Back To CoursePSAT Prep: Help and Review
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Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.
Outside of mathematics, we use the word 'irrational' to mean crazy or illogical; however, to a mathematician, irrational refers to a kind of number that cannot be written as a fraction (ratio) using only positive and negative counting numbers (integers). For example, you can write the rational number 2.11 as 211/100, but you cannot turn the irrational number 'square root of 2' into an exact fraction of any kind.
Don't assume, however, that irrational numbers have nothing to do with insanity. Legend suggests that, around 500 B.C., a guy named Hippasus was thrown overboard from a ship by the Pythagoreans, a group of Greek philosophers, as punishment for proving that the square root of 2 is irrational.
A mental trick you can use to help you visualize whether a number is rational or irrational is to think of the number in terms of cutting pizzas. For instance, if a number is rational, you can imagine cutting pizzas into equal-sized slices described by the denominator of a fraction and then eating the number of slices described by the numerator. For example, 6/8 can be found by cutting a pizza into 8 slices and then consuming 6 of those slices.
For a number like 3.95, you imagine cutting pizzas into a hundred slices each and then taking 395 slices. While you'll probably never be quite that hungry, you can imagine it. A negative number like -3/10 is a little tougher, but you could still visualize it if you slice pizzas into tenths and then give back 3 slices.
It's impossible to think of the square root of 5 that way. While there might be some other way to figure out how to get exactly the square root of 5 pizzas, you can't do it by cutting the pizza into any set number of equal slices and then taking the correct share of those.
In most cases, the best we can do to visualize an irrational number is approximate it with a decimal number.
Let's look at some common irrational numbers.
Some of the most common irrational numbers are roots, such as the square root of 5 or the cube root of 7. Square roots, cube roots, and roots of any higher power are often irrational, as long as they can't be simplified in a way that the radical (square root) symbol vanishes.
Sometimes we write irrational numbers approximately as decimal numbers, but we can never do it exactly because the decimal places go on forever and never fall into a repeating pattern.
As the unlucky Hippasus demonstrated, there is no way to write the square root of 2 as an exact fraction. It is irrational. (Square root of 2 = 1.41421356...).
On the other hand, -5.2 can be written as -52/10, which means that it's a rational number, and even the Pythagoreans wouldn't issue a death sentence over it.
The circumference of a circle divided by its diameter is always a little more than 3. In fact, the result of this division is an irrational number that we commonly refer to as pi.
Pi is part of a group of special irrational numbers that are sometimes called transcendental numbers. These numbers cannot be written as roots, like the square root of 11.
Many people remember the first few digits of pi: 3.14. Remembering those digits can be helpful, but it is not exact since pi goes on indefinitely (pi = 3.141592...). As of 2011, people have discovered more than 5 trillion digits of pi, but we'll never get to the end of it, because there is no end!
Sometimes you might see pi written as 22/7; however, be aware that, like 3.14, 22/7 is only an approximation. It is close to pi, but it's not equal. There is no fraction that exactly equals pi.
To mathematicians, e is more than just a letter in the alphabet. The irrational number e is formally named Napier's constant, but it is commonly called Euler's number, after Leonhard Euler (pronounced 'Oiler'). Just like pi, e occurs commonly in the real world.
Briefly, e is the result of adding a tiny bit to 1 and then raising that to a really big power. The resulting value (2.7182818284...) is irrational. The decimals go on forever without falling into a repeating pattern.
E is a very useful number in the worlds of science and business. It helps us calculate how things grow over time - the number of bacteria in a petri dish, the size of rabbit populations, or the interest your money earns in a savings account.
Another transcendental irrational number is derived from the ratios of the sides of certain geometric shapes. It is sometimes called the golden ratio, golden mean, or divine proportion, and it's represented by the Greek letter phi.
The ratio of longer to shorter sides of a five-pointed star (pentagram) represent phi in several ways as shown by the colors in the picture below. If you divide any colored side by the next shorter colored side, you'll get phi.
Just like the other irrational numbers we've discussed, phi's decimal places go on forever (phi = 1.618033988...). In spite of the fact that it is based on a ratio, phi is not based on a ratio of integers, so you wouldn't be able to make exact pizza slices out of it.
There have been many claims of the golden ratio appearing in nature, the human body, art, and architecture. The golden ratio is considered very pleasing to the human eye, as shown by the Mona Lisa, our galaxy, and the Egyptian pyramids, all of which have dimensions that are close to phi.
Let's review. Irrational numbers are those that can't be written as a fraction comprised of only integers. Think of a pizza - it's a rational number if you can cut the pizza into equal-sized slices determined by the denominator and then eat the number of slices determined by the numerator. It's an irrational number if you cannot. Common irrational numbers include roots, pi, phi, and Euler's number.
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Back To CoursePSAT Prep: Help and Review
18 chapters | 194 lessons