Brief Notes on Significant Errors, Figures, Exact Numbers

Introduction

While making measurements in practical and theory work, we tend to encounter errors. The errors might be due to technical or human errors. No physical quantities have accurate values. To get accurate results, it is important to understand proper measurement. This is likely to differ from the unknown true value of the quantity and the objective of significant figures measurement. 

Significant figures

Significant figures are figures or digits of a numerical amount of the actual distribution. 

Significant figures in a number are the digits that are as close to the actual accuracy or exactness of measurement. They include any non-zero digits. The considerably significant is the digit with the highest proponent value, barely the left-most significant figure. The small sign is the digit with the lowest exponent value or barely the right-most significant figure.

The objective of significant figures measurement resolution

Suppose a number indicating the result of distributions like length, pressure, volume, or has more amounts than the number of digits enabled by the measurement conclusion. In that case, the completed different numbers as allowed by the measurement are useful in calculations. 

Only these digits can be significant figures.

For example, consider the height proportion as 449.7 mm.

This implies the smallest interval between marks on the ruler is 4 mm. The first three digits, 4, 4, and 9, of 449 mm are specific. They are significant figures. 

The numbers that are uncertain but useful are also considered significant figures. 

In the above example, the last digit 7 is added as 0.7 mm. It is a significant figure despite being uncertain.

The rule for significant figures is that the last digit in a measurement determination is the first digit with some uncertainty.

To deduce the digit of considerable digits in a value, begin with the first estimated value on the left side and count the number of digits till the last digit on the right side. 

Significant figures denote the accuracy of the measuring tool used to calculate a consequence.

Rules to specify the significant number in the exact number 

Following rules guiding the significant figures in a number require the knowledge of which digits are useful. 

• Non-zero digits within the given measurement or reporting resolution are significant.

For example, 236.250007 has seven significant figures if the resolution is 0.0001: 2, 3, 6, 2, 5, 0, and 0.

• Zeros between two significant non-zero digits are significant trapped zeros.

For example, 3.400 has four significant figures (3, 4, 0, and 0) if the measurement resolution authorises them.

• Trailing zeros in an integer may not be significant, depending on the measurement or resolution. The initial three non-zero numbers are significant since the trailing zeros are neither valid nor necessary in an example.  

• An exact number has an infinite number of significant figures. If the number of apples in a bag is 5 exact numbers, this number is 5.0000, with infinite trailing zeros to the right of the decimal point. As a result, 5 does not affect the digit of significant figures or digits in the result of calculations with it.

• The significance of trailing zeros in a number not containing a decimal point can be ambiguous.

Types of Zeros’ significant figures 

Zero Type 1: 

Space carrying zeros in amounts smaller than one. 

Let us consider two numbers, including zeros:

0.00500 

0.03040 

These zeros assure only space occupants. 

Zeros are there to put the decimal point in its exact area. 

They do not implicate measurement determinations. 

Zero Type 2:

The zeros are drawn to the left side of the decimal point on digits less than one. 

To simplify this, consider a number like 0.00500.

The initial zero to the left of the decimal point is put there by a pattern in this number. Its single function is to indicate that there is a decimal point.

If written without a decimal, the number would be 00500;  it can be an error because zero should not be excluded.

Zero Type 3: Trailing zeros in a whole number. 

By definition, it does not need a measurement resolution; therefore, they are irrelevant. 

So we can say that any zeros between two considerable digits are significant.

To specify the number of significant figures, utilise these 3 rules:

1. i) Non-zero digits are always significant.
2. ii) Any zeros between two significant digits are significant.

iii) A final zero or trailing zeros in the decimal portion are the only ones significant.

Conclusion

We know that no device can give 100% accuracy. To gain accuracy and precision, we have to study the significant figure. Significant figures play an important role in physics and mathematics.

A significant figure helps to differentiate between accuracy and exactness. Significant figures indicate the accurateness of the calculated value. Using significant figures allows the scientists to know how accurate the answer is or how much the uncertainty is.