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Negative numbers and Integers subtopics - operation on negative numbers and Integers using number line conditions of negative and positive number with respect to 0
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(Integers) Positive and Negative numbers What are INTEGERS? Integers are whole numbers that describe opposite ideas in mathematics. Integers can either be negative (-), positive (+) or zero. The integer zero is neutral. It is neither positive nor negative, but is an integer. Integers can be represented on a number line, which can help us understand the value of the integer. POSITIVE INTEGERS Are numbers to the right of zero. Are valued greater than zero. Express ideas of up, a gain or a profit. The sign for a positive integer is (+), however the sign is not always needed. Meaning +3 is the same value as 3. NEGATIVE INTEGERS Are numbers to the left of zero. Are valued less than zero. Express ideas of down or a loss. The sign for a negative integer is (-). This sign is always needed. Opposite Numbers/Integers – are the pairs of integers that have the same absolute value or have the same distance away from zero. ABSOLUTE VALUE The distance of a number from the origin (0) regardless of direction is called absolute value. The absolute value of a number is never negative. The symbol for absolute value is two straight lines surrounding the number or expression for which you wish to indicate absolute value. Examples: I 4 I = 4, +4 is read “ the absolute value of 4 is 4 “ I -3 I = 3, -3 is read “ the absolute value of -3 is 3” - I 3 I = -3, means “ the negative of the absolute value of 3 is -3 “ COMPARING AND ARRANGING INTEGERS Integers can be compared using a number line. As you move to the left along the number line, the integers decrease in value. On the other hand, integers increase in value as you move to the right along the number line. To arrange integers in ascending order is to arrange them from least to greatest. This means that when you use the number line, the smallest the integer is to the left of 0 on the number line. To arrange integers in descending order is to arrange them from greatest to least. This means that when you use the number line, the largest the integer is to the right of 0 on the number line. This is read as “nine is greater than negative 12.” This is read as “negative thirteen is less than negative 5.” This is read as “negative eight is greater than negative 18.” RWhat are INTEGERS? Integers are whole numbers that describe opposite ideas in mathematics. Integers can either be negative (-), positive (+) or zero. The integer zero is neutral. It is neither positive nor negative, but is an integer. Integers can be represented on a number line, which can help us understand the value of the integer. POSITIVE INTEGERS Are numbers to the right of zero. Are valued greater than zero. Express ideas of up, a gain or a profit. The sign for a positive integer is (+), however the sign is not always needed. Meaning +3 is the same value as 3. NEGATIVE INTEGERS Are numbers to the left of zero. Are valued less than zero. Express ideas of down or a loss. The sign for a negative integer is (-). This sign is always needed. Opposite Numbers/Integers – are the pairs of integers that have the same absolute value or have the same distance away from zero. ABSOLUTE VALUE The distance of a number from the origin (0) regardless of direction is called absolute value. The absolute value of a number is never negative. The symbol for absolute value is two straight lines surrounding the number or expression for which you wish to indicate absolute value. Examples: I 4 I = 4, +4 is read “ the absolute value of 4 is 4 “ I -3 I = 3, -3 is read “ the absolute value of -3 is 3” - I 3 I = -3, means “ the negative of the absolute value of 3 is -3 “ COMPARING AND ARRANGING INTEGERS Integers can be compared using a number line. As you move to the left along the number line, the integers decrease in value. On the other hand, integers increase in value as you move to the right along the number line. To arrange integers in ascending order is to arrange them from least to greatest. This means that when you use the number line, the smallest the integer is to the left of 0 on the number line. To arrange integers in descending order is to arrange them from greatest to least. This means that when you use the number line, the largest the integer is to the right of 0 on the number line. This is read as “nine is greater than negative 12.” This is read as “negative thirteen is less than negative 5.” This is read as “negative eight is greater than negative 18.” CONCEPT OF INTEGERS What are INTEGERS? Integers are whole numbers that describe opposite ideas