Q.A function $f(\theta)$ is defined as: $f(\theta) = 1 - \theta + \dfrac{\theta^2}{2!} - \dfrac{\theta^3}{3!} + \dfrac{\theta^4}{4!}$. Why is it necessary for $\theta$ to be a dimensionless quantity?

The key idea is that the terms in a series expansion must all have the same dimensions, which forces the argument to be dimensionless.

Step 1: Write the series explicitly:

Step 2: The constant term $1$ is dimensionless. For the sum to be physically meaningful, every term must also be dimensionless.

Step 3: The second term is $\theta$ itself. If $\theta$ had any physical dimension (e.g., length, time), then $\theta$ would carry that dimension, while $1$ is dimensionless — you cannot add quantities of different dimensions. …

The key idea here is dimensional homogeneity. In any valid physical equation, you can only add or subtract quantities that have the same dimensions. If you mix metres with seconds squared, the result is nonsense — and the same principle applies here.

Look at $f(\theta)$:

The first term is the number $1$. A pure number has no dimension — it's dimensionless. That means every other term in the sum must also be dimensionless, otherwise you'd be adding apples to oranges.

Let's check term by term:

1. The constant term $1$ is dimensionless. So the entire expression must be dimensionless.

2. The term $-\theta$ must have the same dimension as $1$. Since $1$ is dimensionless, $\theta$ itself must be dimensionless. If $\theta$ had dimensions (say, length or time), then $-\theta$ would have those dimensions — and you cannot add a length to a pure number.

3. The term $\frac{\theta^2}{2!}$ is $\theta$ squared divided by $2$. If $\theta$ had dimensions, $\theta^2$ would have those dimensions squared. That would be a different dimension from $\theta$ itself, and certainly not dimensionless. The only way $\theta^2$ can be dimensionless is if $\theta$ itself is dimensionless.

4. The same logic applies to $\frac{\theta^3}{3!}$ and $\frac{\theta^4}{4!}$. Each successive power would introduce a new dimension (length$^3$, length$^4$, etc.) unless $\theta$ is dimensionless. …

Concept: Arguments of Transcendental Functions Must Be Dimensionless

When we write functions like $e^x$, $\sin x$, $\cos x$, or in this case a power series expansion, the variable $x$ must be dimensionless (a pure number). This is because these functions are defined using infinite series that add terms of different powers.

Method: Dimensional Analysis of a Power Series

Step 1: Write the given series clearly

This is the first five terms of the Maclaurin series for $e^{-\theta}$.

Step 2: Check the dimensions of each term

Let $[\theta]$ denote the dimension of $\theta$.

• Term 1: $1$ is dimensionless → $[1] = 1$
• Term 2: $\theta$ → dimension $[\theta]$
• Term 3: $\dfrac{\theta^2}{2!}$ → dimension $[\theta]^2$
• Term 4: $\dfrac{\theta^3}{3!}$ → dimension $[\theta]^3$
• Term 5: $\dfrac{\theta^4}{4!}$ → dimension $[\theta]^4$

Step 3: Apply the principle of dimensional homogeneity

In any valid physical equation, you can only add quantities that have the same dimensions. Here, we are adding:

For this sum to be meaningful:

• $[\theta]$ must equal $[1]$ (dimensionless)
• $[\theta]^2$ must also equal $[1]$
• $[\theta]^3$ must equal $[1]$
• $[\theta]^4$ must equal $[1]$

Step 4: Conclude the only possibility …

Let’s break down the common mistakes students make and how to avoid them.

✗ Mistake 1: Thinking $\theta$ can have units because it’s “just a variable”

Why it’s wrong:

The series is:

If $\theta$ had units (say meters), then:

• $1$ is dimensionless
• $\theta$ has dimension $[L]$
• $\theta^2$ has dimension $[L^2]$
• $\theta^3$ has dimension $[L^3]$
• $\theta^4$ has dimension $[L^4]$

You cannot add quantities with different dimensions. That would be like adding 1 meter + 1 square meter — physically meaningless.

✓ How to avoid:

Always check: every term in a sum must have the same dimension. In a power series, the only way this works is if the variable is dimensionless (or you multiply each term by a matching dimensional constant — which isn’t the case here).

✗ Mistake 2: Confusing $\theta$ with an angle in degrees/radians

Why it’s wrong:

Angles in radians are dimensionless (they are a ratio of arc length to radius). But many students think “angle = has units.” Actually:

• Radians are dimensionless
• Degrees are not dimensionless in the same sense (they are a scaled unit)

If $\theta$ were in degrees, the series would still be dimensionally inconsistent unless you convert to radians first.

✓ How to avoid:

Remember: radians are dimensionless. If a problem says “$\theta$ is an angle,” check whether it’s in radians. If it’s in degrees, convert before plugging into a series.

✗ Mistake 3: Ignoring dimensional analysis entirely

Why it’s wrong:

Students often jump into algebra without checking dimensions. They see a neat series and assume it’s always valid. But the series for $e^x$, $\sin x$, $\cos x$ — all require $x$ to be dimensionless.

✓ How to avoid:

Make dimensional analysis a first step, not an afterthought. Before doing any calculation, ask:

“Can I add these terms? Do they all have the same dimension?”

If the answer is no, the expression is physically impossible — unless there’s a hidden constant with units (not the case here).

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