Problem 54 - Precalculus: Functions and Graphs

Question

The mass of an electron is approximately \(9.1 \times 10^{-31}\) kilogram. Express this number in decimal form.

Step-by-Step Solution

Answer

0.0000000000000000000000000000091

Step 1: Understanding the Scientific Notation

The given mass of an electron is in scientific notation: \(9.1 \times 10^{-31}\). This means that \(9.1\) should be multiplied by \(10^{-31}\). In scientific notation, \(10^{-31}\) means moving the decimal point 31 places to the left.

Step 2: Converting Scientific Notation to Decimal

To convert \(9.1 \times 10^{-31}\) to decimal form, move the decimal point in \(9.1\) 31 places to the left. Start with \(9.1\), which has one digit to the left of the decimal point. After moving 31 places to the left, there will be 30 zeros between the decimal point and the number 9.

Step 3: Writing the Decimal Form

After moving the decimal point 31 places to the left in \(9.1\), it becomes \(0.0000000000000000000000000000091\). Ensure that there are precisely 30 zeros following the decimal point before the digits '91'.

Key Concepts

Understanding Electron Mass

Electrons are tiny particles that orbit the nucleus of an atom. They are fundamental components in the structure of atoms and play a crucial role in many physical phenomena such as electricity and magnetism. The mass of an electron is incredibly small. It weighs about \(9.1 \times 10^{-31}\) kilograms. This small mass is much less than that of protons and neutrons, which make up the nucleus of an atom.

• An electron's mass is significant because it affects how electrons interact with other particles and forces.
• Despite their small mass, electrons are essential in chemical reactions and conductivity.
• Understanding the mass of an electron helps us grasp why they move quickly and are easily influenced by electric fields.

Electrons might seem tiny, but their effects are monumental in the world of physics and chemistry.

Steps for Decimal Conversion

When handling small or large numbers, scientists often use scientific notation. This notation is useful, but understanding the decimal version is equally important, especially for calculations in precalculus. Decimal conversion transforms scientific notation into a more familiar format. For \(9.1 \times 10^{-31}\), follow these steps to convert to decimal form:

• Start with the number 9.1.
• The exponent in the scientific notation, \'-31\', tells us to move the decimal point 31 places to the left.
• This results in the number becoming incredibly small: \(0.0000000000000000000000000000091\).

These decimals often represent very small values for entities like electron mass, making it easy to be overlooked how such small numbers can have big impacts.

Importance of Precalculus Education

Precalculus forms a foundational pillar in understanding more advanced mathematical concepts, like calculus and analytical geometry. It encompasses the basics of functions, trigonometry, and algebra, all crucial for higher educational learning.

• Precalculus provides the tools needed to manipulate and understand scientific notations and transformations into decimals.
• It teaches students how to analyze and interpret functions, which are essential in understanding various scientific phenomena, including physics and chemistry.
• Mastery of precalculus can hugely impact how you approach problems in fields like engineering and technology.

By focusing on core concepts in precalculus, students create a robust mathematical framework that aids in tackling complex real-world problems in science and mathematics.