![]() His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. ![]() Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. In the 21st century, game theory applies to a wide range of behavioral relations it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. It has applications in all fields of social science, as well as in logic, systems science and computer science. Hence, $3.025 \gt 3.003$.Game theory is the study of mathematical models of strategic interactions among rational agents. Since the ones and tenth place of both the numbers are the same. Hence, $−15 \lt 0 \lt 5 \lt 10$ (ascending order). Since a positive integer is always greater than a negative integer, $−15$ is smaller than all of the given numbers.Īlso, negative integers are less than 0 and positive integers are greater than 0. Thus, any two real numbers can be compared based on their position on the number line.Įxample: Compare – 9 and 8 on the number line.įinally, we replace the numbers with fractions. Thus, the number on the right is always greater than the number on the left. On a number line, the numbers increase as we go from left to right. In other words, we can say that ordering numbers means arranging numbers either in increasing order or decreasing order.Īrrangement of numbers in increasing order, from the smallest to the greatest, is called ascending order.Īrrangement of numbers in decreasing order, from the greatest to smallest, is called descending order.įor example, $10 \gt 8 \gt 6 \gt 4 \gt 2$ Comparing and Ordering Real Numbers Using a Number Line Ordering a list of numbers means arranging the numbers from least to greatest or greatest to least. Step 2: If the number of digits are the same, we start by comparing the highest place values.Įxample: In the following example, $312521 \lt 312712$ since $5 \lt 7$ at the hundreds place. Note that 0 placed at the beginning of a number is insignificant. The number with the more digits, is greater. We can also say that 3 is less or smaller than 6 by $6 \ −\ 3 = 3$. ![]() In the second part, $3 \lt 6$ since the red bar is shorter than the yellow bar. ![]() We can also say that 7 is more or greater than 3 by $7 \ −\ 3 = 4$. In the first example, $7 \gt 3$ since the green bar is taller than the pink bar.On comparing two numbers, we can find out by how much a number is greater or smaller. We can compare the numbers by comparing the heights of the bars. We also use the following symbols for comparing two quantities.Įxample: Let’s understand this visually using vertical bars composed of blocks. The symbols used for comparing numbers are $\gt,\ \lt,$ or $=$. ![]()
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