Integers
Integers - Related Videos
02:43
Quantum Numbers
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2020
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, etc. 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...
02:18
The Bohr Model
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2020
Following the work of Ernest Rutherford and his colleagues in the early twentieth century, the picture of atoms consisting of tiny dense nuclei surrounded by lighter and even tinier electrons continually moving about the nucleus was well established. This picture was called the planetary model since it pictured the atom as a miniature “solar system” with the electrons orbiting the nucleus like planets orbiting the sun. The simplest atom is hydrogen, consisting of a single proton as the nucleus...
02:32
The de Broglie Wavelength
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2020
In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...
02:19
Triggering a Closed-Loop Stimulation by Neuron Spiking Activity
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2025
This video demonstrates the operation of closed-loop stimulation in a rat with a surgically implanted device to record neural activity. Neuronal signals are first recorded and analyzed. Template waveforms are then uploaded, and parameters are defined to trigger closed-loop stimulation.
02:18
X-ray Crystallography
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2020
The size of the unit cell and the arrangement of atoms in a crystal may be determined from measurements of the diffraction of X-rays by the crystal, termed X-ray crystallography. Diffraction Diffraction is the change in the direction of travel experienced by an electromagnetic wave when it encounters a physical barrier whose dimensions are comparable to those of the wavelength of the light. X-rays are electromagnetic radiation with wavelengths about as long as the distance between neighboring...
FAQs
Opposite integers have equal distance from zero but lie on different sides, so they represent balanced changes in opposite directions. This relationship makes zero the natural reference point for comparing gains and losses, positive and negative elevations, or increases and decreases. Recognizing opposites also helps explain why combining a number with its opposite returns to zero.
Integer calculations depend on both direction, represented by the sign, and size, represented by magnitude. Addition combines directed amounts, while multiplication follows a sign pattern in which two negative factors produce a positive result. Keeping these roles separate helps distinguish whether an answer reflects a change in direction, a change in size, or both.
Divisibility examines whether one integer can be divided by another without a remainder. Factors are integers that combine through multiplication to produce a specified integer, while a remainder records what is left when division is not exact. Together, these ideas organize integer relationships and support calculations involving factorization, divisibility patterns, and quantitative classification.
Integers are separated by equal intervals rather than varying continuously, giving them a discrete structure suited to exact counting and stepwise changes. That structure supports reasoning about factors, remainders, and ordered values without relying on intermediate quantities. It also makes integers central to number theory and to mathematical proofs built from precise, countable relationships.
To compare integers, locate them on the number line and use their positions from left to right. A value farther to the right represents the greater quantity, even when one value is negative. The same visual procedure helps track addition: moving right corresponds to a positive change, whereas moving left corresponds to a negative change.
Assigning signs to real-world changes turns situations into consistent mathematical models. Gains, elevations, and increases can be represented in one direction, while losses, lower elevations, and decreases use the opposite direction. Arithmetic then combines those changes, producing a final integer that can describe a balance, position, temperature, or other quantity relative to a chosen reference.
In algebra, integers provide values for equations and relationships involving unknown quantities. In number theory, their factors, divisibility, and remainders reveal structural patterns. Their exact, discrete behavior also supports computer science, where quantities and states can be represented without fractional intermediate values. Across these areas, integer reasoning contributes to calculation, classification, and mathematical proof.