Orders of magnitude.

Step 1 of 5

The atom

The scene goes through six stops, from an atom to the Milky Way, and draws each stop at a comfortable size for the screen. The real sizes are written in the labels, in words. An atom is about a tenth of a billionth of a metre across: a billionth is $\frac{1}{10^9} = 10^{-9}$$\frac{1}{10^9} =$$10^{-9}$⁠, and a tenth of it is another $10$⁠ times smaller.

Write the size of the atom as a power of $10$⁠.

Step 2 of 5

From atom to cell

The order of magnitude of a number is the exponent of $10$⁠ in its scientific notation. An atom, $10^{-10}$⁠ metres, is of order of magnitude $-10$⁠. A cell is about a hundredth of a millimetre across, that is $10^{-5}$⁠ metres. To find how many times bigger one thing is than another you divide, and dividing powers of $10$⁠ subtracts their exponents, the small raised numbers.

Predict how many times wider the cell is than the atom, then see the cell.

Step 3 of 5

The Sun

To compare sizes you need the same unit. The Sun’s width is given in kilometres, and each kilometre is $10^3$⁠ metres. Multiplying powers of $10$⁠ adds the exponents: $10^2 \times 10^3 = 100 \times 1000 = 10^5$$10^2 \times 10^3 =$$100 \times 1000 =$$10^5$⁠.

Step through the stops, and write the Sun’s width in metres.

Step 4 of 5

The galaxy

A light year is the distance light travels in a year, about $9.5 \times 10^{15}$⁠ metres, close to $10^{16}$⁠. The Milky Way is about a hundred thousand light years across, $10^5$⁠ light years.

How many metres is that?

Step 5 of 5

The whole ladder

From the atom to the Milky Way: how many orders of magnitude apart are they? On a ladder of powers of $10$⁠, the distance between two things is the difference of their exponents. Watch the atom’s minus sign, as when subtracting a negative number on the number line.