Positive & negative integers
🎯 In this topic you will
- Estimate, add, and subtract large integers accurately.
- Add and subtract positive and negative integers confidently.
- Find the difference between two integers using efficient methods.
- integer
- negative number
- positive number
Show Definitions
Integer: A whole number with no decimal or fraction part. Integers include negative numbers, zero, and positive numbers.
Negative number: A number less than zero. It is usually written with a minus sign, for example $-3$.
Positive number: A number greater than zero. For example $5$ is a positive number.
This diagram shows the maximum known diving depths for human and other marine mammals.
Question
How much deeper can the sperm whale dive compared to the elephant seal?
Answer
The sperm whale can dive to about $3000$ m and the southern elephant seal to about $1500$ m.
Difference = $3000 - 1500 = 1500$ m
So, the sperm whale can dive $1500$ m deeper.
❓ EXERCISES
🧠 Reasoning Tip
Remember to estimate your answer before you do the calculation.
1. The table shows the flight distance from Dubai to five destinations.
a. Liu flies from Dubai to Singapore. Nadim flies from Dubai to Cairo. What is the difference in their flight distances?
b. Mary lives in Dubai. In one year she travels to Buenos Aires, Cairo and Karachi. She returns home after each trip. How many kilometres does she fly altogether?
👀 Show answer
b. Total one-way distance $= 13656 + 2419 + 1190 = 17265$ km.
Round trips: $2 \times 17265 = 34530\text{ km}$.
2. Ravi completes a subtraction calculation, but his answer is wrong.
$36804 - 5724 = 31180$
Write an explanation to convince Ravi that he is wrong. Correct his answer.
👀 Show answer
$36804 - 5724 = 31080$.
So Ravi’s answer is incorrect.
3. Faizah plays a game. She has $5398$ points.
She scores another $465$ points.
Her target is $6000$ points.
How many more points does she need to reach her target?
👀 Show answer
Needed: $6000 - 5863 = 137$ points.
4. Here is part of a temperature scale showing the temperature at Coldpark.
The temperature falls by $3^\circ\text{C}$. What is the new temperature?
👀 Show answer
New temperature: $-6 - 3 = -9^\circ\text{C}$.
5. The temperature at $8$ a.m. is $-1^\circ\text{C}$.
By midday it is $4$ degrees warmer. What is the temperature at midday?
👀 Show answer
6. The maximum temperature in Mongolia during October is $15^\circ\text{C}$.
The temperature decreases by $12^\circ\text{C}$ each month from October to February.
What is the maximum temperature in January?
👀 Show answer
November: $15 - 12 = 3^\circ\text{C}$
December: $3 - 12 = -9^\circ\text{C}$
January: $-9 - 12 = -21^\circ\text{C}$.
7. The thermometers show the temperatures in Ulaanbaatar and Montreal on the same day.
What is the difference in temperature between Ulaanbaatar and Montreal?
👀 Show answer
Difference: $\lvert -18 - (-5) \rvert = 13^\circ\text{C}$.
🧠 Reasoning Tip
Remember that a difference is always positive.
8. Here is a table of temperatures in five cities for one morning.
a. What is the difference in temperature between London and Oslo?
b. What is the difference in temperature between Moscow and Ulaanbaatar?
c. What is the difference in temperature between the warmest city and the coldest city?
👀 Show answer
b. $\lvert -10 - (-29) \rvert = 19^\circ\text{C}$
c. Warmest $= 4^\circ\text{C}$, coldest $= -29^\circ\text{C}$
Difference: $33^\circ\text{C}$.
9. Find the difference between each pair of numbers.
👀 Show answer
b. Difference $= 60$
🧠 Think like a Mathematician
Investigation:
Choose three digits.
Arrange them in order, biggest first.
Reverse the digits.
Find the difference between the two numbers.
Reverse the answer.
Add the two numbers.
Do this several times. What do you notice?
Start with four digits. What do you notice?
Predict what the answer will be if you start with two digits. Test your prediction. Was your prediction correct?
Follow-up Questions:
Show Answers
- 1: Starting with three digits always leads to $1089$ after following the steps (provided the first and last digits are different).
- 2: With four digits, a consistent pattern still appears, but the final result depends on the digits chosen; it is not always the same fixed number.
- 3: With two digits, the process always leads to a multiple of $9$. Testing confirms the prediction is correct.