Standard calculus is developed using standard real numbers R, usually plotted on the number line.... more Standard calculus is developed using standard real numbers R, usually plotted on the number line. In Leibniz's vision of calculus, there are infinitesimally small numbers dx, that is |dx| < for all n = 1, 2, 3, . . . , yet dx ≠ 0. This requires an extension of the standard reals R called the hyperreal numbers, denoted often by R*. Then the standard real numbers R form a proper subset of the hyperreal numbers R*, that is, R ⊂ R* with R ≠ R*. Leibniz himself could not formally establish the theory of hyperreal numbers, which was done much later in the 1960s by Abraham Robinson, using a more general theory of mathematical logic called Nonstandard Analysis, an advanced subject area.
Standard calculus is developed using standard real numbers R, usually plotted on the number line.... more Standard calculus is developed using standard real numbers R, usually plotted on the number line. In Leibniz’s vision of calculus, there are infinitesimally small numbers dx, that is |dx| < 1/n for all n = 1, 2, 3, . . ., yet dx ≠ 0. This requires an extension of the standard reals R called the hyperreal numbers, denoted often by R*. Then the standard real numbers R form a proper subset of the hyperreal numbers R*, that is, R ⊂ R* with R ≠ R*. Leibniz himself could not formally establish the theory of hyperreal numbers, which was done much later in the 1960s by Abraham Robinson, using a more general theory of mathematical logic called Nonstandard Analysis, an advanced subject area.
In our modern world, we often don't realize the importance of friction. Every day, when millions ... more In our modern world, we often don't realize the importance of friction. Every day, when millions of people travel to work, their mode of transportation is dependent on friction. A common example is when a car is moving on a road, the tire-material (rubber) is acting against the concrete/asphalt. As most roads are concrete, we will study the effect of the surface roughness of rubber on the coefficient of static friction against concrete. This has many practical applications when determining what's the optimal roughness of a tire to suit different conditions. By taking into consideration both wet and dry surfaces and finding the most efficient combination of traction, we can find out what surface roughness the most efficient tire would have. For experimental investigation, we used a concrete slab and small rubber tires with varying surface roughness. The proper relationship between surface roughness & coefficient of static friction is established in this experiment.
Standard calculus is developed using standard real numbers R, usually plotted on the number line.... more Standard calculus is developed using standard real numbers R, usually plotted on the number line. In Leibniz's vision of calculus, there are infinitesimally small numbers dx, that is |dx| < for all n = 1, 2, 3, . . . , yet dx ≠ 0. This requires an extension of the standard reals R called the hyperreal numbers, denoted often by R*. Then the standard real numbers R form a proper subset of the hyperreal numbers R*, that is, R ⊂ R* with R ≠ R*. Leibniz himself could not formally establish the theory of hyperreal numbers, which was done much later in the 1960s by Abraham Robinson, using a more general theory of mathematical logic called Nonstandard Analysis, an advanced subject area.
Standard calculus is developed using standard real numbers R, usually plotted on the number line.... more Standard calculus is developed using standard real numbers R, usually plotted on the number line. In Leibniz’s vision of calculus, there are infinitesimally small numbers dx, that is |dx| < 1/n for all n = 1, 2, 3, . . ., yet dx ≠ 0. This requires an extension of the standard reals R called the hyperreal numbers, denoted often by R*. Then the standard real numbers R form a proper subset of the hyperreal numbers R*, that is, R ⊂ R* with R ≠ R*. Leibniz himself could not formally establish the theory of hyperreal numbers, which was done much later in the 1960s by Abraham Robinson, using a more general theory of mathematical logic called Nonstandard Analysis, an advanced subject area.
In our modern world, we often don't realize the importance of friction. Every day, when millions ... more In our modern world, we often don't realize the importance of friction. Every day, when millions of people travel to work, their mode of transportation is dependent on friction. A common example is when a car is moving on a road, the tire-material (rubber) is acting against the concrete/asphalt. As most roads are concrete, we will study the effect of the surface roughness of rubber on the coefficient of static friction against concrete. This has many practical applications when determining what's the optimal roughness of a tire to suit different conditions. By taking into consideration both wet and dry surfaces and finding the most efficient combination of traction, we can find out what surface roughness the most efficient tire would have. For experimental investigation, we used a concrete slab and small rubber tires with varying surface roughness. The proper relationship between surface roughness & coefficient of static friction is established in this experiment.
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