Question

Rewrite the following numbers in scientific notation as indicated: (a) 630,000 with five significant figures (b) 1300 with three significant figures (c) 794,200,000,000 with four significant figures

Short answer

(a) \(6.3000 \times 10^5\), (b) \(1.30 \times 10^3\), (c) \(7.942 \times 10^{11}\).

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small in the form of a product involving a power of ten. The formula is typically written as: \[a \times 10^{n}\], where \(a\) is a number between 1 and 10, and \(n\) is an integer.

Convert 630,000 to Scientific Notation

Identify the significant figures and place the decimal after the first digit to achieve five significant figures: 6.3000. The number is 630,000, or \(6.3000 \times 10^5\), because you count five places to the right to fall on the end zero.

Convert 1300 to Scientific Notation

Identify the significant figures (the first three digits 130) and place the decimal after the first non-zero digit: 1.30. The number is 1300, so \(1.30 \times 10^3\), because you move the decimal three places to the left to reach 1.30.

Convert 794,200,000,000 to Scientific Notation

Identify the significant figures (the first four significant figures are 7.942). Place the decimal after the first digit to get 7.942. Since the original number is 794,200,000,000, express it as \(7.942 \times 10^{11}\), because we moved the decimal 11 places to reach the significant figure.

Key concepts

Understanding Significant Figures

Significant figures are all the non-zero numbers, any zeros between them, and any trailing zeros in the decimal part. They tell us how precise a number is in scientific notation. Consistently using significant figures ensures clarity and accuracy in mathematical and scientific calculations. For example, the number 630,000 expressed with five significant figures becomes 6.3000.

• Each digit in a number that isn't a leading or trailing zero (unless those zeros are after a decimal point) is significant.
• In the numbers like 1300, when expressed in scientific notation, trailing zeros become important to maintain three significant figures, turning into 1.30.
• This importance arises particularly in measurements and calculations where precision is crucial.

Always remember, significant figures are key to representing the true value with the correct precision.

The Power of Ten in Scientific Notation

The power of ten is what sets scientific notation apart and acts as a multiplier. It indicates where the decimal place is relative to the original number. The exponent tells you how many places you move the decimal point.

• For large numbers, a positive power of ten shifts the decimal to the right. For example, in 630,000, we use 10 raised to the power of 5 because we move five places to reach the end zeroes: 6.3000 x 10^5.
• For smaller numbers, a negative power would shift the decimal to the left.
• In the case of 1300 being 1.30 x 10^3, the exponent '3' represents moving the decimal back three places to its original position.

This flexibility in altering the location of the decimal point using powers of ten makes scientific notation a versatile tool for handling vast numerical ranges.

Representing Large Numbers

When faced with very large numbers, scientific notation simplifies their representation. This method turns cumbersome and overwhelming figures into a manageable format by focusing on significant figures and powers of ten.

• Take 794,200,000,000, for example. In scientific notation, it becomes 7.942 x 10^11. Here, 7.942 retains the important digits, while 10^11 captures the magnitude.
• Such a way of expressing numbers helps avoid errors in reading or writing down long strings of digits.
• Also, calculations become easier as it allows operations such as multiplication and division to be performed more straightforwardly.

By reducing the complexity associated with large numbers, scientific notation helps scientists, mathematicians, and students better manage and understand vast quantities in computations.