Question

What is the general form for writing numbers in scientific notation?

Short answer

Scientific notation is written as a number between 1 and 10 multiplied by a power of ten.

Step-by-step solution

Identify the Components

Scientific notation consists of two main components: a number between 1 and 10 and a power of ten. For example, in the number 3.45 × 10^6, 3.45 is the coefficient and 10^6 is the power of ten.

Writing the Coefficient

The coefficient must be a number greater than or equal to 1 and less than 10. This means if you have a number like 45, you would write it as 4.5. If you have a number like 0.0056, you would write it as 5.6.

Determining the Exponent

The exponent indicates how many times the coefficient must be multiplied by 10 to get the original number. For example, 4.5 × 10^1 equals 45 and 5.6 × 10^{-3} equals 0.0056.

Combining the Components

To write the number in scientific notation, combine the coefficient and the power of ten. For example, 4500 would be written as 4.5 × 10^3.

Key concepts

coefficients

When dealing with scientific notation, the coefficient is a crucial part of the format. The coefficient must be a number that is greater than or equal to 1 and less than 10. This helps in standardizing the representation of numbers, making them easier to compare and compute. For instance, let's consider 3.45 × 10^6. Here, 3.45 is the coefficient. It's a number between 1 and 10. If you have a number like 45, you would convert it to a coefficient of 4.5 because that follows the rule of being less than 10. Similarly, for a small number like 0.0056, the conversion would give you 5.6 as the coefficient.

exponents

The exponent in scientific notation helps you understand the scale of the number. It indicates how many times the coefficient must be multiplied by 10 to result in the original number. In the example 4.5 × 10^1, the exponent is 1, meaning you multiply 4.5 by 10 once to get 45. Similarly, in 5.6 × 10^{-3}, the exponent is -3. This tells you to divide 5.6 by 10 three times, resulting in 0.0056. The exponent is essential for expressing both very large and very small numbers in a compact form.

power of ten

The power of ten is the part of scientific notation that includes the base (10) and an exponent. It's what gives scientific notation its flexibility in representing a wide range of numbers. In 3.45 × 10^6, the power of ten is 10^6. It tells you that you need to multiply 3.45 by 10 six times. For small numbers, like 5.6 × 10^{-3}, the negative exponent (-3) signals that you divide 5.6 by 10 three times. Understanding the power of ten is vital for grasping how scientific notation condenses very large or very small numbers into a manageable form.

standard form

Standard form is another term for scientific notation. It involves writing a number in a way that highlights both a single-digit coefficient and a power of ten. This form makes it easier to handle extremely large numbers like the distance between stars or tiny numbers like the mass of a molecule. For example, instead of writing 4500, you would express it as 4.5 × 10^3. This standardization helps in scientific calculations by providing a uniform way to represent numbers, which is especially useful when comparing or multiplying them.