in mathematics. Integers can either be negative (-), positive (+) or zero. The integer zero is neutral. It is neither positive nor negative, but is an integer. Integers can be represented on a number line, which can help us understand the value of the integer. POSITIVE INTEGERS Are numbers to the right of zero. Are valued greater than zero. Express ideas of up, a gain or a profit. The sign for a positive integer is (+), however the sign is not always needed. Meaning +3 is the same value as 3. NEGATIVE INTEGERS Are numbers to the left of zero. Are valued less than zero. Express ideas of down or a loss. The sign for a negative integer is (-). This sign is always needed. Opposite Numbers/Integers – are the pairs of integers that have the same absolute value or have the same distance away from zero. ABSOLUTE VALUE The distance of a number from the origin (0) regardless of direction is called absolute value. The absolute value of a number is never negative. The symbol for absolute value is two straight lines surrounding the number or expression for which you wish to indicate absolute value. Examples: I 4 I = 4, +4 is read “ the absolute value of 4 is 4 “ I -3 I = 3, -3 is read “ the absolute value of -3 is 3” - I 3 I = -3, means “ the negative of the absolute value of 3 is -3 “ COMPARING AND ARRANGING INTEGERS Integers can be compared using a number line. As you move to the left along the number line, the integers decrease in value. On the other hand, integers increase in value as you move to the right along the number line. To arrange integers in ascending order is to arrange them from least to greatest. This means that when you use the number line, the smallest the integer is to the left of 0 on the number line. To arrange integers in descending order is to arrange them from greatest to least. This means that when you use the number line, the largest the integer is to the right of 0 on the number line. This is read as “nine is greater than negative 12.” This is read as “negative thirteen is less than negative 5.” This is read as “negative eight is greater than negative 18.”Place value, number notation, fractions, decimals, percentages, average, simpl interest, ratio, proportion, factors, multiples, properties of 2D and 3D shapes, area of shapes, addition and subtraction of fractions, angles, perfect squares, cost and selling price, equivalent fractuions, circle, addition of decimals, perimeter of shapes, mean, median, mode, algebra, increment, discount, multiplication of decimals, positive and negative integers, order of fractions, area of circle, time interval and difference, odd and even numbers, pythagoras theorem, average speed, distance and time, probability, roman numerals Positive and negative opposites compare and order integers place numbers on number line absolute value grade 6Understanding Quantum Theory of Electrons in Atoms The goal of this section is to understand the electron orbitals (location of electrons in atoms), their different energies, and other properties. The use of quantum theory provides the best understanding to these topics. This knowledge is a precursor to chemical bonding. As was described previously, electrons in atoms can exist only on discrete energy levels but not between them. It is said that the energy of an electron in an atom is quantized, that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels. The energy levels are labeled with an n value, where n = 1, 2, 3, …. Generally speaking, the energy of an electron in an atom is greater for greater values of n. This number, n, is referred to as the principal quantum number. The principal quantum number defines the location of the energy level. It is essentially the same concept as the n in the Bohr atom description. Another name for the principal quantum number is the shell number. The shells of an atom can be thought of concentric circles radiating out from the nucleus. The electrons that belong to a specific shell are most likely to be found within the corresponding circular area. The further we proceed from the nucleus, the higher the shell number, and so the higher the energy level (Figure 9.4.1). The positively charged protons in the nucleus stabilize the electronic orbitals by electrostatic attraction between the positive charges of the protons and the negative charges of the electrons. So the further away the electron is from the nucleus, the greater the energy it has. This quantum mechanical model for where electrons reside in an atom can be used to look at electronic transitions, the events when an electron moves from one energy level to another. If the transition is to a higher energy level, energy is absorbed, and the energy change has a positive value. To obtain the amount of energy necessary