The proof for this requires some algebra. For example, √4 is 2 because 2×2 = 4, i.e., two equal numbers that multiply together to make 4 are 2.īut √2 has no fraction answer. Finding the square root of a number means finding two numbers that are equal and, when you multiply them together, create the original number. But are these all the possible numbers? The answer is no, but let me show you why, by way of an example. There are many numbers we can make with rational numbers. Like with Z for integers, Q entered usage because an Italian mathematician, Giuseppe Peano, first coined this symbol in the year 1895 from the word “quoziente,” which means “quotient.” The denominator 1 has no effect on the fraction, so we omit it, leaving us with the integer.īecause of this property, you will find all the integers in the rational numbers. ![]() In the case of our example, we get -1, 3, and 0 whenever we have -1/1, 3/1, and 0/1. Aren’t they integers? This is just an area where the integers overlap with the rational numbers (think of this as analogous to how the natural numbers 0, 1, 2, 3, 4, 5, … overlap with the integers …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …). You might have wondered how -1 and 3 and 0 are rational numbers. You can pick any two numbers from …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … and put them together to make a rational number. The number a is called the numerator and b the denominator. This is the familiar fraction we know from school. We form them by taking integers and making every possible fraction out of them-i.e., a/b, where a is an integer and b is an integer (but b is not equal to zero). The next level of number is built out of integers. Recall that integers, Z, are all the negative numbers that go all the way to the left (to “negative infinity”) joined to all the natural numbers, N, that extend forever to the right (positive infinity): Let’s take a fresh look at fractions, though, in light of what we learned yesterday. ![]() Now that we’ve been introduced to the natural numbers and integers, it’s time to learn about the further complexities of numbers.įractions are often a source of stress in math because of how difficult the rules can be for adding and multiplying them. Welcome back to the foundations of mathematics. Episode #2 of the course Foundations of mathematics by John Robin
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