Reading a Thermometer

A thermometer shows $-8$ degrees Celsius. What does this mean?

Identify the sign: The minus sign ($-$) tells us this is a negative number = Negative

Understand the meaning: Negative temperature means below zero = $-8°C$ is 8 degrees below zero

Answer: The temperature is 8 degrees below the freezing point (0°C). This is very cold - water would be frozen!

Placing Numbers on a Number Line

Place these numbers on a number line: $-4$, $2$, $-1$, $0$, $3$

Draw a number line with zero in the middle: Mark zero and draw arrows on both ends = Number line ready

Place positive numbers to the right of zero: $2$ is 2 steps right, $3$ is 3 steps right = $2$ and $3$ placed

Place negative numbers to the left of zero: $-1$ is 1 step left, $-4$ is 4 steps left = $-1$ and $-4$ placed

Answer: From left to right: $-4$, $-1$, $0$, $2$, $3$

Comparing Negative Numbers

Which is greater: $-3$ or $-7$?

Visualize on a number line: $-3$ is 3 steps left of zero, $-7$ is 7 steps left = $-3$ is closer to zero

Remember the rule: Numbers further right on the number line are greater = $-3$ is to the right of $-7$

Write the comparison: $-3 > -7$ = $-3$ is greater than $-7$

Answer: $-3$ is greater than $-7$. Even though 7 is bigger than 3, when we add the negative sign, $-3$ is actually larger because it is closer to zero.

Finding Opposites

What is the opposite of $5$? What is the opposite of $-9$?

Understand opposites: Opposite numbers are the same distance from zero but on different sides = Opposites add up to zero

Find the opposite of $5$: $5$ is 5 steps right of zero, so its opposite is 5 steps left = Opposite of $5$ is $-5$

Find the opposite of $-9$: $-9$ is 9 steps left, so its opposite is 9 steps right = Opposite of $-9$ is $9$

Answer: The opposite of $5$ is $-5$, and the opposite of $-9$ is $9$.

Real-World Application: Elevator Floors

A building has floors numbered from $-2$ (second basement) to $10$ (top floor). You are on floor $-1$ and want to go to floor $3$. How many floors do you need to go up?

Identify starting and ending floors: Start: $-1$, End: $3$ = Positions identified

Count floors to zero: From $-1$ to $0$ is $1$ floor up = $1$ floor

Count floors from zero to destination: From $0$ to $3$ is $3$ floors up = $3$ floors

Add the floors together: $1 + 3 = 4$ = $4$ floors total

Answer: You need to go up $4$ floors to get from floor $-1$ to floor $3$.

Mistake: Thinking $-7$ is greater than $-3$ because 7 is bigger than 3

Why: With negative numbers, the number closer to zero is actually greater. Think of temperature: $-3°C$ is warmer (greater) than $-7°C$.

Correct: $-3 > -7$ because $-3$ is closer to zero and further right on the number line.

Mistake: Forgetting that zero is neither positive nor negative

Why: Zero is the dividing point between positive and negative numbers. It has no sign.

Correct: Zero is neutral - neither positive nor negative. It separates the two types of numbers.

Mistake: Confusing the minus sign with subtraction

Why: The minus sign in front of a number (like $-5$) means negative, not subtract. Context matters!

Correct: In $-5$, the minus means negative five. In $8 - 5$, the minus means subtract.

Mistake: Placing negative numbers to the right of zero on a number line

Why: Negative numbers are less than zero, so they belong on the left side of the number line.

Correct: Negative numbers go LEFT of zero, positive numbers go RIGHT of zero.

Weather and Temperature

Meteorologists use negative numbers every day when reporting temperatures, especially in winter months. Understanding negative temperatures helps us prepare for cold weather and stay safe.

If today's high is $-2°C$ and tonight's low will be $-15°C$, the temperature will drop $13$ degrees.

Bank Accounts and Debt

Banks use negative numbers to show when you owe money (debt). A negative balance means you have spent more than you had and need to deposit money to get back to zero or positive.

If your account shows $-25$ euros, you owe the bank 25 euros. You need to deposit at least 25 euros to have a balance of zero.

Elevation and Geography

Geographers measure elevation relative to sea level, which is defined as zero. Mountains have positive elevations, while valleys and underwater locations have negative elevations.

Mount Everest is about $8,849$ meters above sea level. The Mariana Trench is about $-10,994$ meters (below sea level).

Q: What is the difference between a negative number and subtraction?

A: The minus sign can mean two different things depending on context. When it appears in front of a number by itself (like $-5$), it means the number is negative (5 units below zero). When it appears between two numbers (like $8 - 5$), it means subtraction (take away). Context helps you know which meaning applies.

Q: Why is $-3$ greater than $-7$?

A: Think of it like temperature: $-3°C$ is warmer than $-7°C$. On a number line, $-3$ is to the right of $-7$, and numbers to the right are always greater. Another way to think about it: $-3$ is only 3 away from zero, while $-7$ is 7 away from zero. Being closer to zero makes a negative number greater.

Q: Is zero a positive or negative number?

A: Zero is neither positive nor negative - it is neutral. Zero is the special point on the number line that separates positive numbers (on the right) from negative numbers (on the left). This is why we sometimes say a number is non-negative (zero or positive) or non-positive (zero or negative).

Q: Where do negative numbers come from? Who invented them?

A: Negative numbers have a long history! Ancient Chinese mathematicians used red and black rods to represent positive and negative numbers over 2,000 years ago. Indian mathematicians like Brahmagupta formally defined rules for negative numbers around 628 CE. European mathematicians were slower to accept them - some called them absurd or fictitious numbers well into the 1700s!

Q: What are integers?

A: Integers are the set of all whole numbers, including positive numbers (1, 2, 3, ...), negative numbers (..., $-3$, $-2$, $-1$), and zero. The word integer comes from Latin meaning whole or untouched. Fractions and decimals are not integers.