Order of operations with negative numbers

When you reach for it

Reach for this whenever an expression mixes negative numbers with more than one operation: a calculation like -3 - 4(-5), a formula with a negative number put in for a letter, or a fraction whose top and bottom both have signs in them.

Nothing about the order of operations changes when negatives appear. Grouping still comes first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right, exactly as in the lesson on powers. What changes is the reading. Two questions decide almost every expression with negatives in it:

• Is this minus sign a subtraction, or does it mean “the opposite of”?
• What exactly does this power apply to?

Once those are answered, each step is a calculation you already know: adding and subtracting signed numbers from Adding and subtracting integers, and the signs of products and quotients from Multiplying and dividing integers.

The procedure

One sign, two jobs

A minus sign written between two quantities is subtraction: in 7 - 3 it takes 3 away from 7. A minus sign written in front of a single quantity, with nothing before it, or just after another operation sign or an opening bracket, means the opposite of that quantity: in -3, 4 \cdot (-3) and 5 + (-2) it is attached to the 3 or the 2.

The two jobs are tied together. Subtracting a number is the same as adding its opposite,

which is how subtracting integers works. And taking the opposite of a number is the same as multiplying it by -1,

which is one of the sign rules for products. That second fact settles where “the opposite of” sits in the order: it is a multiplication, so it comes after powers and, like any multiplication, before addition and subtraction.

Where a power reaches

A power applies only to its base, the single thing written immediately to the left of the exponent. That thing is a number, a letter, or a whole bracket, and nothing more.

In (-3)^2 the base is the bracket, so the base is -3 and the square is (-3)(-3) = 9. In -3^2 the base is just 3: the minus sign is not part of it. The expression means the opposite of 3^2, so

This reading is a convention, and it is the one that keeps the rest of algebra consistent. In -x^2 + 5x the term -x^2 is read as -1 times x^2, the same way 3x^2 is 3 times x^2: a number multiplying a power. Powers come before multiplication, so the square is taken first and the opposite after. Reading -3^2 as 9 would make the first term of -x^2 + 5x mean something different from every other term.

Whether a power of a negative number comes out positive or negative is settled by the sign rules for products: an even number of negative factors gives a positive product, an odd number a negative one. Put in this lesson’s terms, that is the result the rest of the lesson leans on:

So for an odd exponent the bracket makes no difference to the value, (-5)^3 = -5^3 = -125, and for an even exponent it changes the sign of the answer, (-5)^4 = 625 but -5^4 = -625.

Minus signs on brackets, and grouping without brackets

A minus sign in front of a bracket takes the opposite of the bracket’s whole value. In -(4 - 9) the bracket comes first, 4 - 9 = -5, and its opposite is 5. Since the minus sign is multiplication by -1, the distributive property gives the same answer another way: -(4 - 9) = -4 + 9 = 5.

A fraction bar groups everything above it and everything below it, as though each were in its own bracket. Work out the top and the bottom completely, then divide.

Absolute value bars group what is between them: |2 - 7| is worked out inside first, giving |-5| = 5, before anything outside. Fraction bars and absolute value bars are both Grouping symbols, as firm as any bracket.

A full pass

Take 7 - 2(-3)^2. The bracket holds a single number, -3, so there is nothing to work out inside it; it is there to make -3 the base of the power.

The minus sign after the 7 is subtraction, since it sits between 7 and 2(-3)^2. It waits until the end, because the term after it has to be found first. Drawn as a tree, the work starts at the bottom and climbs.

Writing negatives so they read one way

Two operation signs are never written side by side. Write 5 - (-2), not 5 - -2, and 4 \cdot (-3), not 4 \cdot -3. When a negative number is put in place of a letter, it goes in brackets: x^2 at x = -3 is (-3)^2 = 9. Without the bracket it would read -3^2, which is -9, a different number.

