Question

Diameter of an Atom The diameter of an atom is about \(1 \times 10^{-10}\) meter. \(^{*}\) Express this diameter as a decimal.

Short answer

0.0000000001

Step-by-step solution

Understanding Scientific Notation

The diameter of the atom is given in scientific notation as \(1 \times 10^{-10}\). This notation is a way of expressing very large or very small numbers.

Convert Scientific Notation to Decimal

To convert \(1 \times 10^{-10}\) to decimal form, move the decimal point 10 places to the left because the exponent is \(-10\).

Moving the Decimal Point

Starting with 1, moving the decimal 10 places to the left, we get: \[ 0.0000000001 \]

Key concepts

converting scientific notation to decimal

Scientific notation is a way to express very large or very small numbers in a compact form. It's especially useful in fields like science and engineering where such numbers are common. For instance, the diameter of an atom is often given in scientific notation because it's incredibly small.
To convert a number from scientific notation to a decimal, follow these steps:

• Identify the coefficient (the number before the multiplication sign) and the exponent (the power of 10).
• Move the decimal point in the coefficient to the right if the exponent is positive, and to the left if it's negative.
• Fill in with zeros as needed.

For example, to convert the scientific notation value of an atom’s diameter, which is given as \(1 \times 10^{-10} \), move the decimal point 10 places to the left (since the exponent is -10). This gives the decimal representation: 0.0000000001. Keep this method in mind for any similar conversion tasks.

understanding exponents

Exponents are a shorthand way of expressing repeated multiplication. In scientific notation, the exponent indicates how many times the base (usually 10) is multiplied by itself. For example, \(10^3\) means \(10 \times 10 \times 10\), which equals 1000.
When the exponent is negative, it means you're dividing by that number of tens. So \(10^{-3} \) means \(\frac{1}{10 \times 10 \times 10}\), or 0.001.
This helps to express very small numbers. For instance, \(10^{-10}\times 1 \) means moving the decimal point of 1 to the left by 10 places, which results in 0.0000000001. This relation between the exponent and the movement of the decimal point is crucial in converting scientific notation to decimal form.

decimal representation

Decimal representation is the expression of numbers in the base-10 number system, which is the standard system for denoting integer and non-integer numbers. In base-10, each digit represents a power of 10.
The decimal point separates the integer part from the fractional part of the number. For example, the number 123.45 is in decimal form, with 123 being the integer part and 45 being the fractional part.
When we convert a number from scientific notation to its decimal form, we get a direct and easily interpretable representation of that number. For instance, the scientific notation \(1 \times 10^{-10} \) converts to the decimal format 0.0000000001. This conversion makes it easier to understand and use the number in practical situations, such as measuring the diameter of an atom.