Writing big numbers · Code with Martin

Millions and billions without counting zeros: writing numbers with powers of ten.

A billion without counting zeros

The Earth–Sun distance is about 150,000,000 km. Try reading those zeros. And now imagine you have to copy them into your notebook without a mistake.

Scientists do not do it that way. They write 1.5 · 10⁸ km. Short, exact, and nobody counts zeros with a finger on the screen.

The trick stands on what you already know: 10⁸ is a one and eight zeros, so 100,000,000. And 1.5 · 100,000,000 = 150,000,000. It fits.

This notation is called writing with powers of ten. You split the number into two parts: how much (say 1.5) and how big (say 10⁸).

Today you will learn to translate both ways: from a long number into a power and back. It helps in physics, in geography and in the news — everywhere millions and billions fly around.

Adam and Jonáš are doing a report on space. Adam dictates: “Light travels three hundred thousand kilometres per second… write 300,000.” Jonáš writes and gets lost in the zeros: once he writes four, next time six. Adam finds the notation 3 · 10⁵ km/s in the textbook. “Look, they have it differently here. A three and then ten to the power of five.” Jonáš counts: ten to the power of five is a one with five zeros, times three… 300,000. “That is better. I remember five zeros more easily than how many times I tapped zero.”

💡

The exponent on the ten = how many places the decimal point moves. 2.5 · 10⁴: move the point 4 places to the right → 25,000.

How to translate a number into a power of ten

From the short notation to a long number: 3.2 · 10⁶. The exponent 6 says: move the decimal point 6 places to the right. 3.2 → 32 → 320 → … → 3,200,000. Fill missing places with zeros.

From a long number to the short notation:

Find the first non-zero digit and put a point after it: 47,000,000 → 4.7.
Count how many places the point moved. From 47,000,000 to 4.7 that is 7 places.
Write: 4.7 · 10⁷.

Always check backwards: 4.7 · 10,000,000 = 47,000,000. It fits.

The agreement is that the first part is a number from 1 to 10 (say 4.7, not 47 or 0.47). The notation 47 · 10⁶ is the same value, but the standard form is 4.7 · 10⁷ — that is how textbooks and calculators write it.

Names for orientation: 10³ a thousand, 10⁶ a million, 10⁹ a billion, 10¹² a trillion. A million has six zeros, a billion nine.

Example 1: 150,000,000 into a power

Earth–Sun distance: 150,000,000 km.

Step 1: a point after the first digit → 1.5.

Step 2: how many places did it move? From 150,000,000 to 1.5 that is 8 places.

Step 3: the notation 1.5 · 10⁸ km.

Check: 1.5 · 100,000,000 = 150,000,000. It fits. I did not have to count the eight zeros one by one even once — moving the point is more reliable.

Example 2: 6.378 · 10³ back

The Earth’s radius is about 6.378 · 10³ km. How many kilometres is that?

Exponent 3 → move the point 3 places to the right: 6.378 → 63.78 → 637.8 → 6378 km.

No zeros were added, because there were enough digits after the point. If there were not, you add zeros: 6.4 · 10³ = 6400.

Example 3: compare 8 · 10⁵ and 2 · 10⁶

Which number is bigger?

First the exponents: 10⁶ is ten times more than 10⁵. A higher exponent almost always wins.

I check: 8 · 10⁵ = 800,000. And 2 · 10⁶ = 2,000,000. Two million is more than eight hundred thousand — 2 · 10⁶ won, even though it has a smaller digit at the front.

The rule: first compare the exponents, deal with the front digits only when the exponents are the same.

⚠️

10⁶ is not 60 or 10 · 6. It is a million — a one and six zeros. The exponent counts zeros, it does not multiply the ten.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

The number is a power of ten. Write only the exponent — that is, the number of zeros.

Practise the questions

On paper

Translate both ways until moving the point feels boring.

Translate into notation with a power of ten: 7000, 250,000, 3,600,000,000. For each one write a check backwards too.
Translate into a long number: 9 · 10⁴, 1.25 · 10⁷, 5.03 · 10⁵.
Order from smallest: 3 · 10⁶, 9 · 10⁵, 1.1 · 10⁷. Write what you ordered by first.
Find one big number in a geography textbook or on the internet (with a parent) — say the population of China — and write it with a power of ten.

Now you 💪

I can move the point both ways according to the exponent.
I know the first part of the notation should be a number from 1 to 10.
When comparing I look at the exponents first.

✅

Done when: You have translated six numbers both ways, ordered a triple, and a run of five correct in practice.

What to take from this lesson

Big numbers are written as a number from 1 to 10 times a power of ten: 4.7 · 10⁷.
The exponent = the shift of the decimal point.
A million is 10⁶, a billion 10⁹.
When comparing, the exponent decides first.
The check is translating back.