Rounding and Significant Figures

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2.3Rounding and Significant Figures


In these revision notes for Rounding and Significant Figures, we cover the following key points:

  • What is rounding? Why do we use it?
  • How many types of rounding are there? What are the corresponding digits that are associated to each of them?
  • How do we round numbers to the nearest ten?
  • How do we round numbers to the nearest hundred/thousand and more.
  • How to perform rounding in a number axis?
  • How to round non-whole numbers?
  • What are significant figures?
  • How to find the number of significant figures in a given value?
  • How are significant figure used in practice?

Rounding and Significant Figures Revision Notes

Rounding means replacing a number with an approximate value that has a shorter and simpler representation.

The value of a rounded number is slightly different from the original number. A rounded integer always ends with zero. We use the symbol (≈) to represent rounding in expressions.

There are two types of rounding: (1) rounding up, and (2) rounding down. When a number is rounded up, its value increases to become the nearest number ending with zero to the original number. On the other hand, when a number is rounded down, its value decreases to become the nearest number ending with zero to the original number.

Rule: "A number is rounded down when the digit that becomes zero in the original number was 1, 2, 3 and 4, while a number is rounded up when the digit that becomes zero after rounding was previously 5, 6, 7, 8 and 9".

When a number is rounded to the nearest ten it has at least one zero at the end. When a number ends with 0, it has no need to be rounded to the nearest ten. When a number ends with 1, 2, 3, 4, it becomes smaller after rounded to the nearest ten. On the other hand, when a number ends with 5, 6, 7, 8 and 9, it becomes bigger after rounded to the nearest ten.

The number will have at least two zeroes at the end after being rounded to the nearest hundred. We have to check the value of tens to known whether the number must be rounded up or down. The same rules are applied, i.e. when the tens digit ends with 0, 1, 2, 3 and 4, the number is rounded down to the nearest hundred, so its value decreases. On the other hand, if the value of tens digit is 5, 6, 7, 8 and 9, the number increases, as it takes the value of the upper nearest hundred.

The same procedure is also used when rounding a number to the nearest thousand. In this case, we look at the hundreds digit (the third from the right). If it is 0, 1, 2, 3 and 4, then the number is rounded down to the nearest thousand. This means the rounded number becomes smaller than the original. On the other hand, when the hundreds digit is 5, 6, 7, 8 and 9, the number is rounded up to the nearest thousand and the number therefore becomes bigger. obviously, the number will have at least 3 zeroes at the end.

Rounding process can be understood easier if numbers are shown in the number axis.

When numbers involved are not integers, we may need to round them to the nearest integer. In order to do this, the original number must have at least one digit after the decimal point. Then, the known rules are used.

Another way of rounding numbers is to count only the first few digits (we call them "figures") that have a value attached to them. This method of rounding is called significant figures and it's often used with larger numbers, or very small numbers.

The following rules are applied to find the number of significant figures in a certain number:

  1. The zeroes at the end of whole numbers are not counted as significant figures.
  2. The zeroes at the beginning of any decimal number are not counted as significant figures.
  3. Any zero in-between two non-zero digits (both in whole number as well as in decimals) are always counted as significant figures.

We can identify the number of significant figures by writing the number in the standard form. In such cases, only the part of the number comes before powers of ten is considered as significant.

Significant figures have a wide range of applications in practice, such as in calculating the dimensions of objects, etc.

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