Negative Numbers

On the field, on the market, below zero

Got a 15-yard penalty, then another 5? You already know the drive lost 20 yards. The only question is how to write it.

Same idea when a stock drops $3, then another $2. Or when a timezone jumps to UTC−5. Negative numbers are not a classroom trick. They are how you record “backward,” “down,” and “the other way.”

The sticky part is almost never the definition. It is which rule fires when — especially with minus a minus, or with two negatives multiplied.

Keep the sign as its own step. Then do the arithmetic.

How it plays on the field and the tape

Flag on the play. You were on the 50. Ten-yard penalty. You do not go to the 60.

You go to the 40.

Figure

Football field markers: ball on the 50, arrow left 10 yards to the 40 after a penalty

50 + (−10) = 50 − 10 = 40

Another flag at the 40, five yards:

40 + (−5) = 40 − 5 = 35

Now the messy one: the call gets overturned. That is not another penalty. That is removing a penalty — subtracting a negative — so the ball moves forward again:

35 − (−5) = 35 + 5 = 40

Minus a minus is a double reversal. Undo a loss and you gain.

Track the signs as direction cues. The arithmetic follows.

The sign rules, without the fog

A deadline moves three days earlier. That is not poetry. It is a signed shift on a schedule — same machinery as every other negative operation.

Assume you already add and multiply positive whole numbers. What changes is the sign step.

Adding and subtracting

Team has 10 points. A 3-point swing against them:

10 + (−3) = 7

Adding a negative is subtracting its absolute value.

Different beat: you sit at −5 yards from an earlier flag. The booth takes away a bad 2-yard call. You are not losing more yards. You are clawing two back:

−5 − (−2) = −3

Multiplying and dividing

Account loses 2 followers a day for 3 days:

3 × (−2) = −6

Different signs → negative product (or quotient).

Same signs → positive. The classic “why is (−)×(−) positive?” lands cleaner as language: I can’t not go means you go. Two reversals face forward.

Figure

Two reverse arrows cancel into one forward arrow: negative times negative

Rewind a clip of someone walking backward. Negative time on a negative direction. On screen they walk forward:

(−4) × (−5) = 20

Same signs multiply positive. Different signs multiply negative. Memorize that pair, not a speech.

Same pattern for division. These integer rules carry to decimals and fractions once the sign habit is solid.

Easy traps (where people slip)

A few spots keep catching people. Not “advanced.” Just sharp corners.

Watch for these:

• Double negatives written as a − (−b)
• Exponents with the minus inside vs outside parentheses
• Ordering negatives (closer to zero is larger)
• Absolute-value bars mixed with other ops

Double negative. You see 5 − (−3) and want to keep subtracting. Don’t. Minus a minus flips to plus — same as “don’t not go.”

a − (−b) = a + b

Game score: you have 5. A 3-point penalty gets overturned:

5 − (−3) = 5 + 3 = 8

Eight. Not two.

Exponents and parentheses. Squaring a negative with the minus inside the base:

(−4)² = (−4) × (−4) = 16

Without those parentheses, the exponent hits 4 first, then the leading minus:

−4² = −(4 × 4) = −16

Same digits. Opposite story.

Order feels upside-down. With positives, bigger digit wins. With negatives, closer to zero is larger. Golf relative to par: −3 beats −7. Fewer strokes under.

Figure

Golf scores on a line: -7 left of -3; -3 closer to zero and greater

−3 > −7

Absolute value and order. Bars mean distance from zero — after you finish what is inside.

Parentheses and absolute-value bars are traffic lights. Finish inside before you move on.

Pick a tool that fits the job

You do not need one holy method. Match the tool to the mess.

What you are optimizing for

Speed, accuracy, or “show me why.” Quick mental check? Speed. Messy homework? Accuracy and process. Learning day? The why.

Side by side

• Scoreboard intuition. Team loses 7 yards, then 3. You feel −10 without a ceremony. Follower count drops 15, then 10 → −15 + (−10) = −25.
• Formal sign rules. Same signs multiply/divide positive; different signs negative; minus a minus is plus. Example: (−8) × (−4) = 32.
• Calculator. Best when the digits get ugly — still only as good as the signs you type. Example: 12.5 − (−3.7) + (−8.1) → 8.1.

Figure

Three lanes — intuition, rules, calculator — pick by job not habit

Mental check → picture. Process you must defend → rules. Ugly digits → calculator, then sanity-check the sign.

Quick close

Negatives are direction and change, not a personality test for numbers.

1. Isolate the sign.
2. Do the arithmetic.
3. Check with a second method (scoreboard rewrite or calculator).

Try two mixed problems tonight — one field-or-followers story, one pure a − (−b) line — and check each answer a second way.