Integers and Modern Applications for the Uninitiated

The value of each digit depends on its place. The rightmost digit is the ones digit, the second digit from the right is the 10s digit, the third digit from the right is the hundreds digit, the fourth digit from the right is the thousands digit, and so on. When reading a number, we multiply the rightmost digit by \(1=10^0\text{,}\) the second digit from the right by \(10=10^1\text{,}\) the third digit by \(100=10^2\text{,}\) the fourth digit from the right by \(1000=10^3\text{,}\) and so on. We call \(1,10,100,1000,\dots\) the values of the places of the digits. The place value of the \(n\)-th digit from the right is \(10^{n-1}\) (remember that the place value of the rightmost digit is \(10^0=1\)). So the values of the places of a number with \(n\) (decimal) digits are

\(10^{n-1}\text{,}\) \(10^{n-2}\text{,}\) \(\dots\text{,}\) \(10^4=10\,000\text{,}\) \(10^3=1000\text{,}\) \(10^2=100\text{,}\) \(10^1=10\text{,}\) \(10^0=1\text{.}\)

When we want to emphasize the value of the place of each digit we give the base 10 expansion:

\(a_{n-1}a_{n-2}\dots a_3 a_2 a_1 a_0\) \(= (a_{n-1}\cdot 10^{n-1})+(a_{n-2}\cdot10^{n-2})+\dots\)\(+ (a_3\cdot 10^3)+ (a_2\cdot 10^2)+(a_1\cdot 10^1)+ (a_0\cdot 10^0)\)

The digit \(a_0\) is the first digit from the right. Because it is in the “ones place”, it has the value 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. The second digit from the right, which we called \(a_1\text{,}\) is in the “tens place”, so it has the value 0, 10, 20, 30, 40, 50, 60, 70, 80, or 90. The third digit from the right (\(a_3\)) is in the “hundreds place”, so it has the value 0, 100, 200, 300, 400, 500, 600, 700, 800, or 900. The pattern continues.

In the video in Figure 11.1.2 we recap the material covered above and present examples.

We give the extended base \(10\) expansion of three numbers.