Opening the book…
After completing this section, you should be able to:
- Evaluate an exponential expression.
- Convert a Hindu-Arabic numeral to expanded form.
- Convert a number in expanded form to a Hindu-Arabic numeral.
The modern system of counting and computing isn’t necessarily natural. The fact that different symbols are used to indicate different quantities or amounts is a relatively new invention. Simple marking by scratches or dots, one for each item being counted, was the norm long into human history. The modern system doesn’t use repeated symbols to indicate more than one thing. It uses the place of a digit in a numeral to determine what that digit represents. A numeral is a symbol used to represent a number. A number is an abstract idea that represents quantity or amount.
Being clear about the difference between a numeral and a number is important. Just like a person can be called by various names, such as brother, father, husband, uncle, they all represent the same person, John Smith. The person John Smith is the number, and the names brother, father, husband, and uncle are the numerals.
The numerals we currently use are referred to as Hindu-Arabic numerals, although they have changed as time has passed. Early forms of the numerals for \(0, 1, 2, 3, 4, 5, 6, 7, 8,\) and \(9\) began in India, and passed through Persia to the Middle East. Place value was also employed in India's early systems. Once this system was in North Africa and the Middle East, it spread to Europe, eventually replacing Roman numerals. Over time, the original symbols transformed into our modern ones.
Place values in the Hindu-Arabic System rely on powers of \(10.\) They are
\begin{align*}
&~~~~~...,10^6~~~10^5~~~10^4~~~10^3~~~10^2~~~10^1~~~10^0\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{Or}\\
&...,\text{1,000,000}~~~\text{100,000}~~~\text{10,000}~~~\text{1,000}~~~\text{100}~~~10~~~1\\
\end{align*}
The system we use for counting and computing uses place values based on powers of 10. In this section, we review exponents and our positional system.
Evaluating Exponential Expressions
Most modern numerical systems depend on place values, where the quantity represented depends not only on the digit, but also on where the digit is in the number. The place value is a power of some specific number, which means most numbering systems are actually exponential expressions. An exponential expression is any mathematical expression that includes exponents. So, evaluating such an expression means performing the calculation.
In this chapter, we will be using exponents that are positive integer values. Before we do so, let’s remind ourselves about exponents and what they represent. Suppose you want to multiply a number. Let’s label that number , by itself some number of times. Let’s label the number of times . We denote that as \(a^b\). We say , or the base, raised to the th power, or the exponent. For example, if we are multiplying \(13\) by itself eight times, we write and say \(13\) to the eighth power.
When computing exponential expressions, we should be careful to remember the order of operations. Using the order of operation rules, calculations inside the parentheses are done first, then exponents are calculated, then multiplication and division calculations are performed, and then addition and subtraction.
Example \(\PageIndex{2}\): Evaluating an Exponential Expression
- \(4\times5^2+2\times6^3\)
- \(6\times8^2+3\times8^1+4\times 8^0\)
- \(3\times10^2+0\times10^1+6\times 10^0\)
- Answer 1
-
To evaluate or calculate this expression, we use the order of operations, which means the exponents are done first, then multiplications, and then addition.
\[4\times5^2+2\times6^3=4\times25+2\times216=100+432=532\nonumber\]
- Answer 2
-
To evaluate the expression, we use the order of operations, which means the exponents are done first, then the multiplications, then the additions. Remember that any base raised to the exponent \(0\) is \(1.\) \[6\times8^2+3\times8^1+4\times8^0=6\times 64+3\times 8+4\times1=384+24+4=412\nonumber\]
- Answer 3
-
To evaluate the expression, we use the order of operations, which means the exponents are done first, then the multiplications, and then the additions. Remember that any base raised to the power \(0\) is \(1.\) \[3\times10^2+0\times10^1+6\times10^0=3\times 100+0\times 10+6\times1=306\nonumber\]
Converting Hindu-Arabic Numerals to Expanded Form
When you see the number \(738,\) and you speak the number out loud, what do you say? You probably said “seven hundred thirty-eight” while wondering what point could possibly be made by asking this. What you didn’t say was “seven, and three, and eight.” A pre-K student might say that. Which should make you wonder, why?
The reason is that you’ve been taught place values, or the positions of digits in a number that determine the values of those digits. You know that in a three-digit number, the first digit is hundreds, the second digit is tens, and the last digit is ones. These place values rely on powers of \(10,\) making this system a base \(10\) system.
This sense of place value makes our system of numbers so useful. You’ve also been taught the Hindu-Arabic numeration system. This system, which uses the digits \( 0, 1, 2, 3, 4, 5, 6, 7, 8,\) and \(9,\) and also employs place value based on powers of \(10,\) is in use today.
