What Are Real Numbers?

By: Jesslyn Shields
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Math is pretty confusing — at least to people who don’t understand it. Which is most of us.

The thing about math is you have to learn terms in order to understand what numbers are, what kinds of numbers are out there, and the characteristics of each type. Numbers are just mathematical symbols that are used to count and measure. But not all numbers are created equal.

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For instance, take the concept of “real numbers.” If numbers can be real, are there also fake numbers? Well, yes — at least there are real numbers and imaginary numbers. But what does that mean?

Real numbers are basically all the numbers you could think of if somebody told you to think of a number. Real numbers are based on the concept on the number line: the positive numbers sitting to the right of zero, and the negative numbers sitting to the left of zero. Any number that can be plotted on this number line is a real number. The numbers 27, -198.3, 0, 32/9 and 5 billion are all real numbers. Strangely enough, numbers such as √2 (the square root of 2, the value of which is 1.14142…) and π (3.1415…) can be plotted on a number line as well, even though they are nonterminating decimal numbers. So, even though the number after the decimal never ends, they can still be plotted on the number line.

Real numbers can also be described as all the numbers that are either rational or irrational. Rational numbers are numbers that can be written as a fraction, which includes whole numbers, all of which can be written as a fraction: 3/8, 5/1, 9/10, etc. Decimals can be rational as well — they’re just numbers that have either terminating or repeating decimals. So, 8.372 is a terminating decimal and 5.2222222… is a repeating decimal. These are rational numbers, which are also real numbers. Irrational numbers are also real numbers: those are decimals that are nonterminating like π and √2.

In contrast, an imaginary number is the value of the square root of a negative number. You may remember this special little math rule, but there is no number that, when squared, will produce a negative number. But that doesn’t stop mathematicians from doing it, as long as they admit the result is imaginary. Infinity is also an imaginary number.

Real numbers were just “numbers” until the 1500s, when the Italian polymath Girolamo Cardano invented imaginary numbers in order to solve polynomial equations.

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What Are Real Numbers?

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