-$ are different, so they become a minus. The calculation becomes $-2 - 3$ .
Step 2: Use the number line. Start at $-2$ and move $3$ spaces to the left.
Answer: $-5$

Worked Example: Subtraction (Double Negative)

Calculate $7 - (-9)$

Step 1: Simplify adjacent signs. The two minus signs are the same, so they become a plus. The calculation becomes $7 + 9$ .
Step 2: Use the number line. Start at $7$ and move $9$ spaces to the right.
Answer: $16$

Multiplying and Dividing Directed Numbers

Think of playing a video: if you rewind (a negative action) a video of someone walking backwards (a negative direction), they appear to be moving forwards. This explains why a negative times a negative makes a positive!

The rules for multiplication and division rely on the signs interacting to form a product or quotient.
Same signs: Multiplying or dividing two positive or two negative numbers always gives a positive result.
Different signs: Multiplying or dividing a positive and a negative number always gives a negative result.
For continuous strings of multiplication, an even number of negative signs gives a positive answer, while an odd number gives a negative answer.

Worked Example: Multiplication

Calculate $5 \times (-4)$

Step 1: Multiply the magnitudes. $5 \times 4 = 20$
Step 2: Apply the sign rule. Positive $\times$ Negative = Negative.
Answer: $-20$
(Number Line Method: To show $3 \times -2$ , interpret this as "three lots of $-2$ " using repeated addition. Start at $0$ and jump $2$ units left three times: $0 \rightarrow -2 \rightarrow -4 \rightarrow -6$ .)

Worked Example: Division

Calculate $-42 \div (-7)$

Step 1: Divide the magnitudes. $42 \div 7 = 6$
Step 2: Apply the sign rule. Negative $\div$ Negative = Positive.
Answer: $6$

Combined Operations with Directed Numbers

Just like assembling a flat-pack wardrobe, executing mathematical steps in the wrong order will leave you with a complete mess.

When a calculation involves mixed operations, you must follow the standard order of operations (BIDMAS).
Apply the specific sign rules for directed numbers at each individual stage of the calculation.

Worked Example: Combined Operations (BIDMAS)

Work out $-12 \div (-3) + (-2) \times 5$

Step 1: Division. $-12 \div -3 = 4$ (applying the same signs rule).
Step 2: Multiplication. $-2 \times 5 = -10$ (applying the different signs rule).
Step 3: Substitute back. Replace the calculated parts into the expression to get $4 + (-10)$ .
Step 4: Addition (Simplify signs). The $+(-)$ becomes a $-$ , so the final calculation is $4 - 10$ .
Answer: $-6$

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Exam Tips

Common Mistake
Students heavily confuse the rules for addition/subtraction with those for multiplication. Remember that 'two minuses make a plus' applies ONLY to multiplication/division or when two signs are directly next to each other (e.g., 10 - (-2) = 10 + 2), not to additions like -5 - 3.
2
In multi-step 'show that' questions or algebraic substitutions, explicitly write out the simplification of double signs (e.g., changing 10 - (-2) to 10 + 2) to ensure you secure method marks.
3
On Calculator papers (Papers 2 and 3), use the (-) key (unary minus) to input negative numbers rather than the standard subtraction operator key to prevent syntax errors.
4
Always use brackets around negative numbers when squaring on a calculator; entering (-3)^2 outputs 9, whereas -3^2 incorrectly outputs -9 because the calculator squares the number before applying the negative sign.

Key Terms(8)

Directed number: A number, such as an integer, decimal, or fraction, that has both a size and a specific direction (positive or negative) from zero.
Integer: A whole number that can be positive, negative, or zero.
Magnitude: The size or numerical value of a number, regardless of its positive or negative sign.
Absolute value: The distance a number is from zero on a number line, representing its size without regard to its sign.
Inverse operation: An operation that reverses or 'undoes' another operation, such as using addition to check a subtraction.
Product: The result obtained from multiplying two or more numbers together.
Quotient: The result obtained from dividing one number by another.
Number line: A horizontal line used to represent numbers, where zero is the origin, positive numbers are to the right, and negative numbers are to the left.

Previous NoteThe Four Operations and Formal Written Methods Next NotePlace Value Concepts

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Key Terms(8)

Directed number: A number, such as an integer, decimal, or fraction, that has both a size and a specific direction (positive or negative) from zero.
Integer: A whole number that can be positive, negative, or zero.
Magnitude: The size or numerical value of a number, regardless of its positive or negative sign.
Absolute value: The distance a number is from zero on a number line, representing its size without regard to its sign.
Inverse operation: An operation that reverses or 'undoes' another operation, such as using addition to check a subtraction.
Product: The result obtained from multiplying two or more numbers together.
Quotient: The result obtained from dividing one number by another.
Number line: A horizontal line used to represent numbers, where zero is the origin, positive numbers are to the right, and negative numbers are to the left.

Calculating with Positive and Negative Numbers

What are Directed Numbers?

Every time you check a bank balance that has dipped into an overdraft, you are working with negative numbers.

Directed numbers are numbers that have both a size and a specific direction (positive or negative) from zero.
They are represented on a number line, where zero is the origin. Zero is a unique integer because it is neither positive nor negative.
The size of the number, regardless of its sign, is called its magnitude or absolute value. For example, the magnitude of both $-5$ and $+5$ is $5$ .

