In math, minus minus minus equals one minus: three negatives leave a negative sign, while two negatives cancel to a plus.
Seeing three minus signs in a row can feel like your brain hit a speed bump. You’re not alone. The good news: the math behind it is steady once you separate two ideas. A minus sign can mean “take away,” or it can mean “make this value negative.”
This guide shows what “minus minus minus” can mean in everyday arithmetic, in algebra, and in calculator-style expressions. You’ll get a repeatable method to simplify long chains of minus signs, plus practice problems you can check in one glance.
What Minus Minus Minus Equals Means
People use the phrase in a few different ways. Sometimes they mean the sign of a number like -(-(-5)). Sometimes they mean a subtraction that includes a negative number like 8 – (-3). Sometimes they typed three hyphens in a row and want to know what a worksheet expects.
The clean fix is to look for parentheses. Parentheses tell you whether the minus sign is acting on a value (unary minus) or sitting between two values (subtraction).
Unary Minus: The “Negative Of” Sign
When a minus sign is stuck to a single number or variable, it means “the negative of that value.” In algebra, -x means the opposite of x. In arithmetic, -5 means five units to the left of zero on the number line.
Now stack those signs and you get the pattern that makes triple minus easy: every pair of negatives flips back to positive.
Two Negatives Cancel, Three Negatives Leave One Negative
Start with a double negative. -(-5) is the negative of negative five, so it becomes +5. Add one more negative sign: -(-(-5)). The inside turns into +5, then the outside minus flips it back to -5.
That’s the core rule you’ll use again and again: an even count of leading unary minus signs gives a positive value, an odd count gives a negative value.
What Three Minus Signs Equal In Real Math
When you see three minus signs that all act as unary minus, you can treat them like three multiplications by -1. This mental move works because sign rules for multiplication are consistent.
-(-(-x)) equals (-1)·(-1)·(-1)·x. Since the product of three -1 values is -1, the whole expression becomes -x. Same idea with numbers: -(-(-12)) becomes -12.
Here’s the sticky-note version: — cancels to +. Add one more minus and you’re back to –. That’s why folks say a triple minus leaves a minus when all three signs attach to the value.
| Pattern You See | How To Read It | Simplified Result |
|---|---|---|
| -(-a) | negative of negative a | a |
| -(-(-a)) | negative of (negative of negative a) | -a |
| –a | two unary minus signs | a |
| —a | three unary minus signs | -a |
| a – (-b) | subtract negative b | a + b |
| a – -b | subtract (negative b) with missing parentheses | a + b |
| (-a)(-b) | multiply two negatives | ab |
| (-a)(-b)(-c) | multiply three negatives | -abc |
| a – (-b) – (-c) | subtract two negatives in a row | a + b + c |
When Triple Minus Shows Up On Screens
On paper, teachers usually avoid writing a raw “—” unless they also add parentheses. On a screen, you might see it when you backspace, when you copy a formula, or when you type a subtraction and then decide the second number should be negative.
The safest habit is simple: if a number is meant to be negative, wrap it in parentheses right away. That turns a hard-to-read string of symbols into something you can verify at a glance.
Typing Patterns That Stay Clear
Use spacing and parentheses so the expression reads the same way for you and for the tool that’s evaluating it.
- Unary negatives: write -(-(-5)), not —5, when you want three unary minus signs.
- Subtracting a negative: write 8 – (-3), not 8–3, even if a calculator accepts it.
- Chained subtraction: write 7 – (-3) – (-2) to show each step, then simplify.
- Products with signs: write (-2)(-4)(-5) or (-2)·(-4)·(-5) so you can count negatives quickly.
A Fast Self-Check Before You Hit Enter
Read the expression out loud in plain words. If you can’t say it cleanly, rewrite it. Then do one last sign check: count the unary negatives attached to each number or factor and see whether the count is even or odd.
Rule Of Signs For Multiplication
Multiplication has a clean sign rule: multiply magnitudes as usual, then decide the sign by counting negatives. An even count of negative factors gives a positive product. An odd count gives a negative product.
If you want a short, formal statement of this sign rule, Wolfram MathWorld’s Rule of Signs page is a solid reference.
Why Counting Negatives Works
Think of each negative sign as a “flip.” Multiply by -1 once and the number flips to the other side of zero. Multiply by -1 twice and it flips back. Three flips land you on the negative side again.
This flip idea lines up with what you already know from opposites: the opposite of 7 is -7, and the opposite of -7 is 7.
Subtracting A Negative Adds
Subtraction is where people often type a run of minus signs. The key identity is simple: subtracting a negative turns into addition.
8 – (-3) becomes 8 + 3. The minus sign between the numbers stays a subtraction sign, but the negative sign on the 3 flips it into a plus.