for the transition to a higher energy level, a photon is absorbed by the atom. A transition to a lower energy level involves a release of energy, and the energy change is negative. This process is accompanied by emission of a photon by the atom. The following equation summarizes these relationships and is based on the hydrogen atom: The values nf and ni are the final and initial energy states of the electron. The principal quantum number is one of three quantum numbers used to characterize an orbital. An atomic orbital, which is distinct from an orbit, is a general region in an atom within which an electron is most probable to reside. The quantum mechanical model specifies the probability of finding an electron in the three-dimensional space around the nucleus and is based on solutions of the Schrödinger equation. In addition, the principal quantum number defines the energy of an electron in a hydrogen or hydrogen-like atom or an ion (an atom or an ion with only one electron) and the general region in which discrete energy levels of electrons in a multi-electron atoms and ions are located. Another quantum number is l, the angular momentum quantum number. It is an integer that defines the shape of the orbital, and takes on the values, l = 0, 1, 2, …, n – 1. This means that an orbital with n = 1 can have only one value of l, l = 0, whereas n = 2 permits l = 0 and l = 1, and so on. The principal quantum number defines the general size and energy of the orbital. The l value specifies the shape of the orbital. Orbitals with the same value of l form a subshell. In addition, the greater the angular momentum quantum number, the greater is the angular momentum of an electron at this orbital. Orbitals with l = 0 are called s orbitals (or the s subshells). The value l = 1 corresponds to the p orbitals. For a given n, p orbitals constitute a p subshell (e.g., 3p if n = 3). The orbitals with l = 2 are called the d orbitals, followed by the f-, g-, and h-orbitals for l = 3, 4, 5, and there are higher values we will not consider. There are certain distances from the nucleus at which the probability density of finding an electron located at a particular orbital is zero. In other words, the value of the wavefunction ψ is zero at this distance for this orbital. Such a value of radius r is called a radial node. The number of radial nodes in an orbital is n – l – 1. Consider the examples in Figure 9.4.2. The orbitals depicted are of the s type, thus l = 0 for all of them. It can be seen from the graphs of the probability densities that there are 1 – 0 – 1 = 0 places where the density is zero (nodes) for 1s (n = 1), 2 – 0 – 1 = 1 node for 2s, and 3 – 0 – 1 = 2 nodes for the 3s orbitals. The s subshell electron density distribution is spherical and the p subshell has a dumbbell shape. The d and f orbitals are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found. Principal quantum number (n) & Orbital angular momentum (l): The Orbital Subshell: https://youtu.be/ms7WR149fAY If an electron has an angular momentum (l ≠ 0), then this vector can point in different directions. In addition, the z component of the angular momentum can have more than one value. This means that if a magnetic field is applied in the z direction, orbitals with different values of the z component of the angular momentum will have different energies resulting from interacting with the field. The magnetic quantum number, called ml, specifies the z component of the angular momentum for a particular orbital. For example, for an s orbital, l = 0, and the only value of ml is zero. For p orbitals, l = 1, and ml can be equal to –1, 0, or +1. Generally speaking, ml can be equal to –l, –(l – 1), …, –1, 0, +1, …, (l – 1), l. The total number of possible orbitals with the same value of l (a subshell) is 2l + 1. Thus, there is one s-orbital for ml = 0, there are three p-orbitals for ml = 1, five d-orbitals for ml = 2, seven f-orbitals for ml = 3, and so forth. The principal quantum number defines the general value of the electronic energy. The angular momentum quantum number determines the shape of the orbital. And the magnetic quantum number specifies orientation of the orbital in space, as can be seen in Figure 9.4.3. Figure 9.4.4 illustrates the energy levels for various orbitals. The number before the orbital name (such as 2s, 3p, and so forth) stands for the principal quantum number, n. The letter in the orbital name defines the subshell with a specific angular momentum quantum number l = 0 for s orbitals, 1 for p orbitals, 2 for d orbitals. Finally, there are more than one possible orbitals for l ≥ 1, each corresponding to a specific value of ml. In the case of a hydrogen atom or a one-electron ion (such as He+, Li2+, and so on), energies