Worked examples

Boundaries

The reading of -3^2 is a convention of written mathematics. It is the convention everywhere in algebra, for the reason given above, but software does not always follow it. Some spreadsheet programs give 9 for -3^2. When it matters, write the bracket: -(3^2) or (-3)^2 cannot be misread by anyone.

Two operation signs need a bracket between them. 4 \cdot -3 and 5 - -2 are not standard, because the reader cannot see at a glance which sign belongs to the number. Write 4 \cdot (-3) and 5 - (-2).

Division written next to a bracket can be read two ways. In -12 \div 3(-2), strict left to right gives -4 \cdot (-2) = 8, but some readers take 3(-2) as one product and get -12 \div (-6) = 2. Do not write a quotient that way; use a fraction bar or another bracket so only one reading exists.

A bottom that works out to 0 has no value. In \dfrac{6}{-3 + 3} the bottom is 0, and division by zero is not defined, whatever the signs elsewhere.

Left to right applies within a rank, not across ranks. Multiplication and division are done from left to right among themselves, and so are addition and subtraction, but a product is still done before a sum even if the sum comes first in the line. Once every subtraction is rewritten as adding the opposite, the additions may be done in any order, by the commutative and associative properties of addition; subtractions themselves cannot be reordered.

A minus sign up in the exponent is a different matter. In 2^{-3} the minus sign belongs to the exponent, not to the base, and a negative exponent has its own meaning. This lesson deals only with exponents that are counting numbers.

Where this goes next

Every expression in algebra with a negative coefficient is read with this lesson’s two questions. Combining like terms with negative coefficients treats -x^2 and -3x as -1 times x^2 and -3 times x, exactly as here, and distributing a minus sign across a bracket is the move used in Example 4.

Evaluating formulas at negative values is where the bracket around a substituted number earns its place: x^2 at x = -3 is 9, and leaving the bracket out turns it into -9. The sharpest case is the quadratic formula, which asks for -b and b^2; when b is itself negative, -b is positive and b^2 is still positive, and both follow from the same readings as Example 5(a), whose top is exactly a b^2 - 4ac.

Reading -a as (-1) \cdot a, so that (-a)^n is (-1)^n a^n, is a first case of the Laws of exponents, where a power of a product splits into a product of powers in general.

Exercises

Where the power reaches

Evaluate.

1. -9^2
2. (-9)^2
3. -(-1)^4
4. (-1)^7 - (-1)^8
5. -3^3 - (-3)^3

Signs in the order of operations

Evaluate.

6. -9 - 3(-4)
7. -36 \div (-4) \cdot 3
8. -36 \div (-4 \cdot 3)
9. 5 - (-2)(-7) + (-10)
10. -2(-5) - (-3)^2

Brackets, bars and absolute value

Evaluate.

11. -(8 - 13) - (-2)^3
12. -3[4 - (1 - 6)^2]
13. \dfrac{-15 + 3^2}{-2 - 4}
14. \dfrac{(-4)^2 - 2(-4)}{-4 + 1}
15. -|2 - 7| + (-2)^2

Negative values in place of letters

16. Evaluate x^2 - 4x when x = -3.
17. Evaluate -x^2 - x when x = -5.
18. Evaluate a^3 - 2ab when a = -2 and b = 3.
19. Evaluate \dfrac{b^2 - 4ac}{2a} when a = -1, b = -4 and c = 5.

From words

20. A lift in a mine stops 15 meters below ground level, at -15 meters. It goes down 6 meters four times, then up 20 meters. Write one expression for where it ends and evaluate it.
21. Take the difference of -4 and 6, square it, and then take the opposite of the result. Write this as one expression and evaluate it.

Reasoning

22. Explain why -a^2 is never positive and (-a)^2 is never negative, whatever number a is.
23. Explain why (-a)^3 and -a^3 are equal for every number a, but (-a)^4 and -a^4 are equal only when a = 0.
24. Explain why -2^2 and 0 - 2^2 have the same value, and use this to say why the power is worked out before the opposite is taken.
25. Is -(a - b) equal to b - a for every pair of numbers a and b? Prove your answer.