Mathematics Test on Directed Numbers
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Introduction to Directed Numbers
What are directed numbers?
Directed numbers are numbers that have a direction as well as a size (magnitude). They can be positive or negative.
$$ \text{Positive numbers} \rightarrow \text{greater than zero (e.g., }+5\text{)} $$
$$ \text{Negative numbers} \rightarrow \text{less than zero (e.g., }-5\text{)} $$
Directed numbers are used to describe real-world situations such as temperature, altitude, money owed, and movement on a number line.
1. Addition of Directed Numbers
Rule: When adding numbers with the same sign, add their values and keep the sign.
When adding numbers with different signs, subtract the smaller value from the larger value and take the sign of the larger value.
Worked Example 1: Calculate
$$ -4+7 $$
Step 1: The numbers have different signs, so subtract the smaller value from the larger value.
$$ 7-4=3 $$
Step 2: Take the sign of the larger value (7 is positive).
$$ -4+7=3 $$
Answer: 3
Worked Example 2: Calculate
$$ -6+(-9) $$
Step 1: Both numbers are negative (same sign), so add the values and keep the sign.
$$ 6+9=15 $$
Step 2: Since both numbers were negative, the answer is negative.
$$ -6+(-9)=-15 $$
Answer: -15
2. Subtraction of Directed Numbers
Rule: Subtracting a number is the same as adding its opposite.
$$ a-b=a+(-b) $$
$$ a-(-b)=a+b $$
Worked Example 3: Calculate
$$ 5-9 $$
Step 1: Rewrite subtraction as adding the opposite.
$$ 5-9=5+(-9) $$
Step 2: Add the directed numbers (different signs, so subtract and take the sign of the larger value).
$$ 5+(-9)=-4 $$
Answer: -4
Worked Example 4: Calculate
$$ -3-(-8) $$
Step 1: Subtracting a negative number is the same as adding a positive number.
$$ -3-(-8)=-3+8 $$
Step 2: Add the directed numbers.
$$ -3+8=5 $$
Answer: 5
3. Multiplication of Directed Numbers
Rule:
$$ (+)\times(+)=(+) $$
$$ (-)\times(-)=(+) $$
$$ (+)\times(-)=(-) $$
$$ (-)\times(+)=(-) $$
In short: if the signs are the same, the answer is positive. If the signs are different, the answer is negative.
Worked Example 5: Calculate
$$ -7\times4 $$
Step 1: Multiply the values.
$$ 7\times4=28 $$
Step 2: The signs are different (negative × positive), so the answer is negative.
$$ -7\times4=-28 $$
Answer: -28
Worked Example 6: Calculate
$$ -6\times(-5) $$
Step 1: Multiply the values.
$$ 6\times5=30 $$
Step 2: The signs are the same (both negative), so the answer is positive.
$$ -6\times(-5)=30 $$
Answer: 30
4. Division of Directed Numbers
Rule: Division follows the same sign rules as multiplication.
$$ (+)\div(+)=(+) $$
$$ (-)\div(-)=(+) $$
$$ (+)\div(-)=(-) $$
$$ (-)\div(+)=(-) $$
Worked Example 7: Calculate
$$ -36\div4 $$
Step 1: Divide the values.
$$ 36\div4=9 $$
Step 2: The signs are different, so the answer is negative.
$$ -36\div4=-9 $$
Answer: -9
Worked Example 8: Calculate
$$ -45\div(-9) $$
Step 1: Divide the values.
$$ 45\div9=5 $$
Step 2: The signs are the same (both negative), so the answer is positive.
$$ -45\div(-9)=5 $$
Answer: 5
5. Real-World Application
Question: A submarine is at a depth of -120 m. It descends a further 35 m, then rises 60 m. What is its final depth?
Step 1: Start at the initial position.
$$ -120 $$
Step 2: Descending means going further negative, so subtract 35.
$$ -120-35=-155 $$
Step 3: Rising means adding a positive value, so add 60.
$$ -155+60=-95 $$
Answer: The submarine's final depth is -95 m.
Key Takeaways
1. When adding or subtracting directed numbers, think of movement along a number line.
2. When subtracting, always rewrite it as "adding the opposite."
3. When multiplying or dividing, same signs give a positive answer, and different signs give a negative answer.
4. Practising directed numbers with both number problems and real-world word problems builds a stronger, more intuitive understanding.
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