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Whole Numbers

Definitions


  • When we have 10 ones=, we group them into 1 ten=.
  • When we have 10 tens=, we group them into 1 hundred=.
  • When we have 10 hundreds = , we group them into 1 thousand =
  • When we have 10 thousands = , we group them into 1 ten thousand=
  • To count how many ten thousands, thousands, hundreds, tens, and ones there are, we can make a table:
    Ten Thousands Thousands Hundreds Tens Ones
    1 2 1 2 3
    The table tells us we have 1 ten thousands, 2 thousands, 1 hundred, 2 tens, and 3 ones, which we can write in positional notation as \(12\,123\).

Definition Base 10 system
In the base 10 system, the place of a digit in a number determines its value. We can represent the number \(32\,354\) in several ways:
  • With digits: $$32\,354$$
  • In expanded form:$$\begin{aligned}[t]3\text{ ten-thousands } &+ &2\text{ thousands } &+ &3\text{ hundreds } &+ &5\text{ tens } &+ &4\text{ ones }\\30\,000 &+ &2\,000 &+ &300 &+ &50 &+ &4 \\3\times 10\,000&+& 2\times 1\,000&+& 3\times 100&+&5\times 10&+&4\times 1\\\end{aligned}$$
  • With words:
    thirty-two thousand three hundred fifty-four
  • In a table:
    Ten-Thousands Thousands Hundreds Tens Ones
    3 2 3 5 4
  • With cubes:

On the Number Line


  • A number line shows numbers like 0, 1, 2, 3, and so on in order.
  • Let’s make counting easier by counting by tens on our number line. Now we jump 10 at a time: 0, 10, 20, 30.

Definition Number Line
A number line is a line that shows numbers in order. Moving right adds by same number.

Big Numbers


Have you ever wondered how big numbers like millions or billions work? Let’s explore together:
  • A million is \(1\,000\) thousands, or \(1\,000\,000\).
  • A billion is \(1\,000\) millions, or \(1\,000\,000\,000\).
Here’s a fun way to think about these numbers:
  • A stack of \(\dollar 1\,000\) made of \(\dollar 1\) bills is as tall as a small tree (0.1 meters).
  • A stack of \(\dollar 1\,000\,000\) made of \(\dollar 1\) bills is as tall as a skyscraper (100 meters).
  • A stack of \(\dollar 1\,000\,000\,000\) made of \(\dollar 1\) bills is as tall as a hill (0.1 kilometers).
  • Elon Musk’s fortune, \(\dollar 340\,000\,000\,000\) (340 billion dollars) in \(\dollar\) 1 bills, could touch space!
Understanding these numbers helps us think about big things like money, populations, or stars in the sky. Let’s see how we can use them!

Definition Understanding Place Value
Big numbers are written using the place value system:
  • A thousand is \(1\,000\) ones, or \(1\,000\).
  • A million is \(1\,000\) thousands, or \(1\,000\,000\).
  • A billion is \(1\,000\) millions, or \(1\,000\,000\,000\).
We can write a number in three ways:
  • With digits: $$340\,120\,000\,000$$
  • With words:
    three hundred forty billion one hundred twenty million
  • In a table: