Question

Write, in your own words, the steps involved in expressing a number in standard scientific notation.

Short answer

To express a number in standard scientific notation, follow these steps: (1) Understand that scientific notation represents numbers as a decimal between 1 and 10 multiplied by a power of 10. (2) Identify the given number's decimal point, and if it's a whole number, assume the decimal is at the end. (3) Move the decimal point to create a number between 1 and 10 and note the number of places it moved (left for positive exponents and right for negative exponents). (4) Determine the exponent of 10 based on the number of places the decimal point moved. (5) Write the final expression in the form \(a \times 10^b\), where \(a\) is the adjusted decimal number and \(b\) is the exponent of 10.

Step-by-step solution

Understand the concept of scientific notation

Standard scientific notation is a way of representing large or small numbers more concisely by expressing them as a product of a decimal number between 1 and 10, and a power of 10. This notation helps make large or small numbers more manageable and easier to read and compare.

Identify the given number and its decimal point

To convert a given number into scientific notation, find the decimal point in the number. For whole numbers, assume the decimal point is at the end of the number. For decimals, the decimal point is already present.

Move the decimal point to create a number between 1 and 10

Shift the decimal point to the left or right so that you have a number between 1 and 10. Count the number of places you need to move the decimal point to achieve this. Moving the decimal point to the left will result in a positive exponent, and moving the decimal point to the right will result in a negative exponent.

Determine the exponent of 10

The number of places you moved the decimal point will be the exponent of 10. If you moved the decimal point to the left, the exponent will be positive, and if you moved the decimal point to the right, the exponent will be negative.

Express the number in standard scientific notation

Combine the new decimal number (between 1 and 10) with the appropriate power of 10 to write the number in standard scientific notation. The final notation will be in the form of \(a \times 10^b\), where \(a\) is the decimal number between 1 and 10, and \(b\) is the exponent of 10.

Key concepts

Expressing Numbers in Scientific Notation

Scientific notation is a shorthand method used to write very large or very small numbers with ease. To express a number in scientific notation, start by locating the decimal point. If the number is a whole number, imagine the decimal point is immediately to the right of the last digit. The second step is moving this decimal point to the place right after the first non-zero digit, which will give you a coefficient between 1 and 10. Count the number of places the decimal has moved; this will determine the exponent of 10. For example, the number 5300 would become 5.3 after moving the decimal two places to the left, making the scientific notation form to be 5.3 times 10 to the power of 2.

Remember, if the decimal is moved to the left, the exponent is positive, and moving it to the right results in a negative exponent. This process simplifies large numbers into a product of a number between 1 and 10 and an exponent of 10, making them easier to handle, especially in calculations.

Decimal Point Placement

The exact placement of the decimal point is crucial when expressing numbers in scientific notation. It ensures that the coefficient (the number we multiply by the power of 10) is between 1 and 10. This is an important rule. If the coefficient is not between 1 and 10, the notation will be incorrect. To illustrate, 0.045 becomes 4.5 after moving the decimal three places to the right, resulting in a coefficient within the accepted range. The scientific notation is then 4.5 times 10 to the power of -3. Placing the decimal point correctly is essential to maintaining the integrity of the original number while converting it to scientific notation.

Exponent of 10

The exponent of 10 in scientific notation indicates how many times you need to multiply or divide the number by 10 to return to its original value. The direction in which you moved the decimal point earlier will dictate whether the exponent is positive or negative. Moving it to the left (for numbers greater than 10) gives a positive exponent, while moving it to the right (for numbers less than 1) gives a negative exponent. It's a logarithmic scale; each change of one in the exponent represents a tenfold increase or decrease. For example, 10 to the power of 3 (10^3) stands for 1000, and 10 to the power of -3 (10^-3) corresponds to 0.001. This exponent is the tool that allows you to scale numbers up or down efficiently in scientific notation.

Standard Form in Mathematics

The standard form in mathematics, often referred to as scientific notation, is a system of writing numbers that is particularly useful for representing very large or very small values. When a number is written in standard form, it is easy to see at a glance the scale of the number, whether it is in the millions, billions, or much smaller like thousandths or millionths. For instance, 0.00000072 in standard form would be 7.2 times 10 to the power of -7. This standardization makes comparison and mathematical operations much simpler and reduces the chances of errors when processing or communicating these numbers. It's a concise, efficient, and universally understood way of representing quantities in the realm of science and mathematics.