of all the orbitals with the same n are the same. This is called a degeneracy, and the energy levels for the same principal quantum number, n, are called degenerate energy levels. However, in atoms with more than one electron, this degeneracy is eliminated by the electron–electron interactions, and orbitals that belong to different subshells have different energies. Orbitals within the same subshell (for example ns, np, nd, nf, such as 2p, 3s) are still degenerate and have the same energy. While the three quantum numbers discussed in the previous paragraphs work well for describing electron orbitals, some experiments showed that they were not sufficient to explain all observed results. It was demonstrated in the 1920s that when hydrogen-line spectra are examined at extremely high resolution, some lines are actually not single peaks but, rather, pairs of closely spaced lines. This is the so-called fine structure of the spectrum, and it implies that there are additional small differences in energies of electrons even when they are located in the same orbital. These observations led Samuel Goudsmit and George Uhlenbeck to propose that electrons have a fourth quantum number. They called this the spin quantum number, or ms. The other three quantum numbers, n, l, and ml, are properties of specific atomic orbitals that also define in what part of the space an electron is most likely to be located. Orbitals are a result of solving the Schrödinger equation for electrons in atoms. The electron spin is a different kind of property. It is a completely quantum phenomenon with no analogues in the classical realm. In addition, it cannot be derived from solving the Schrödinger equation and is not related to the normal spatial coordinates (such as the Cartesian x, y, and z). Electron spin describes an intrinsic electron “rotation” or “spinning.” Each electron acts as a tiny magnet or a tiny rotating object with an angular momentum, even though this rotation cannot be observed in terms of the spatial coordinates. The magnitude of the overall electron spin can only have one value, and an electron can only “spin” in one of two quantized states. One is termed the α state, with the z component of the spin being in the positive direction of the z axis. This corresponds to the spin quantum number ms=12. The other is called the β state, with the z component of the spin being negative and ms=−12. Any electron, regardless of the atomic orbital it is located in, can only have one of those two values of the spin quantum number. The energies of electrons having ms=−12 and ms=12 are different if an external magnetic field is applied. Figure 9.4.5 illustrates this phenomenon. An electron acts like a tiny magnet. Its moment is directed up (in the positive direction of the z axis) for the 12 spin quantum number and down (in the negative z direction) for the spin quantum number of −12. A magnet has a lower energy if its magnetic moment is aligned with the external magnetic field (the left electron) and a higher energy for the magnetic moment being opposite to the applied field. This is why an electron with ms=12 has a slightly lower energy in an external field in the positive z direction, and an electron with ms=−12 has a slightly higher energy in the same field. This is true even for an electron occupying the same orbital in an atom. A spectral line corresponding to a transition for electrons from the same orbital but with different spin quantum numbers has two possible values of energy; thus, the line in the spectrum will show a fine structure splitting. The Pauli Exclusion Principle An electron in an atom is completely described by four quantum numbers: n, l, ml, and ms. The first three quantum numbers define the orbital and the fourth quantum number describes the intrinsic electron property called spin. An Austrian physicist Wolfgang Pauli formulated a general principle that gives the last piece of information that we need to understand the general behavior of electrons in atoms. The Pauli exclusion principle can be formulated as follows: No two electrons in the same atom can have exactly the same set of all the four quantum numbers. What this means is that electrons can share the same orbital (the same set of the quantum numbers n, l, and ml), but only if their spin quantum numbers ms have different values. Since the spin quantum number can only have two values (±12), no more than two electrons can occupy the same orbital (and if two electrons are located in the same orbital, they must have opposite spins). Therefore, any atomic orbital can be populated by only zero, one, or two electrons. The properties and meaning of the quantum numbers of electrons in atoms are brieflyNegative Numbers and Absolute Value Review