Question

Convert each number from enginecring notation to decimal notation. $$\left(1.557 \times 10^{2}\right)+\left(9.000 \times 10^{-1}\right)$$

Short answer

The sum of \(1.557 \times 10^{2}\) and \(9.000 \times 10^{-1}\) in decimal notation is 156.6.

Step-by-step solution

Understanding Engineering Notation

Engineering notation is a version of scientific notation in which the exponent of ten must be divisible by three. However, in this exercise, we are using standard scientific notation, where any exponent is allowed. Our task is to convert numbers from scientific notation to decimal notation.

Convert the First Term to Decimal Notation

The first term is given as \(1.557 \times 10^{2}\). To convert it to decimal notation, we need to move the decimal point two places to the right since the exponent of ten is 2. The result is \(155.7\).

Convert the Second Term to Decimal Notation

The second term is \(9.000 \times 10^{-1}\). To convert it to decimal notation, we move the decimal point one place to the left because the exponent is -1. The resulting number is \(0.9000\).

Sum the Converted Numbers

After converting both terms to decimal notation, we simply add them together. \(155.7 + 0.9000 = 156.6\). Note that we have maintained all significant figures in the second term for accuracy.

Key concepts

Understanding Scientific Notation

Scientific notation is a concise way to express very large or very small numbers. It's particularly handy in engineering and science to simplify measurements and calculations.

Essentially, a number in scientific notation is written as the product of two numbers: a coefficient and a power of 10. The coefficient is a number greater than or equal to 1 and less than 10, and the exponent on the 10 tells you how many places to move the decimal point. For instance, in the term \(1.557 \times 10^{2}\), \(1.557\) is the coefficient and \(10^{2}\) indicates that the decimal point in \(1.557\) should move 2 places to the right, transforming it into \(155.7\).

When working with scientific notation, it's important to be comfortable with powers of 10 and the movement of decimal points, as these are the key elements at play in the notation system.

Mastering Decimal Conversion

Converting a number from scientific notation to decimal notation involves moving the decimal point. This might seem daunting at first, but once you understand the rule — move the point to the right for positive exponents and to the left for negative exponents — it becomes straightforward.

For example, with \(1.557 \times 10^{2}\), our positive exponent of 2 tells us to shift the decimal two places right, resulting in \(155.7\). Conversely, with \(9.000 \times 10^{-1}\), we deal with a negative exponent, which means we move the decimal one place to the left, giving us \(0.9000\).

Remembering the direction to move the decimal based on the sign of the exponent is key to mastering decimal conversion from scientific notation.

The Role of Significant Figures

Significant figures play a crucial role in maintaining the precision of calculations in science and engineering. They indicate how many digits in a number are meaningful and should be retained throughout calculations.

For instance, in the number \(9.000\), all four digits are significant because the trailing zeros indicate measured precision. When converting to decimal notation, all significant figures must carry over. So, \(9.000 \times 10^{-1}\) becomes \(0.9000\), where we keep all the zeros after the decimal to reflect precision. Similarly, adding \(155.7\) and \(0.9000\) ensures we respect the significance by considering all digits, leading to the correct sum of \(156.6\).

The concept of significant figures is critical for accurate scientific communication, ensuring that any operation, including conversions or calculations, doesn't artificially inflate or understate the precision of the figures involved.