How do you show the group of digits that repeat in a repeating decimal??
21 answers
To show the group of digits that repeat in a repeating decimal, you can use a bar over the digits that repeat. This bar, known as a vinculum, indicates that the digits are repeating infinitely. For example, if you have the decimal 0.33333..., you can write it as 0.3̅, where the digit 3 repeats infinitely. Similarly, 0.153153153... can be written as 0.15̅3̅.
How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?
The length of the repeating cycle of a repeating decimal depends on the prime factors of the denominator. In this case, since the denominator is 7, which is a prime number, the repeating cycle can be at most 6 digits long.
Let's take an example to illustrate this. Suppose we have the rational number 1/7. When we divide 1 by 7, the decimal representation is 0.142857142857..., and the cycle of repeating digits is 142857. As 7 is a prime number, the repeating cycle cannot be longer than the number of digits in the denominator (7 in this case), which means that the repeating cycle can be at most 6 digits long.
Therefore, for a rational number with a denominator of 7, the repeating cycle can be 1 to 6 digits long.
Let's take an example to illustrate this. Suppose we have the rational number 1/7. When we divide 1 by 7, the decimal representation is 0.142857142857..., and the cycle of repeating digits is 142857. As 7 is a prime number, the repeating cycle cannot be longer than the number of digits in the denominator (7 in this case), which means that the repeating cycle can be at most 6 digits long.
Therefore, for a rational number with a denominator of 7, the repeating cycle can be 1 to 6 digits long.
Convert 0.27 repeating, to a rational number in simplest form
To convert the repeating decimal 0.27̅ (where the digits 27 repeat infinitely) to a rational number in simplest form, we can use algebraic manipulation.
Let's represent the repeating decimal as x:
x = 0.27̅
To eliminate the repeating digits, we can multiply both sides of the equation by 100 (to shift the decimal point two places to the right):
100x = 27.27̅
Now, let's subtract the original equation from the new equation to eliminate the repeating part:
100x - x = 27.27̅ - 0.27̅
99x = 27
To get x by itself, divide both sides of the equation by 99:
x = 27/99
Now, simplify the fraction by finding the greatest common divisor (GCD) of 27 and 99, which is 9:
x = (27/9) / (99/9)
x = 3/11
Therefore, the rational number equivalent to 0.27̅ is 3/11 in simplest form.
Let's represent the repeating decimal as x:
x = 0.27̅
To eliminate the repeating digits, we can multiply both sides of the equation by 100 (to shift the decimal point two places to the right):
100x = 27.27̅
Now, let's subtract the original equation from the new equation to eliminate the repeating part:
100x - x = 27.27̅ - 0.27̅
99x = 27
To get x by itself, divide both sides of the equation by 99:
x = 27/99
Now, simplify the fraction by finding the greatest common divisor (GCD) of 27 and 99, which is 9:
x = (27/9) / (99/9)
x = 3/11
Therefore, the rational number equivalent to 0.27̅ is 3/11 in simplest form.
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator?
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of 999.
Let's take an example to understand this. Suppose we have the repeating decimal 0.abc̅, where the digits abc repeat infinitely (c̅ indicates that c is the repeating digit). To convert this repeating decimal to a rational number, we can write it as:
0.abc̅ = abc/999
The denominator 999 is used because it is a three-digit number with all 9s, and when written as a fraction, it has three 9s in the numerator and three 9s in the denominator.
Therefore, if a repeating decimal has a repeating cycle of three digits, its equivalent rational number will have a denominator of 999.
Let's take an example to understand this. Suppose we have the repeating decimal 0.abc̅, where the digits abc repeat infinitely (c̅ indicates that c is the repeating digit). To convert this repeating decimal to a rational number, we can write it as:
0.abc̅ = abc/999
The denominator 999 is used because it is a three-digit number with all 9s, and when written as a fraction, it has three 9s in the numerator and three 9s in the denominator.
Therefore, if a repeating decimal has a repeating cycle of three digits, its equivalent rational number will have a denominator of 999.
Which statement is from neils bohrs proposed model of atom
The statement "Electrons revolve around the nucleus in specific energy levels" is from Niels Bohr's proposed model of the atom.
If an atomic nucleus were the size of a dime, how far might one of the electrons be?
