Rounding to the Greatest Place Value

When you express a fraction in decimal form, it may be accurate to more places than you need or are able to use. Long decimals are unwieldy, so scientists often round them to make them easier to handle, even though this sacrifices accuracy. They also round large whole numbers that have too many digits to manage. When rounding to the greatest place value, you basically keep one number – the farthest non-zero one to the left – and you make all the numbers to the right of it zero.

TL;DR (Too Long; Didn’t Read)

The greatest place value of a number is the first non-zero digit on the left in that number. You round up or down according to which numeral is to the right of the greatest place value.

Rounding Rules

When you round a digit in a number series, you don’t have to look at all the digits that follow it. The only one that’s important is the one immediately to the right. If it’s 5 or larger, you add one to the digit you’re rounding and you make all the digits to the right of it zero. This is called rounding up. For example, you would round 5,728 up to 6,000. If the digit to the right of the one you’re rounding is smaller than 5, you leave the one you’re rounding as it is. This is called rounding down. For example, 5,213 would round down to 5,000.

The Greatest Place Value

In any number, whether it’s a decimal fraction or a whole integer, the non-zero digit farthest to the left is the one with the greatest place value. In a decimal fraction, this digit is the first non-zero one to the right of the decimal, and in a whole integer, it’s the first digit in the number series. For example, in the fraction 0.00163925, the digit with the greatest place value is 1. In the whole integer 2,473,981, the digit with the greatest place value is 2. When you round the digit with the greatest place value in these two examples, the fraction becomes 0.002 and the integer becomes 2,000,000.

Scientific Notation

Another way to make large numbers more manageable is to express them in scientific notation. To do this, you write the number as a single digit followed by a decimal with all the rest of the digits following the decimal, and then you multiply by a power of 10 equal to the number of digits. For example, the number 2,473,981 when expressed in scientific notation becomes 2.473981 x 106. You can also express fractions in scientific notation. The decimal fraction 0.000047039 becomes 4.7039 x 10-5. Note that for fractions, you count the digits to the left of the decimal, including the digit with greatest place value, when calculating the power, and you make the power negative.

It’s common to round numbers in scientific notation, and when you round to the greatest place value, you round the digit before the decimal and omit all the other digits. Thus, 2.473981 x 106 becomes simply 2 x 106. Similarly, 4.7039 x 10-5 becomes 5 x 10-5.