Khan Academy walks through this idea in its lessons on negative-number operations inside Negative Numbers: Multiply And Divide.
What About A Chain Like 10 – – -2?
Writing three minus signs with no parentheses can look messy, but it can still be simplified if you decide what each sign is doing. In typical arithmetic notation, the middle minus is subtraction, and the signs attached to the 2 are unary minus signs.
Start by grouping the number: 10 – ( -(-2) ). The inside becomes 10 – ( +2 ), so the final result is 8.
Turning Subtraction Into Addition
A helpful rewrite is a – b = a + (-b). Once you do that, minus signs stop feeling random. You’re always adding signed numbers.
Try these rewrites on paper, then simplify:
- 5 – 12 becomes 5 + (-12)
- 5 – (-12) becomes 5 + (12)
- -5 – (-12) becomes -5 + 12
Minus Signs In Front Of Parentheses
You’ll also meet stacked minus signs when a minus sits in front of a whole group, like -(a – b). That first minus sign applies to every term inside the parentheses. It flips the signs inside the group.
Write it out step by step: -(a – b) becomes -a + b. If the inside is (a + b), then -(a + b) becomes -a – b.
Why This Creates Triple Minus
Combine group flipping with negative numbers and you can get a cluster like -(a – (-b)). Inside the parentheses, subtracting negative b turns into a + b. Then the outside minus flips the whole result to -a – b.
If you see three minus signs, pause and add parentheses until you can point to what each minus sign is attached to. After that, simplify with pairs.
Order Of Operations And Parentheses
Minus signs can interact with exponents and other operators. This is where a tiny pair of parentheses saves a lot of confusion.
Exponents Bind Tightly
On most calculators and in many textbooks, -3^2 means “negative of 3^2,” which is -9. If you mean “square negative three,” you write (-3)^2, which is 9.
This also matters with stacked signs. -(-3)^2 is the negative of 9, so it is -9. In contrast, (-(-3))^2 turns the inside into 3, then squares to 9.
Minus Signs With Fractions And Decimals
A negative sign belongs to the number it touches. So -(1/2) and -1/2 mean the same thing, while – (1/2)^2 is negative one quarter because the square happens first.
When you mix subtraction and a negative decimal, parentheses keep it readable:
- 6 – (-0.5) becomes 6 + 0.5, so it equals 6.5.
- -(-0.5) equals 0.5 because two unary negatives cancel.
How To Simplify Any String Of Minus Signs
You don’t need a special trick for three minus signs. You need a routine that works for two, three, or ten, even when the expression is long.
- Label each minus sign. Is it subtraction between two values, or unary minus attached to one value?
- Add parentheses. Group the number or group the whole chunk the unary minus acts on.
- Cancel in pairs. Two unary negatives become a plus. Odd leaves one minus.
- Re-check the sign. Ask if an odd count of unary negatives is still attached to that value.
This routine keeps you out of trouble when the symbols feel cramped or when you’re copying from a board and spacing gets lost.
Practice: Simplify These Without Guessing
Grab paper for a minute. Write each one, add parentheses where you need them, then simplify. After you finish, compare your work to the simplified column.
| Expression | Simplified | Reason |
|---|---|---|
| -(-(-6)) | -6 | three unary negatives leave one negative |
| 14 – (-9) | 23 | subtracting a negative turns into addition |
| -(-8) + (-3) | 5 | double negative becomes positive, then add a negative |
| (-2)(-4)(-5) | -40 | odd count of negative factors gives a negative product |
| 7 – -3 – (-2) | 12 | subtract negative 3, then subtract negative 2 |
| -(3^2) | -9 | exponent happens before unary minus without parentheses |
| (-3)^2 | 9 | parentheses attach the sign to the base |
| -(a – (-b)) | -a – b | inside becomes a + b, then the outside minus flips both terms |
Slip-Ups That Cause Wrong Signs
Mixing Up “Take Away” And “Negative Of”
A minus sign between two numbers is subtraction. A minus sign glued to one number is a sign. When you slow down and label each one, the rest falls into place.
Dropping Parentheses Too Early
If you remove parentheses while the sign is still acting on a whole group, you can flip the result by accident. Keep parentheses until you finish the sign work, then simplify.
Losing Track After A Long Chain
Long strings of minus signs are tiring to read. Use the “pairs cancel” rule, then do a last sign check: even means positive, odd means negative.
A Quick Rule You Can Trust
Here’s the whole idea in one line: pair off unary negatives. If one negative is left unpaired, the value stays negative.
If you’re checking homework, circle each minus sign and label it S for subtraction or N for negative. Then simplify on paper.
So, minus minus minus equals minus when those signs all attach to the same value. If one minus sign is subtraction and the next is a negative sign, the expression can turn into addition.