Directed Numbers (Integers)

Introduction to Directed Numbers

Directed numbers, also called integers, are numbers that have both a size (magnitude) and a direction, represented by a positive (+) or negative (−) sign. Directed numbers include all positive whole numbers, all negative whole numbers, and zero. They are called "directed" because the sign shows a direction relative to zero: positive numbers lie to the right of zero on a number line, while negative numbers lie to the left. Understanding directed numbers is essential for representing real-life quantities such as temperature, altitude, debt, and financial gain or loss, and forms a key foundation for algebra.

The Number Line

The number line is a straight line on which numbers are placed at equal intervals, extending infinitely in both directions. Zero sits at the centre of the number line, with positive numbers increasing to the right and negative numbers decreasing (becoming more negative) to the left. The number line is a powerful visual tool for understanding the size and order of directed numbers, as well as for performing addition and subtraction by moving left or right along the line.

Positive and Negative Numbers in Real Life

Directed numbers appear frequently in real-life contexts: temperatures below zero are expressed as negative numbers (e.g., −5°C means 5 degrees below zero), altitudes below sea level are negative (e.g., −200 m means 200 metres below sea level), financial debts are often represented as negative amounts, while profits or gains are positive, and in football, a negative goal difference shows more goals conceded than scored, while a positive goal difference shows the opposite.

Comparing Directed Numbers

On the number line, numbers increase in value as you move to the right and decrease as you move to the left. This means that any positive number is always greater than any negative number, and among negative numbers, the one closer to zero is the larger value. For example, −2 is greater than −5, because −2 lies to the right of −5 on the number line, even though 5 is a larger digit than 2.

Addition of Directed Numbers

To add directed numbers, several rules apply depending on the signs involved. When adding two positive numbers, simply add them as usual, and the result is positive (e.g., 4 + 3 = 7). When adding two negative numbers, add their magnitudes and keep the negative sign (e.g., −4 + (−3) = −7). When adding a positive and a negative number, subtract the smaller magnitude from the larger magnitude, and give the answer the sign of the number with the larger magnitude (e.g., −7 + 3 = −4, since 7 is larger than 3, and 7 was negative; or 8 + (−3) = 5, since 8 is larger than 3, and 8 was positive).

Subtraction of Directed Numbers

Subtracting a directed number is the same as adding its opposite (additive inverse). In other words, subtracting a positive number is the same as adding a negative number, and subtracting a negative number is the same as adding a positive number. This can be summarised as: a − b = a + (−b). For example, 5 − 8 = 5 + (−8) = −3, and 5 − (−3) = 5 + 3 = 8. This rule, sometimes remembered as "two negatives make a positive" (when they are adjacent, as in subtracting a negative), is one of the most important rules to master with directed numbers.

Multiplication of Directed Numbers

When multiplying directed numbers, the sign of the result depends on the signs of the numbers being multiplied, following these rules: positive × positive = positive (e.g., 4 × 3 = 12); positive × negative = negative (e.g., 4 × (−3) = −12); negative × positive = negative (e.g., (−4) × 3 = −12); and negative × negative = positive (e.g., (−4) × (−3) = 12). A helpful way to remember this is: if the two signs are the same, the answer is positive; if the two signs are different, the answer is negative.

Division of Directed Numbers

Division of directed numbers follows exactly the same sign rules as multiplication: positive ÷ positive = positive (e.g., 12 ÷ 3 = 4); positive ÷ negative = negative (e.g., 12 ÷ (−3) = −4); negative ÷ positive = negative (e.g., (−12) ÷ 3 = −4); and negative ÷ negative = positive (e.g., (−12) ÷ (−3) = 4). As with multiplication, matching signs give a positive result, and different signs give a negative result.

Order of Operations with Directed Numbers

When simplifying expressions involving directed numbers, the standard order of operations (BODMAS) still applies: work out brackets first, then powers, then multiplication and division (from left to right), and finally addition and subtraction (from left to right), taking care to apply the correct sign rules at every step. For example, to evaluate −3 + 4 × (−2), multiplication is performed first: 4 × (−2) = −8, then addition: −3 + (−8) = −11.

Absolute Value

The absolute value of a directed number is its distance from zero on the number line, regardless of direction, and is always positive or zero. It is written using two vertical bars, such as |−7| = 7 and |7| = 7. Absolute value is useful for comparing the magnitude (size) of numbers without regard to their sign, such as comparing the size of a debt or a temperature difference.

Directed Numbers in Coordinate Systems

Directed numbers are also used to describe positions on a coordinate grid, where both the horizontal (x) and vertical (y) positions can be positive or negative depending on which direction they are measured from the origin (0, 0). This concept extends the number line idea into two dimensions and is fundamental to graphing in coordinate geometry.

Common Errors with Directed Numbers

Common mistakes when working with directed numbers include forgetting to change the sign when subtracting a negative number, incorrectly applying sign rules during multiplication and division (especially forgetting that a negative times a negative gives a positive), misreading the number line when comparing negative values, and losing track of signs during multi-step calculations. Careful attention to signs at every step, and checking answers against the number line when in doubt, helps avoid these errors.

Summary

Directed numbers, or integers, include positive numbers, negative numbers, and zero, and are used to represent quantities that have both size and direction, such as temperature, altitude, and financial gain or loss. The number line provides a helpful visual tool for comparing and calculating with directed numbers. Specific rules govern addition, subtraction, multiplication, and division of directed numbers, with sign rules being especially important to master, since they determine whether the result of a calculation is positive or negative.

Why Directed Numbers Matter for Algebra

A firm grasp of directed numbers is one of the most important building blocks for success in algebra and beyond. Nearly every algebraic manipulation — solving equations, simplifying expressions, substituting values, or working with coordinates and graphs — depends on correctly handling positive and negative signs. Students who struggle with directed number rules often find that the errors follow them into far more advanced topics, so it is well worth practising these rules repeatedly, using the number line as a visual check, until sign changes become automatic and confident rather than a source of hesitation.

Worked Examples

Example 1: Add −8 and 5. Since the signs are different, subtract the magnitudes and keep the sign of the larger: −8 + 5 = −3.

Example 2: Subtract −9 from −6. Subtracting a negative is the same as adding a positive: −6 − (−9) = −6 + 9 = 3.

Example 3: Multiply −7 by −4. Same signs give a positive result: (−7) × (−4) = 28.

Example 4: Divide −36 by −9. Same signs give a positive result: (−36) ÷ (−9) = 4.

Example 5: Evaluate −5 + 6 × (−3) using BODMAS. Multiplication first: 6 × (−3) = −18. Then addition: −5 + (−18) = −23.

Student Exercise

Solve the following problems, showing all your working:

1. Add −12 and 7.
2. Subtract −4 from 9.
3. Multiply −8 by 6.
4. Divide −45 by 5.
5. Divide −15 by −3.
6. Evaluate 8 + (−3) × 2 using BODMAS.
7. Find the absolute value of −15.
8. State which is greater: −9 or −4, and explain why.
9. Simplify (−3) + (−7) − (−2).
10. The temperature at midnight was −3°C and it dropped by a further 4°C by dawn. What was the temperature at dawn?