If we scale down the size of an atomic nucleus to that of a dime (which has a diameter of 17.91 mm), the electron would be located significantly farther away. In atomic terms, electrons are found in energy levels or orbitals. The distance between the nucleus and the electrons in an atom is not fixed but varies depending on the specific energy level.
For example, in the hydrogen atom (the simplest atom), the average distance of the electron from the nucleus in its ground state (lowest energy level) is approximately 0.053 nm (nanometers) or 0.053 x 10^(-9) meters.
Scaling this down to the size of a dime, which has a diameter of 17.91 mm (or 17.91 x 10^(-3) meters), the electron would be much farther away. The exact distance depends on the specific atom and energy level being considered. However, on average, if the atomic nucleus were the size of a dime, one of the electrons might be several centimeters or even meters away.
For example, in the hydrogen atom (the simplest atom), the average distance of the electron from the nucleus in its ground state (lowest energy level) is approximately 0.053 nm (nanometers) or 0.053 x 10^(-9) meters.
Scaling this down to the size of a dime, which has a diameter of 17.91 mm (or 17.91 x 10^(-3) meters), the electron would be much farther away. The exact distance depends on the specific atom and energy level being considered. However, on average, if the atomic nucleus were the size of a dime, one of the electrons might be several centimeters or even meters away.
If an atomic nucleus were the size of a dime, how far might one of the electrons be?
Answers
1 inch
1 foot
1 mile
1 yard
Answers
1 inch
1 foot
1 mile
1 yard
If an atomic nucleus were the size of a dime, the distance of one of the electrons would be approximately 1 mile away.
What particles are found in the nucleus of an atom
The particles found in the nucleus of an atom are protons and neutrons. Protons have a positive charge (+1) and contribute to the atomic mass of the atom, while neutrons have no charge (neutral) and also contribute to the atomic mass. The number of protons determines the atomic number and the element the atom belongs to, while the sum of protons and neutrons gives the atomic mass of the atom. Electrons, which have a negative charge (-1), are found outside the nucleus in electron shells or energy levels.
Pick two
Sure! The two particles found in the nucleus of an atom are:
1) Protons: Protons have a positive charge (+1) and contribute to the atomic mass of the atom. The number of protons determines the atomic number and the element the atom belongs to.
2) Neutrons: Neutrons have no charge (neutral) and also contribute to the atomic mass of the atom. They help stabilize the nucleus by counteracting the repulsive forces between protons.
1) Protons: Protons have a positive charge (+1) and contribute to the atomic mass of the atom. The number of protons determines the atomic number and the element the atom belongs to.
2) Neutrons: Neutrons have no charge (neutral) and also contribute to the atomic mass of the atom. They help stabilize the nucleus by counteracting the repulsive forces between protons.
What occupies most of the volume of a atom?
Most of the volume of an atom is occupied by the electron cloud or electron shells. The electron cloud is the region surrounding the atomic nucleus where electrons are found. Although the electrons themselves have negligible mass, they occupy a significant amount of space due to their rapid movement. The electron cloud is where the probability of finding an electron is highest, and it determines the size of the atom. It is important to note that the nucleus, which contains protons and neutrons, has a significantly smaller volume compared to the electron cloud.
Rutherford suggested that?
Rutherford suggested that atoms have a small, dense, positively charged nucleus at the center. This proposal came after the famous gold foil experiment carried out by Rutherford and his colleagues, where they bombarded a thin sheet of gold foil with alpha particles. The majority of the alpha particles passed straight through the foil, while a few were deflected at large angles or even bounced back.
Based on these observations, Rutherford concluded that the positive charge and most of the mass of an atom are concentrated in a tiny, dense region at the center called the nucleus. He also proposed that the nucleus is surrounded by a large empty space where the negatively charged electrons move. This model became known as the Rutherford model of the atom and laid the foundation for our understanding of atomic structure.
Based on these observations, Rutherford concluded that the positive charge and most of the mass of an atom are concentrated in a tiny, dense region at the center called the nucleus. He also proposed that the nucleus is surrounded by a large empty space where the negatively charged electrons move. This model became known as the Rutherford model of the atom and laid the foundation for our understanding of atomic structure.
2 answers