At first glance, writing down a physical quantity like –3.4 cm/s or +0.1 mmol seems simple: just slap a plus or minus sign on the number, write the unit, and you’re done. But does it actually matter whether you interpret “–3.4 cm/s” as “–3.4, then multiply by cm/s” or as “the negative of 3.4 cm/s”? And what about the role of units and signs—do they interact, or are they kept strictly separate? The answer to these questions reveals not just a convention, but a deep mathematical structure underlying all of science. Let’s dig into why the order of calculation for signs and units matters—and why, in practice, it almost never changes the value of a physical quantity, as long as you follow the rules.
Short answer: The order in which plus or minus signs and units are applied to physical quantities does not affect their value. The sign (positive or negative) is always associated with the numerical part (the scalar), not with the unit. Whether you write –3.4 cm/s, (–3.4)(cm/s), or –(3.4 cm/s), they all mean the same thing: a velocity of 3.4 centimeters per second in the negative direction. Units themselves are unsigned references; the sign only modifies the number. This is an internationally agreed-upon convention, and it ensures clarity in calculations and communication.
Let’s break down why this is the case, what the rules are, and how these conventions make scientific measurement and calculation possible.
The Anatomy of a Physical Quantity
Every measured value in physics—be it length, mass, speed, or energy—is expressed as a number (the magnitude) multiplied by a unit (the reference standard). For example, in “–3.4 cm/s,” the “–3.4” is the magnitude (including its sign), and “cm/s” is the unit. Openstax.org and texasgateway.org both emphasize this basic structure, noting that “the value of a quantity is generally expressed as the product of a number and a unit.” This product is not just a matter of notation; it reflects how we interpret and use physical quantities in equations and experiments.
Why Only the Number Gets the Sign
A key principle, according to physics.stackexchange.com and supported by the Bureau International de Poids et Mesures (BIPM), is that “a unit name or symbol cannot have a sign, the number yes.” Units are reference standards—they’re always positive, because you can’t have a negative meter or a negative second as a unit. The sign tells you about the direction or sense of the physical quantity, not about the unit itself.
That’s why, when you write –3.4 cm/s, it’s always understood as (–3.4) × (cm/s). You could also write –(3.4 cm/s); mathematically, these are identical because multiplication of real numbers is associative and commutative. In both cases, the negative sign operates on the number, not the unit. This is mirrored in the mathematical structure of physical quantities: as one physics.stackexchange.com commenter put it, “quantities of a given dimension constitute a 1-dimensional vector space with the unit of measurement as the basis,” so multiplying by –1 just changes the direction, not the nature of the unit.
Historical Roots of Plus and Minus Signs
The use of plus and minus signs to denote positive and negative quantities goes back centuries, as detailed by grokipedia.com and jordanbell.info. These symbols originally arose in medieval Europe to denote increases and decreases in values—long before they were standardized for arithmetic. The plus sign (+) stood for “more” (Latin: plus), and the minus sign (–) for “less” (minus), especially in mercantile accounting. Over time, their usage became more formalized in mathematics and, by extension, in science and measurement. Today, these signs are “universally recognized in mathematical education and scientific communication,” and their placement is governed by international agreement.
Practical Examples: Positive, Negative, and Zero
Let’s look at some concrete cases, using examples from physics.stackexchange.com and met.reading.ac.uk. Suppose you have:
+0.1 mmol = (+0.1) mmol = +(0.1 mmol) = 0.1 mmol
–0.1 m = (–0.1) m = –(0.1 m) = –0.1 m
+1 m = (+1) m = +(1 m) = 1 m
In every case, the sign is attached to the number, and the resulting quantity is simply that number (positive or negative) times the unit. Writing the sign before the number or before the whole quantity has no effect on the value. In mathematical terms, –(a × u) = (–a) × u, where a is a real number and u is a unit.
Why the Convention Matters
If you were to try to attach a sign to the unit itself—say, writing “3.4 (–cm/s)”—it wouldn’t make sense in the context of measurement. Units are fixed references; the sign has physical meaning only when it changes the direction or sense of the measurement, not the definition of the unit. This is not just a notational convenience; it’s a mathematical necessity. As met.reading.ac.uk explains, “measurable quantities must be expressed in terms of real numbers,” which can be positive or negative, while the units provide the scale.
This separation is what allows us to perform calculations like –3.4 cm/s + 3.4 cm/s = 0 cm/s: the numbers combine algebraically, and the units serve as a consistent frame of reference.
What About Parentheses and Ambiguity?
International standards, as cited by physics.stackexchange.com, explicitly forbid writing things like (–15) °C with parentheses in published work, since it’s redundant. The meaning is clear: –15 °C is always the temperature 15 degrees below zero, and there is no such thing as “–°C.” Parentheses are sometimes used in teaching to emphasize the structure, but not in scientific writing.
Significance in Calculations and Equations
This convention becomes especially important in more complex calculations. For example, if you multiply two velocities, say (–3 m/s) × (2 s), you get –6 m. The sign multiplies with the number, and the units combine according to the rules of algebra. The units retain their dimension; the sign only changes the magnitude’s sense. This is the basis for all dimensional analysis, as explained in openstax.org and met.reading.ac.uk.
Units also allow us to check the plausibility of equations: both sides of a physical equation must have the same units, regardless of the sign of the numbers involved. If you wrote –3.4 cm/s = 3.4 s/cm, the units wouldn’t match, so the equation would be invalid, regardless of the sign.
The Role of Zero
If you write 0 cm/s, or +0 cm/s, or –0 cm/s, all these are mathematically equivalent to zero velocity. The sign doesn’t affect the value, because zero is neither positive nor negative in physical measurement.
Why Notation Is Standardized
The need for standardized notation is fundamental to communication in science. As openstax.org and texasgateway.org highlight, “without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way.” Imagine trying to compare results from different labs where one used “–cm/s” as a unit and another used “cm/s” with negative numbers. Confusion would be rampant, and the results would be impossible to interpret reliably.
In summary, the calculation order of plus/minus signs and units does not affect the numeric value of physical quantities, because the sign is always associated with the number, not with the unit. This is an internationally recognized convention, rooted in both mathematical necessity and practical convenience. It ensures that physical quantities are “expressed as the product of a number and a unit,” as stated by openstax.org and physics.stackexchange.com, and that “the unit does not have a sign”—only the number does.
To quote directly from physics.stackexchange.com: “The unit does not have a sign. The quantity (which may be a scalar with a sign, or a vector with a direction…) can sometimes have a sign. ‘Positive’ and ‘negative’ are properties of the reals… Units are something else entirely.”
So, whether you write –3.4 cm/s, (–3.4) cm/s, or –(3.4 cm/s), you are always describing a velocity of 3.4 centimeters per second in the negative direction. This clarity is what makes modern science, engineering, and mathematics possible—across languages, cultures, and centuries.