Question

Write these numbers in scientific notation by counting the number of places the decimal point is moved. a) 0.000552 b) 1,987 c) 0.00000000887

Short answer

a) \(5.52 \times 10^{-4}\), b) \(1.987 \times 10^{3}\), c) \(8.87 \times 10^{-9}\).

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way to express numbers as a product of a coefficient and a power of 10. The number is typically of the form \(a \times 10^n\), where \(1 \leq |a| < 10\) and \(n\) is an integer.

Convert 0.000552 to Scientific Notation

To convert 0.000552 to scientific notation, start by identifying the first nonzero digit, which is '5'. Move the decimal point right until it is just after the '5'. This requires moving the decimal 4 places to the right.The number is now written as \(5.52\). Hence, the scientific notation is \(5.52 \times 10^{-4}\).

Convert 1,987 to Scientific Notation

For 1,987, identify where the decimal point should be to create a number between 1 and 10. The first digit is '1', so place the decimal right after it, making it \(1.987\). Since we originally have the decimal point after 987 (3 places right), the exponent will be positive 3.The scientific notation is \(1.987 \times 10^{3}\).

Convert 0.00000000887 to Scientific Notation

Similarly, for 0.00000000887, find the first nonzero digit, which is '8', and move the decimal point so it is right after '8'. Move the decimal 9 places to the right.Now it reads \(8.87\). Thus, the scientific notation is \(8.87 \times 10^{-9}\).

Key concepts

Decimal Point

The decimal point is a crucial element in both ordinary numbers and scientific notation. It is a symbol used to separate the whole number part from the fractional part of a number. Understanding where the decimal point is in a number is essential when converting it into scientific notation.
In scientific calculations, moving the decimal point is necessary to adjust the numbers into a standardized form. By relocating the decimal, you can turn a very large or very small number into a more manageable format.
For instance, in the number 0.000552, the decimal point moves four places to the right to simplify it to 5.52. This readability shift aids mental math and aligns with the scientific form by locating the decimal after the first nonzero digit.

Exponents

Exponents play a pivotal role in scientific notation as they indicate the power of 10 that the number should be multiplied by. Essentially, an exponent tells us how many times the base (10) is used as a factor.
In scientific notation, when we move a decimal point, the exponent reflects the shift. A positive exponent signals a leftward shift of the decimal point; a negative exponent indicates a rightward shift.
For example, the number 1,987 converts to 1.987 using a positive exponent 3, as the decimal moves three places left. Conversely, 0.000552 has an exponent of -4, showing a shift four places right.
Understanding exponents helps in estimating magnitudes and eases complex calculations by simplifying the notation of large or tiny numbers.

Nonzero Digits

Nonzero digits are integral to the concept of scientific notation. They determine the number part, which must be between 1 and 10.
Identifying the first nonzero digit in a sequence is the key step in converting any number to scientific notation. This digit dictates where the new position of the decimal point should be.
For instance, with 0.00000000887, the first nonzero digit is '8'. Hence, the decimal is moved to immediately follow the '8', resulting in the significant number 8.87.
Always ensure the part before the multiplication sign in scientific notation belongs to the range of 1 to less than 10, maintaining clarity and efficiency in calculations.

Powers of 10

Powers of 10 are foundational to understanding scientific notation. They reflect the number of tens multiplied together and give the scale factor needed to convert the simplified number back to its ordinary form.
Every movement of the decimal point in a number increases or decreases the power of 10 in its scientific notation form. The versatility of powers of 10 enables us to express very large numbers, like 1,987 or minuscule amounts, such as 0.00000000887, easily.
For example, a positive power of 10, such as the exponent 3 in 1.987 × 10^3, expands the value back to its large counterpart. Meanwhile, negative powers, like -9 in 8.87 × 10^{-9}, shrink the value to its original small size, making scientific notation both comprehensive and manageable.