Comparing Two Negative Numbers

Compare $-4$ and $-9$. Which is greater?

Visualize on the number line: Place both numbers: $-9$ is at position $-9$, $-4$ is at position $-4$ = $-4$ is further right

Apply the rule: Numbers further right are greater = $-4 > -9$

Verify with real-world context: $-4°C$ is warmer than $-9°C$, so $-4$ is greater = Confirmed: $-4 > -9$

Answer: $-4 > -9$ (negative four is greater than negative nine)

Comparing a Positive and Negative Number

Compare $-8$ and $3$. Use $<$, $>$, or $=$.

Identify the signs: $-8$ is negative, $3$ is positive = Different signs

Apply the key rule: Any positive number is always greater than any negative number = $3 > -8$

Write with the correct symbol: From left to right: $-8 < 3$ = $-8 < 3$

Answer: $-8 < 3$ (any negative is less than any positive)

Comparing with Zero

Order these numbers from least to greatest: $0$, $-6$, $4$, $-1$

Identify negative numbers: Negatives: $-6$, $-1$. Non-negatives: $0$, $4$ = Negatives are smallest

Order the negatives: $-6$ is further left than $-1$, so $-6 < -1$ = $-6$, then $-1$

Place zero and positives: Zero is between negatives and positives. $4$ is the only positive. = $-1$, then $0$, then $4$

Write the complete order: From left to right on number line = $-6 < -1 < 0 < 4$

Answer: $-6 < -1 < 0 < 4$

Ordering Multiple Integers

Order from greatest to least: $-12$, $5$, $-3$, $0$, $-7$, $2$

Separate by sign: Positives: $5$, $2$. Zero: $0$. Negatives: $-3$, $-7$, $-12$ = Three groups

Order positives (greatest first): $5 > 2$ = $5$, $2$

Place zero after positives: Zero is less than all positives = $5$, $2$, $0$

Order negatives (closest to zero first): $-3 > -7 > -12$ (closer to zero = greater) = $-3$, $-7$, $-12$

Combine all: Positives, then zero, then negatives = $5$, $2$, $0$, $-3$, $-7$, $-12$

Answer: $5 > 2 > 0 > -3 > -7 > -12$

Mistake: Thinking $-15 > -5$ because $15 > 5$

Why: With negative numbers, the absolute value (digit size) works opposite! $-15$ is further from zero in the negative direction, so it's actually smaller.

Correct: $-5 > -15$ because $-5$ is closer to zero (further right on the number line).

Mistake: Confusing $<$ and $>$ symbols

Why: It's easy to mix up which symbol points which way.

Correct: The symbol always 'eats' the bigger number. Think of it as a hungry alligator that always wants the larger meal: $5 > 3$ and $3 < 5$.

Mistake: Forgetting that zero is greater than all negatives

Why: Zero might seem like 'nothing,' but it's actually greater than every negative number.

Correct: $0 > -1 > -100$. Zero is the boundary between positive and negative.

Temperature Comparison

Weather forecasters compare temperatures to determine which days are colder or warmer.

Moscow recorded $-18°C$ and Helsinki recorded $-12°C$. Since $-12 > -18$, Helsinki was warmer.

Elevation Comparison

Geographers use negative numbers for elevations below sea level.

The Dead Sea is at $-430$ meters and Death Valley is at $-86$ meters. Since $-430 < -86$, the Dead Sea is lower.

Bank Balance Comparison

Banks show negative balances when you owe money (overdraft).

Account A shows $-75$ euros. Account B shows $-20$ euros. Since $-20 > -75$, Account B is in better standing (owes less).

Q: Why is $-3$ greater than $-10$?

A: Think of temperature: $-3°C$ is warmer than $-10°C$. On the number line, $-3$ is closer to zero (further right), so it's greater. The rule is: for negatives, closer to zero = greater.

Q: Is zero positive or negative?

A: Zero is neither positive nor negative. It's the boundary point. But zero IS greater than all negative numbers and less than all positive numbers.

Q: How do I remember which way $<$ and $>$ point?

A: The symbol always opens toward the bigger number, like a hungry mouth eating the larger value. Or remember: the small end points to the small number.

Q: Why do we say $-100$ is less than $-1$?

A: Even though 100 is a bigger digit than 1, we're talking about negative numbers. $-100$ is 100 units left of zero, while $-1$ is only 1 unit left. Further left = smaller, so $-100 < -1$.