4.1 Hindu-Arabic 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 $a$, by itself some number of times. Let’s label the number of times $b$. We denote that as $a^b$. We say $a$, or the base, raised to the $b$th power, or the exponent. For example, if we are multiplying 13 by itself eight times, we write ${13}^8$ 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.

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, which makes this system a base 10 system.

This sense of place value is what 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.

Writing a number using these place values is writing them in expanded form. For a number with $n$ digits, the expanded form is the first digit times 10 raised to one less than $n$, plus each following digit times 10 raised to one less than the previous power of 10. For example, the number 738 would be written as $7\times {10}^2+3\times {10}^1+8\times {10}^0$.

What about a four-digit number, like 5,825? Out loud, we’d say five thousand, seven hundred twenty-five. In expanded form, it would be $5\times {10}^3+8\times {10}^2+2\times {10}^1+5\times {10}^0$. Notice that the largest exponent is one less than the number of digits, and that the exponents go down by one as we move through the number.

Converting Numbers in Expanded Form to Hindu-Arabic Numerals

Converting from expanded form back into a Hindu-Arabic numeral is the reverse process of expanding a number, and is equivalent to evaluating the exponential expression.