# How do constants differ from numbers?

Although I scanned the other version that's not Early Transcendentals, in Calculus Early Transcendentals 7th ed 2011, James Stewart never defines "constant". It first appears on p. 3 but not in this context, and contextually it first appears on p. 27 when he defines "coefficients of a polynomial".

Why not just call "constants" numbers? Isn't "constant" superfluous? Aren't numbers constant?

Kindly see green underline. Why not call $$a$$ a positive number, instead of "positive constant"?

• Constants are numbers that are not variables. – mjw Jan 12 at 22:41
• Consider $\log(ax)$: both $a$ and $x$ are numbers, but $a$ stays constant, whereas $x$ is variable – J. W. Tanner Jan 12 at 22:41
• – amWhy Jan 12 at 22:46
• It's all about preventing people from asking "what kind of number"? To stop this line of questioning, all the text assures is $a$ is positive and has no derivative. The way they used it, we also require that $a \cdot 1/a = 1$. – William Barnes Jan 12 at 22:50

By constant, what it means is the number is fixed. Just as @Alejandro Bergasa Alonso said, $$2$$ is always $$2$$, $$3$$ is always $$3$$. If someone says "$${f(x) = \sin(ax)}$$ where $$a$$ is some constant" that just means that $$a$$ is some fixed, unchanging number.

However, the "$$x$$" in the expression $${f(x) = \sin(ax)}$$ is not a constant, since it varies. We can plug whatever number we like into it. It's a variable.

Usually, whether you can treat a particular symbol as a constant or variable will depend completely upon context. But the key bit is it's called a variable since it varies, and it's called a constant because it's fixed and doesn't change.

I'll try to explain it the best I can.

All numbers "are constant", since a $$2$$ is a $$2$$ always. A $$2$$ is not some variable value that can change.

But you need to understand that, in mathematics, there are way more things than numbers: sets, functions, etc. and quite anything can be defined to be constant (tought often "fixed" is the word used to describe them too), so no, constant isn't equal to number.

In your text, it's obvious that $$a\in\mathbb{R}$$ so it's a number and hence it's constant, so it's irrelevant if they say positive number of positive constant (constant is more correct in my opinion, but in this case it's not important at all).

To sum up, numbers are constants, but a constant does not have to be a number necessarily.

Now, seeing the difference between a variable and a constant, variables are entities that don't have a fixed value. When a function, for example $$f(x)=ax$$ is defined, notice how the $$f$$ has an $$x$$ inside the pharenthesis. That means $$x$$ is a variable that, depending of its value, will give different outcomes through $$f$$. But the $$a$$ isn't inside the $$f$$'s pharenthesis, that's because it's a fixed value, to be said, $$f$$ does not depend on variations of $$a$$ (yes, different values of $$a$$ will also give different results through $$f$$, but $$f$$ is defined depending only on $$x$$, so to it, $$a$$ is fixed).

I hope it's a bit clearer for you, comment if necessary.

• just to be clear, i didn't downvote you. – Intellectually disabled Jan 12 at 22:51
• @amWhy I was answering to her question about why don't people call cosntants just numbers, explaining that not all constants are numbers. I'll edit it. – Alejandro Bergasa Alonso Jan 12 at 22:53
• Fair enough, @AlejandroBergasaAlonso ! – amWhy Jan 12 at 22:55