---
title: "Large Numbers"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 21
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/21-large-numbers
---

# Chapter 21 — Large Numbers

Thousands were the ceiling of [Chapter 14](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#ch-g3-numbers); now the ceiling rises to *millions*. The trick that makes huge numbers readable is to cut their digits into groups of three — the [classes](#def-g4-numbers-classes).

## 21.1 Classes of three digits

**Definition 21.1 (Classes).**

Starting from the right, the digits of a number are grouped in *classes* of three: the class of *units*, the class of *thousands*, the class of *millions*. Each class has its own units, tens and hundreds. For instance

$$
6\,309\,452 = 6 \text{ millions} + 309 \text{ thousands}
+ 452 \text{ units},
$$

read “six million three hundred nine thousand four hundred fifty-two”.

![The class table of 6\,309\,452 (h = hundreds, t = tens, u = units of each class). Reading a large number means reading each class, followed by its name.](https://one-course.com/images/onecourse/chapters/math-1/g4-numbers/fig-cebfe5222dad.svg)

*The class table of $6\,309\,452$ (h $=$ hundreds, t $=$ tens, u $=$ units of each class). Reading a large number means reading each class, followed by its name.*

**Example 21.2 (Writing with the classes).**

“Twelve million forty thousand five hundred” has $12$ in the millions class, $040$ in the thousands class, $500$ in the units class:

$$
12\,040\,500 .
$$

The [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) of $040$ are essential: each class (except the leftmost) must show its three digits.

**Example 21.3 (How many thousands?).**

The number $6\,309\,452$ *contains* $6\,309$ thousands (everything left of the units class): $6\,309\,452 = 6\,309 \times
1\,000 + 452$. Asking “how many thousands” is not asking for one digit, but for a whole prefix of the number.

## 21.2 Comparing and rounding

**Method 21.4 (Comparing large numbers).**

Exactly as in [Method 14.4](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#met-g3-numbers-compare), with the [classes](#def-g4-numbers-classes) as a shortcut: more digits wins; same length, compare from the left — class by class, then digit by digit.

**Example 21.5.**

$2\,145\,890$ and $2\,148\,540$: millions equal ($2$), thousands class $145 < 148$, so

$$
2\,145\,890 < 2\,148\,540 .
$$

**Definition 21.6 (Rounding).**

To *round* a number to the nearest thousand (or ten, hundred, million …), replace it by the closest multiple of $1\,000$. If the number sits exactly halfway, round up. In practice: look at the digit just below the rounding place — $5$ or more rounds up, $4$ or less rounds down.

**Example 21.7.**

$37\,614$ rounded to the nearest thousand: the hundreds digit is $6$, so [round](#def-g4-numbers-round) up: $38\,000$. Rounded to the nearest hundred: $37\,600$. Rounded to the nearest ten thousand: $40\,000$.

![Rounding 37\,614 to the nearest thousand: it lies past the halfway mark 37\,500, so it rounds up to 38\,000.](https://one-course.com/images/onecourse/chapters/math-1/g4-numbers/fig-c6769ea21dd6.svg)

*[Rounding](#def-g4-numbers-round) $37\,614$ to the nearest thousand: it lies past the halfway mark $37\,500$, so it [rounds](#def-g4-numbers-round) up to $38\,000$.*

**Example 21.8 (Orders of magnitude in the world).**

[Rounding](#def-g4-numbers-round) is how large numbers are actually used: a country of $67\,842\,591$ inhabitants is “about $68$ million”; a city of $214\,376$ people is “about $200\,000$”. The exact figure changes every day anyway — the rounded one carries the useful information.

## 21.3 Exercises

**Exercise 21.1 ★.**

Write in digits, with the [classes](#def-g4-numbers-classes) separated: “three million five hundred twenty thousand”; “seven million eight”; “two hundred forty thousand sixty”.

**Solution of Exercise 21.1.**

$3\,520\,000$; $7\,000\,008$; $240\,060$.

**Exercise 21.2 ★.**

Write in words: $5\,204\,300$; $10\,050\,000$; $999\,999$.

**Solution of Exercise 21.2.**

$5\,204\,300$: five million two hundred four thousand three hundred.

$10\,050\,000$: ten million fifty thousand.

