Question

The first six digits of the square root of 2 are \(1.41421\). What is the square root of 2 to four significant figures?

Short answer

The square root of 2 to four significant figures is 1.414.

Step-by-step solution

Understand Significant Figures

Significant figures are the digits in a number that contribute to its accuracy. In this case, we need to round the square root of 2 to four significant figures.

Identify the Relevant Digits

The first six digits of the square root of 2 given are 1.41421. From these digits, we need to identify the first four significant digits.

Rounding to Four Significant Figures

To round 1.41421 to four significant figures, we look at the fifth digit. The number is 1.41421, so the first four significant figures are 1.414. The fifth digit is 2, which is less than 5, so we round down.

Stating the Rounded Number

After rounding, the square root of 2 to four significant figures is 1.414. Thus, the rounded value to four significant figures does not change from the first digits of the square root.

Key concepts

Understanding Square Roots

The square root of a number is another number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. Square roots are useful in many areas of mathematics, especially in solving equations and understanding geometrical concepts. In our problem, we are dealing with the square root of 2. The exact decimal representation of the square root of 2 is an irrational number, which means it goes on forever without repeating. The first few digits of this number are 1.41421, but it actually continues infinitely. Finding square roots is made easier with calculators or tables, but understanding the concept is essential for accurate calculations and problem-solving.

Mastering the Concept of Rounding

Rounding is a numerical method used to reduce the number of digits in a number while maintaining its overall value close to the original. It's important when you need a simpler or more concise representation of a number. In the context of significant figures, rounding helps in maintaining precision while limiting how many digits we keep. For instance, when we rounded 1.41421 to four significant figures, we referred to the next digit, which was 2. If this digit was less than 5, as it indeed was, we left the fourth digit unchanged. This simplifies our number without sacrificing much accuracy, resulting in 1.414 in our example. Understanding rounding is essential in scientific fields, economics, and everyday mathematics, where approximation is necessary.

Significance of Digits in Numerical Precision

Digits play a crucial role in determining the precision of a number. They are all the numbers from 0 through 9, and each digit's position signifies its contribution to the number's value. In any numerical representation, significant figures are those which indicate the number's exactness. When an exercise asks for a number to specific significant figures, like four in our task, we need to identify the first four figures that contribute to its precision. But not all digits are significant; leading zeros are not considered significant, and neither are trailing zeros when they only serve as placeholders. By identifying the significant digits, precision is maintained, supporting accuracy in everything from scientific measurements to financial calculations. This idea was crucial in determining the square root of 2 in our problem, ensuring a balance between accuracy and simplicity.