Infinitesimal facts for kids

Infinitesimals are super tiny quantities, so small they are almost zero but not quite! Imagine a number that is smaller than any fraction you can think of, like 1/1000 or 1/1,000,000. Infinitesimals are even smaller than that!

These incredibly small numbers are very important in a branch of mathematics called Calculus. Calculus helps us understand how things change, like how fast a car is going at an exact moment, or how a curve bends.

Tiny Numbers in Calculus

Long ago, two brilliant mathematicians, Isaac Newton and Gottfried Leibniz, invented Calculus. They used the idea of infinitesimals to figure out things like speed and acceleration. They thought of changes happening in incredibly tiny steps.

Later, other mathematicians, like Karl Weierstraß, found a different way to do calculus using something called "limits". This method was very precise but also a bit harder to understand.

Then, in the 1960s, a mathematician named Abraham Robinson showed that using infinitesimals could also be very precise and rigorous. Many schools now teach calculus using infinitesimals because it can be easier to grasp!

How Infinitesimals Help with Speed

Imagine you want to know the exact speed of a car at a specific moment. In calculus, we use infinitesimals to do this.

We use symbols like `ds` and `dt`.

• `ds` means a super tiny change in s (which stands for position or distance).
• `dt` means a super tiny change in t (which stands for time).

So, the speed, often written as , means "the tiny change in position divided by the tiny change in time." It's like finding the speed over an incredibly short period, so short it's almost an instant!

Let's say a car's position is described by the formula s = t² (where s is distance and t is time). To find its speed at any moment, we look at what happens when time changes by a tiny amount, `dt`.

• At time t, the position is s = t².
• At a slightly later time, t + `dt`, the position becomes s + `ds` = (t + `dt`)².

We can then figure out `ds` (the tiny change in position) by subtracting the first position from the second: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} ds & = (t + dt)^2-t^2 \\ & = t^2 + 2t(dt) + (dt)^2-t^2 \\ & = 2t(dt) + (dt)^2 \end{align}

Now, to find the speed (), we divide `ds` by `dt`: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): v = \frac{ds}{dt} = \frac{2t(dt) + (dt)^2}{dt} = 2t + dt

Since `dt` is an infinitesimal (super, super tiny), it's so small that we can just ignore it when we're looking for the exact instantaneous speed. So, the speed v becomes simply 2t.

This shows how infinitesimals help us find exact speeds or rates of change at any given moment, making complex problems much simpler to solve!

Related pages

• Derivative (mathematics)

Images for kids

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  Infinitesimals (ε) and infinities (ω) on the hyperreal number line (ε = 1/ω)

See also

In Spanish: Infinitesimal para niños