Adding and Subtracting Directed Numbers

Taking away a penalty actually helps your score. In maths, subtracting a negative number works exactly the same way—it increases your total!

When adding and subtracting, think about directional movement on the number line. Adding moves you right, while subtracting moves you left.
If two signs appear directly next to each other in a calculation, you must simplify them using the "Two Sign" rule.
Same signs directly next to each other ( $+(+)$ or $-(-)$ ) become a plus ( $+$ ).
Different signs directly next to each other ( $+(-)$ or $-(+)$ ) become a minus ( $-$ ).
You can check your result using an inverse operation (an operation that "undoes" another, like using addition to check subtraction).

Worked Example: Addition (Different Signs)

Calculate $-2 + (-3)$

Step 1: Simplify adjacent signs. The $+$ and $-$ are different, so they become a minus. The calculation becomes $-2 - 3$ .
Step 2: Use the number line. Start at $-2$ and move $3$ spaces to the left.
Answer: $-5$

Worked Example: Subtraction (Double Negative)

Calculate $7 - (-9)$

Step 1: Simplify adjacent signs. The two minus signs are the same, so they become a plus. The calculation becomes $7 + 9$ .
Step 2: Use the number line. Start at $7$ and move $9$ spaces to the right.
Answer: $16$

Multiplying and Dividing Directed Numbers

The rules for multiplication and division rely on the signs interacting to form a product or quotient.
Same signs: Multiplying or dividing two positive or two negative numbers always gives a positive result.
Different signs: Multiplying or dividing a positive and a negative number always gives a negative result.
For continuous strings of multiplication, an even number of negative signs gives a positive answer, while an odd number gives a negative answer.

Worked Example: Multiplication

Calculate $5 \times (-4)$

Step 1: Multiply the magnitudes. $5 \times 4 = 20$
Step 2: Apply the sign rule. Positive $\times$ Negative = Negative.
Answer: $-20$
(Number Line Method: To show $3 \times -2$ , interpret this as "three lots of $-2$ " using repeated addition. Start at $0$ and jump $2$ units left three times: $0 \rightarrow -2 \rightarrow -4 \rightarrow -6$ .)

Worked Example: Division

Calculate $-42 \div (-7)$

Step 1: Divide the magnitudes. $42 \div 7 = 6$
Step 2: Apply the sign rule. Negative $\div$ Negative = Positive.
Answer: $6$

Combined Operations with Directed Numbers

Just like assembling a flat-pack wardrobe, executing mathematical steps in the wrong order will leave you with a complete mess.

When a calculation involves mixed operations, you must follow the standard order of operations (BIDMAS).
Apply the specific sign rules for directed numbers at each individual stage of the calculation.

Worked Example: Combined Operations (BIDMAS)

Work out $-12 \div (-3) + (-2) \times 5$

Step 1: Division. $-12 \div -3 = 4$ (applying the same signs rule).
Step 2: Multiplication. $-2 \times 5 = -10$ (applying the different signs rule).
Step 3: Substitute back. Replace the calculated parts into the expression to get $4 + (-10)$ .
Step 4: Addition (Simplify signs). The $+(-)$ becomes a $-$ , so the final calculation is $4 - 10$ .
Answer: $-6$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students heavily confuse the rules for addition/subtraction with those for multiplication. Remember that 'two minuses make a plus' applies ONLY to multiplication/division or when two signs are directly next to each other (e.g., 10 - (-2) = 10 + 2), not to additions like -5 - 3.
2
In multi-step 'show that' questions or algebraic substitutions, explicitly write out the simplification of double signs (e.g., changing 10 - (-2) to 10 + 2) to ensure you secure method marks.
3
On Calculator papers (Papers 2 and 3), use the (-) key (unary minus) to input negative numbers rather than the standard subtraction operator key to prevent syntax errors.
4
Always use brackets around negative numbers when squaring on a calculator; entering (-3)^2 outputs 9, whereas -3^2 incorrectly outputs -9 because the calculator squares the number before applying the negative sign.

Key Terms(8)

Directed number: A number, such as an integer, decimal, or fraction, that has both a size and a specific direction (positive or negative) from zero.
Integer: A whole number that can be positive, negative, or zero.
Magnitude: The size or numerical value of a number, regardless of its positive or negative sign.
Absolute value: The distance a number is from zero on a number line, representing its size without regard to its sign.
Inverse operation: An operation that reverses or 'undoes' another operation, such as using addition to check a subtraction.
Product: The result obtained from multiplying two or more numbers together.
Quotient: The result obtained from dividing one number by another.
Number line: A horizontal line used to represent numbers, where zero is the origin, positive numbers are to the right, and negative numbers are to the left.

Previous NoteThe Four Operations and Formal Written Methods Next NotePlace Value Concepts

Back to The Four Operations

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(8)

Directed number: A number, such as an integer, decimal, or fraction, that has both a size and a specific direction (positive or negative) from zero.
Integer: A whole number that can be positive, negative, or zero.
Magnitude: The size or numerical value of a number, regardless of its positive or negative sign.
Absolute value: The distance a number is from zero on a number line, representing its size without regard to its sign.
Inverse operation: An operation that reverses or 'undoes' another operation, such as using addition to check a subtraction.
Product: The result obtained from multiplying two or more numbers together.
Quotient: The result obtained from dividing one number by another.
Number line: A horizontal line used to represent numbers, where zero is the origin, positive numbers are to the right, and negative numbers are to the left.