Every digit in a number has a place value.
In this lesson, through definition and example, we will learn what a place value is and how to find the place value of a specific digit. At the end of the lesson, you can continue practicing with a quiz.
Money and Place Value
Whether we have it or not, we are all very familiar with money and how to write different amounts of money. Consider the amount $154.37. We know these numbers preceded by the dollar sign represents one hundred fifty-four dollars and thirty-seven cents, but have you ever thought about what each of the digit’s values are in a given amount of money? For example, the 5 in $154.37 isn’t just worth five pennies.
Its location is two units to the left of the decimal point, so in terms of money, the 5 is worth fifty dollars. Similarly, the 1 is worth one hundred dollars, the 4 is worth four dollars, the 3 is worth thirty cents, and the 7 is worth seven cents. We see that the value of the digit depends on its location in the number. In mathematics, we call this a place value, and it applies to all numbers, not just money.
Further Exploring Place Value
The place value is the value of the location of a digit in a number. The place values are determined by how many places the digit lies to the right or the left of the decimal point.The place values to the left of the decimal point are increasing powers of 10:
- The first place to the left of the decimal point is the ones place, or 10^0.
- The second place to the left of the decimal is the tens place, or 10^1.
- The third place to the left of the decimal is the hundreds place, or 10^2.
The pattern continues.
In general, the nth place to the left of the decimal has place value 10^(n – 1). For example, consider the number 136,774.8591. The digit 3 in this number falls five places to the left of the decimal. This place is determined by 10^(5-1) = 10^4 = 10,000, so this is the ten-thousands place. Therefore, the 3 in 136,774.
8591 has place value 3 * 10,000 = 30,000.The place values to the right of the decimal are decreasing in powers of 10:
- The first place to the right of the decimal is the tenths place, or 10^-1.
- The second place to the right of the decimal is the hundredths place, or 10^-2.
- The third place to the right of the decimal is the thousandths place, or 10^-3, and so on.
In general, the nth place to the right of the decimal has place value 10^-n. For example, in our number 136,774.8591, the 9 falls three places to the right of the decimal, so it falls in the 10^-3 = 1 / 1000 place, or in the thousandths place. Therefore its place value is 9 / 1000, or nine-thousandths.
The following image illustrates the different place values of a number:
This is valuable knowledge. For instance, assume someone says to choose a digit in the number 12345.
6789, and they will give you that much money. If we were not familiar with place value, we may be inclined to pick the highest digit, which would be 9; however that would mean we would receive 0.0009 cents.
Do they even make a coin that small? Now that we are familiar with place value, we would quickly choose the digit 1, because though it is the smallest digit, it is worth the most money, namely $10,000.
Steps to Finding the Place Value of a Digit
The steps to find the place value of a digit are as follows:
- Count the number of places the digit is away from the decimal point. Call it n places.
- Determine if the digit falls to the right or left of the decimal point.
- If the digit falls to the left of the decimal point, then it is in the 10^(n-1) place, so to find the place value of the digit, we multiply the digit by 10^(n – 1). If the digit falls to the right of the decimal point, then it is in the 10^-n place, so to find the place value of the digit, we multiply the digit by 10^-n.
For example, let’s consider our money offer example with the number 12345.6789. If we chose the digit 4, how much money would we receive? To find this out, we find the place value of 4.
Following our steps, we see that 4 is two places to the left of the decimal, so the place value of 4 is 4 * 10^(2 – 1) = 4 * 10^1 = 40. Thus, if we chose 4, we would get $40.
We may have never thought about why the 4 in $469.23 actually represents $400, but now we know. It all has to do with place value.
The place value of a digit is the value of the location of the digit in a number. We can find the place value of a digit by finding how many places the digit is to the right or left of the decimal point in a number. If it is n places to the left of the decimal point, we multiply the digit by 10^(n – 1) to get the digit’s place value. If it is n places to the right of the digit, we multiply the digit by 10^-n to get the digit’s place value.