Question

1.56 Write the following numbers in scientific notation: (a) 131,\(000.0 ;\) (b) 0.00047 ; (c) 210,006 (d) 2160.5 .

Short answer

(a) \( 1.31 \times 10^5 \), (b) \( 4.7 \times 10^{-4} \), (c) \( 2.10006 \times 10^5 \), (d) \( 2.1605 \times 10^3 \)

Step-by-step solution

Understanding Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10. The general form is: \( a \times 10^n \), where \( 1 \leq a < 10 \).

- Convert 131,000.0 to Scientific Notation

Identify the significant digits which is 1.31. Count the number of places the decimal moves to get from 1.31 to 131,000.0. It moves 5 places to the right. So, \( 131,000.0 = 1.31 \times 10^5 \).

- Convert 0.00047 to Scientific Notation

Identify the significant digits which is 4.7. Count the number of places the decimal moves to get from 4.7 to 0.00047. It moves 4 places to the left. So, \( 0.00047 = 4.7 \times 10^{-4} \).

- Convert 210,006 to Scientific Notation

Identify the significant digits which is 2.10006. Count the number of places the decimal moves to get from 2.10006 to 210,006. It moves 5 places to the right. So, \( 210,006 = 2.10006 \times 10^5 \).

- Convert 2160.5 to Scientific Notation

Identify the significant digits which is 2.1605. Count the number of places the decimal moves to get from 2.1605 to 2160.5. It moves 3 places to the right. So, \( 2160.5 = 2.1605 \times 10^3 \).

Key concepts

significant digits

Significant digits, also known as significant figures, are the meaningful digits in a number. These digits provide information about the precision of the number. When writing numbers in scientific notation, it is essential to identify and use significant digits correctly. For instance, in the number 131,000.0, the significant digits are 1, 3, and 1. Ignoring the trailing zeros, these digits need to be represented in the scientific notation format. For example, 131,000.0 becomes 1.31 in scientific notation because we only focus on the significant digits, which are 1, 3, and 1.

standard form

The standard form is another way of referring to scientific notation. When we convert a number to its standard form, we express it as a product of a number between 1 and 10, and a power of 10. This makes it easier to handle very large or very small numbers. For example, when converting 0.00047 to its standard form, we get 4.7 because 4.7 is a number between 1 and 10. The number of decimal places we move gives us the exponent for the power of 10. Thus, 0.00047 becomes 4.7 \(\times\) 10\(^{-4}\) in scientific notation.

powers of 10

Powers of 10 are a key component of scientific notation. They indicate how many times the base number (10) is multiplied by itself. For positive exponents, such as 10\(^5\), it shows 10 multiplied by itself five times. For negative exponents, such as 10\(^{-4}\), it indicates the reciprocal of 10 multiplied by itself four times. Translating this into scientific notation, consider 210,006, which becomes 2.10006 \(\times\) 10\(^5\). Here, the exponent 5 tells us the decimal point has moved five places to the right. Similarly, 0.00047 becomes 4.7 \(\times\) 10\(^{-4}\), where the exponent -4 indicates a leftward shift of the decimal point.

decimal places

Decimal places refer to the number of digits to the right of the decimal point in a number. Understanding and correctly managing decimal places is crucial for precise calculations and accurate scientific notation. When converting numbers, you must count the decimal places moved. For instance, converting 2160.5 to scientific notation involves identifying 2.1605 as significant digits. You count how many decimal places to shift from 2.1605 to 2160.5, which is three places to the right. Thus, 2160.5 is written as 2.1605 \(\times\) 10\(^3\) in scientific notation.