$999\,999$: nine hundred ninety-nine thousand nine hundred ninety-nine.

**Exercise 21.3 ★.**

In $8\,472\,156$: which digit is the tens of thousands? The hundreds of the millions class? How many *thousands* does the number contain (see [Example 21.3](#ex-g4-numbers-contained))?

**Solution of Exercise 21.3.**

Tens of thousands: $7$. Hundreds of the millions class: there is no hundreds-of-millions digit in a seven-digit number — the millions class of $8\,472\,156$ contains only the digit $8$ (units of millions). Thousands contained: $8\,472$ (since $8\,472\,156 = 8\,472 \times 1\,000 + 156$).

**Exercise 21.4 ★.**

Copy and complete with $<$ or $>$:

$$
980\,000 \;?\; 1\,020\,000, \qquad
3\,456\,000 \;?\; 3\,465\,000, \qquad
7\,090\,050 \;?\; 7\,090\,500 .
$$

**Solution of Exercise 21.4.**

$980\,000 < 1\,020\,000$; $3\,456\,000 < 3\,465\,000$ (thousands class $456 < 465$); $7\,090\,050 < 7\,090\,500$.

**Exercise 21.5 ★.**

Order from smallest to biggest: $2\,300\,000$; $980\,500$; $2\,030\,000$; $2\,299\,999$.

**Solution of Exercise 21.5.**

$980\,500 < 2\,030\,000 < 2\,299\,999 < 2\,300\,000$.

**Exercise 21.6 ★.**

Round $63\,782$: to the nearest ten; to the nearest hundred; to the nearest thousand; to the nearest ten thousand.

**Solution of Exercise 21.6.**

To the ten: $63\,780$. To the hundred: $63\,800$. To the thousand: $64\,000$. To the ten thousand: $60\,000$.

**Exercise 21.7 ★.**

Round to the nearest million: $4\,721\,000$; $6\,499\,999$; $9\,500\,000$.

**Solution of Exercise 21.7.**

$5\,000\,000$; $6\,000\,000$ (the $499\,999$ falls short of [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) a million); $10\,000\,000$ (exactly halfway [rounds](#def-g4-numbers-round) up).

**Exercise 21.8 ★.**

Count by steps: three steps of $1\,000$ from $998\,500$; two steps of $100\,000$ from $950\,000$. What happens to the [classes](#def-g4-numbers-classes) as you cross a million?

**Solution of Exercise 21.8.**

$998\,500 \to 999\,500 \to 1\,000\,500 \to 1\,001\,500$: the second step crosses the million, and a new class lights up on the left.

$950\,000 \to 1\,050\,000 \to 1\,150\,000$: same crossing at the first step.

**Exercise 21.9 ★.**

A stadium holds $81\,338$ spectators. A newspaper writes “more than $80\,000$ fans”; a radio says “about $81\,000$”. Are both right? To which place did each [round](#def-g4-numbers-round)?

**Solution of Exercise 21.9.**

Both are right. “More than $80\,000$” [rounds](#def-g4-numbers-round) down to the ten thousand ($81\,338 \to 80\,000$); “about $81\,000$” [rounds](#def-g4-numbers-round) to the nearest thousand ($81\,338 \to 81\,000$).

**Exercise 21.10 ★★.**

Using each of the digits $1$ to $7$ exactly once, write the biggest possible seven-digit number, the smallest one, and the one closest to $4\,000\,000$.

**Solution of Exercise 21.10.**

Biggest: $7\,654\,321$. Smallest: $1\,234\,567$. Closest to $4\,000\,000$: just below works best — $3\,765\,421$ is $234\,579$ away, while just above, $4\,123\,567$, is only $123\,567$ away: the closest is $4\,123\,567$.

**Exercise 21.11 ★★.**

What is the biggest whole number that [rounds](#def-g4-numbers-round) to $50\,000$ when [rounding](#def-g4-numbers-round) to the nearest thousand? And the smallest?

**Solution of Exercise 21.11.**

[Rounding](#def-g4-numbers-round) to the nearest thousand gives $50\,000$ for all numbers from $49\,500$ (halfway [rounds](#def-g4-numbers-round) up) to $50\,499$. Biggest: $50\,499$; smallest: $49\,500$.