What is Significant Digit in physics simple explanation

WHAT IS SIGNIFICANT DIGITS

When a measurement is made, a numerical value is read generally from some calibrated scale. To measure the length of a body we can place a metre scale  in contact with the body. One end  of the body may be made to coincide with the zero of the metre scale and the reading just in front of the other end is noted from the scale. When an electric current is measured with an ammeter the reading of the pointer on the graduation of the ammeter is noted. The value noted down includes all the digit that can be directly read from the scale and one doubtful digit at the end. The doubtful digit corresponds to the eye estimation within the smallest subdivision of the scale. This smallest subdivision is known as the least count of the instrument. In a metre scale, the major graduations are at an interval of one centimetre and ten subdivisions are made between two consecutive major graduations. Thus, the smallest subdivision measures a millimetre. If one end of the object coincides with the zero of the metre scale, the other end may fall between 10.4 cm and 10.5 cm mark of the scale (figure 4.1). We can estimate the distance between the 10.4 cm mark and the edge of the body as follows.

Figure 4.1

We mentally divide the 1mm division in 10 equal parts and guess on which part is the edge falling. We may note down the reading as 10.46 cm. The digits 1, 0 and 4 are certain but 6 is doubtful. All these digits are called significant digits. We say that the length is measured up to four significant digits. The rightmost or the doubtful digit is called the least significant digit and the leftmost digit is called the most significant digit.

There may be some confusion if there are zeroes at the right end of the number. For example, if a measurement is quoted as 600 mm and we know nothing about the least count of the scale we cannot.    be sure whether the last zeros are significant or not. If the scale had marking only at each metre then the edge must be between the marks 0 m and 1m and the digit 6 is obtained only through the eye estimation. Thus, 6 is the doubtful digit and the zeros after that are insignificant. But if the scale had markings at centimetres, the number read is 60 and these two digit are significant, the last zero is insignificant. If the scale used had markings at millimetres, all the three digits 6, 0, 0 are significant To avoid confusion one may report only the significant digit and the magnitude may be correctly described by proper powers of 10. For example, if only 6 is significant in 600mm we may write it as 6ｘ10^2 mm. If 6 and the first zero are significant we may write it as 6ｘ10^2 mm and if all the three digits are significant we may write it as 6.00 ☓ 10^2 mm.

If the integer part is zero any number of continuous zeros just after the decimal part is insignificant. Thus, the number of significant digits in 0.0023 is two and in 1.0023 is five.

SIGNIFICANT DIGITS IN CALCULATIONS 

When two or more numbers are added, subtracted, multiplied or divided, how to decide about the number of significant digits in the answer? For example, suppose the mass of A is measured to be 12.0 kg and of another body B to be 7.0 kg. What is the ratio of the mass of A to the mass of B ? Arithmetic will give this as 

12.0 / 7.0 = 1.714285.....

However, all the digits of this answer cannot be significant. The zero of 12.0 is a doubtful digit and the zero of 7.0 is also doubtful. The quotient cannot have so many reliable digits. The rules for deciding the number of significant digit in a arithmetic calculation are listed below.

1. In  a multiplication or division of two or more quantities, the number of significant digits in the answer is equal to the number of significant digits in the quantity which has the minimum number of significant digits. Thus, 12.0 / 7.0 will have two significant digits only.

The insignificant digits are dropped from the result if they appear after the decimal point. They are replaced by zeros if they appear to the left of the decimal point. The least significant digit is rounded according to the rules given below. If the digit next to the one rounded is more than 5, the digit to be rounded is increased by 1. If the digit next to the one rounded is less than 5, the digit to be rounded is left unchanged. If the digit next to the one rounded is 5, then the digit to be rounded is increased by 1 if it is odd and is left unchanged if it is even.

2. For addition or subtraction write the numbers one below the other with all the decimal point in one line. Now locate the first column from left that has a doubtful digit. All digits right to this column are dropped from all the numbers and rounding is done to this column. The addition or subtraction is now performed to get the answer.