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Section 11.1 Decimal Representation

In the decimal system, every number is written with the 10 digits

\begin{equation*} 0, 1, 2, 3, 4, 5, 6, 7, 8, \text{ and } 9\text{.} \end{equation*}

The value of each digit depends on its location. The right most digits are the ones, the second digit from the right are the 10s, the third digit from the right are the hundreds, the fourth digit from the right are the thousands and so on. When reading a number we multiply the right most 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 form the right by \(1000=10^3\) 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 \(10^0=1\)). Thus the values of the places of a number with \(n\) (decimal) digits are

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

In the following we denote the digits of an \(n\)-digit decimal number by

\begin{equation*} a_{n-1}, a_{n-2}, \dots, a_3, a_2, a_1, a_0 \end{equation*}

where \(a_0\) is the rightmost digit and \(a_{n-1}\) is the leftmost digit and each digit is in an element \(\Z_{10}=\{0,1,2,3,4,5,6,7,8,9\}\text{.}\)

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

Table 11.1.1. For selected numbers \(a\text{,}\) we give the \(a\) in decimal (base \(10\)) representation, the digits of the decimal representation of \(a\) explicitly by place, and the base \(10\) expansion of \(a\text{.}\) Recall that \(10^1=10\) and \(10^0=1\text{.}\)
\(a\) in base \(10\) digits of \(a\) base \(10\) expansion of \(a\)
base \(10\) \(10^4\) \(10^3\) \(10^2\) \(10^1\) \(10^0\)
\(1\) \(1\) \(1\cdot 1\)
\(10\) \(1\) \(0\) \((1\cdot 10)+(0\cdot 1)\)
\(100\) \(1\) \(0\) \(0\) \((1\cdot 10^2)+(0\cdot 10)+(0\cdot 1)\)
\(562\) \(5\) \(6\) \(2\) \((5\cdot 10^2)+(6\cdot 10)+(2\cdot 1)\)
\(2341\) \(2\) \(3\) \(4\) \(1\) \((2\cdot 10^3)+(3\cdot 10^2)+(4\cdot 10)+(1\cdot 1)\)
\(12004\) \(1\) \(2\) \(0\) \(0\) \(4\) \((1\cdot 10^4)+(2\cdot 10^3)+(0\cdot 10^2)+(0\cdot 10)+(4\cdot 1)\)
\(56784\) \(5\) \(6\) \(7\) \(8\) \(4\) \((5\cdot 10^4)+(6\cdot 10^3)+(7 \cdot 10^2)+(8\cdot 10)+(4\cdot 1)\)

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

Figure 11.1.2. Decimal Numbers by Matt Farmer and Stephen Steward

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

  1. \(562\)\(=5\cdot 10^2+6\cdot 10+2\cdot 1\)\(= 5\cdot 100+6\cdot 10 + 2\cdot 1\)

  2. \(56200\)\(=5\cdot 10^4+6\cdot 10^3+2\cdot 10^2+0\cdot 10^1+0\cdot 10^0\)\(=5\cdot 10000+6\cdot 1000+2\cdot 100+0\cdot 10+0\cdot 1\)

  3. \(2001\)\(=2\cdot 10^3+0 \cdot 10^2+0\cdot 10+1\cdot 1 \)\(= 2\cdot 1000+0 \cdot 100+0\cdot 10+1\cdot 1\)

See Table 11.1.1 for further examples.

Now find the base \(10\) expansion of a natural number.