R1 — Five minutes

Goal: see what Leonardo is for, before learning anything about it.

Open the browser REPL in another tab. There is nothing to install and nothing to configure: type the lines below at the > prompt and compare.

It is a calculator

> 2 + 3 * 4
14.0
> (1 + 2)^10
59049.0
> sin(pi/2)
1.0

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Precedence, brackets and the usual constants — pi and e are built in.

Why 14.0 and not 14? Every ordinary number here is a decimal, so a whole result still prints with its decimal part. That is worth noticing now rather than wondering about later, and R8 shows how to get 14 — and exact fractions — when you want them.

It is not only a calculator

Now ask for something a calculator cannot answer:

> derive(x^2 + 3x, x)
((2.0 * x) + 3.0)

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There is no value of x here, so there is no number to give. Leonardo answered with the derivative itself — an expression, not a measurement.

That is the whole idea, and it is the one thing worth taking from this lesson:

When it can compute a number it does. When it cannot, it does the algebra and hands you back an expression.

Give it numbers and it goes the other way:

> integral(sin(x), x, 0, pi)
2.0

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Same command shape — integral(what, variable, from, to) — and this time everything was known, so the answer is a number.

It does work you would otherwise look up

> integral(x * exp(x), x)
((x * exp(x)) + (-1.0 * exp(x)))
> limit(sin(x)/x, x, 0)
1.0

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The first is an integration by parts, chosen without being asked. The second is the standard first-year limit that 0/0 makes awkward by hand.

And it draws

plot puts a curve on the page — in the browser only, which is why this page shows the command rather than its result:

> plot sin(x) x 0 6.28
(plotted 200 points)

Type it in the tab you opened. Then try plot x^3 - 2x x -3 3. The arguments are the same shape again: what, which variable, from, to.

Try it yourself

What is 2^10?
Differentiate x^3 with respect to x.
Integrate cos(x) from 0 to pi.

Answers

```text > 2^10 1024.0 > derive(x^3, x) (3.0 * (x ^ 2.0)) > integral(cos(x), x, 0, pi) 0.0 ``` *[run this session](https://lycogrypho.github.io/Leonardo/app/#s=precision%205%0Acolors%20dark%0Apretty%20off%0Alatex%20off%0Anames%20off%0Alogic%20minmax%0Alogic%20symmetric%20off%0Aexact%20precision%2030%0Aexact%20off)* The third is zero because the positive and negative halves of the cosine cancel over `0 … π`. Note that `2^10` came back as `1024.0` — the decimal point again.

What you learned

An answer is a number when one exists and an expression when it does not. Nothing failed when you got an expression back.
derive, integral, limit and plot all name the variable explicitly: derive(f, x).
Whole numbers print as 14.0, because ordinary arithmetic here is decimal.
plot is browser-only.

Next: R2 — Expressions and free variables · All lessons