Example \(\PageIndex{3}\): Identify the Place Value
Write the place value of each digit below in the number \(75,140,382\).
- Answer
-
The place values of each digit are shown in the table.
Figure \(\PageIndex{2}\): Place value of \(75,140,382\). Table displaying data with columns labeled: "Activity," "Thousands," "Crime," including various numerical entries with plave values of 75140382.
The Hindu-Arabic numeral system developed in India, and Aryabhata of Kusumapura is credited with the place value notation in the \(5\)th century. However, the system wasn’t as complete as it could be until. Roughly a century later, when Brahmagupta introduced the symbol for \(0\). The \(0\) is necessary to indicate that a given place value has been skipped, as in \(4,098.\) In \(4,098,\) the \(10^2\) power is skipped. Without such a symbol, \(4,098\) and \(498\) look similar. The value of both the place value notation and the introduction of the symbol \(0\) cannot be overstated, for math and the sciences.
Example \(\PageIndex{4}\): Writing a Number in Expanded Form
Write the following in expanded form.
- \(563\)
- \(4,821\)
- \(903,786\)
- Answer 1
-
Multiply each digit by its respective place value and add them. From right to left, the place value of \(3\) is \(10^0\), the place value of \(6\) is \(10^1,\) and the place value of \(5\) is \(10^2.\) So expanded form of \(563\) is
\begin{aligned}
5\times 10^2+6\times 10^1+3 \times10^0
\end{aligned} - Answer 2
-
Multiply each digit by its respective place value and add them. From right to left, the place value of \(1\) is \(10^0,\) the place value of \(2\) is \(10^1,\) the place value of \(8\) is \(10^2,\) and the place value of \(4\) is \(10^3.\) So expanded form of \(4,821\) is
\begin{aligned}
4\times 10^3+8\times 10^2+2\times 10^1+ 1 \times10^0
\end{aligned} - Answer 3
-
Similarly, we can expand \(903,786.\)
\begin{aligned}
9\times 10^5+0\times 10^4+3\times 10^3+ 7\times10^2+8\times 10^1+ 6\times10^0
\end{aligned}
Converting Numbers in Expanded Form to Hindu-Arabic Numerals
Converting from an expanded form back into a Hindu-Arabic numeral is the reverse process of expanding a number and is equivalent to evaluating the exponential expression.
Example \(\PageIndex{5}\): Converting a Number from Expanded Form to a Hindu-Arabic Numeral
Convert the following into Hindu-Arabic numerals.
- \(3\times 10^2+4\times 10^1+8 \times10^0\)
- \(5\times 10^3+0\times 10^2+9\times 10^1+ 9\times10^0\)
- \(6\times10^6+2\times 10^5+0\times 10^4+9\times 10^3+ 1\times10^2+1\times 10^1+ 7\times10^0\)
- Answer 1
-
Evaluating the expression results in:
\begin{align*}
3\times 10^2+4\times 10^1+8 \times10^0=300+40+8=348
\end{align*} - Answer 2
-
Evaluating the expression results in:
\begin{align*}
5\times 10^3+0\times 10^2+9\times 10^1+ 9\times10^0=5000+0+90+9=5,099
\end{align*} - Answer 3
-
Evaluating the expression results in:
\begin{align*}
6\times10^6+2\times 10^5+0\times 10^4+9\times 10^3+ 1\times10^2+1\times 10^1+ 7\times10^0=6,209,117
\end{align*}
Place Value to the Right of the Decimal Point
Place values after the decimal point represent fractions of one and follow a specific order, decreasing in size from left to right. Here's how they go: Tenths (\(\frac{1}{10}\)), Hundredths (\(\frac{1}{100}\)), Thousandths (\(\frac{1}{1000}\)), Ten-thousandths (\(\frac{1}{10000}\)), etc. and those place values are summarized in table.
Example \(\PageIndex{6}\): Identify the Place Value
Which number in \(1642.895\) represents the hundredths place value?
- Answer
-
\(9\) is in the hundredths place.
Videos
Referecnes
- Pink Pencil. (2022, August 6). Decimal Place Value - Explained! YouTube. Decimal Place Value [www.youtube.com]
- Learning Support Virtual. (2017, February 3). Convert Hindu-Arabic to Expanded Form. YouTube. Convert Hindu Arabic to Expanded Form [www.youtube.com]
- Mathispower4u. (2009, August 31). Exponential Notation. YouTube. Exponential Notation [www.youtube.com]
- American Public University System. (2010, January 4). Expanded form of Decimal Numbers. YouTube. Expanded form of Decimal Numbers [www.youtube.com]
This page titled 1.1: Hindu-Arabic Positional System was last modified on Fri, 24 Apr 2026 14:51:08 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Cheban Acharya via source content that was edited to the style and standards of the LibreTexts platform.
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