Rational numbers space whole numbers, fractions, and decimals - the numbers we use in our daily lives. They can be written as a ratio of 2 integers. Reasonable numbers are contrasted with irrational numbers - number such as Pi, √2, √7, other roots, sines, cosines, and logarithms of numbers. This short article concentrates on reasonable numbers.

You are watching: Is 1 11 a rational number

The meaning says the a number is rational if you deserve to write it in a form a/b wherein a and b are integers, and b is no zero. Clearly all fractions are of that form, therefore fractions space rational numbers. Terminating decimal numbers can additionally easily be written in the form: for instance 0.67 = 67/100, 3.40938 = 340938/100000, and so on.

We can illustrate hopeful rational numbers in the coordinate airplane with lines the go v the origin and another allude with entirety number coordinates. For example:
 Line y = 5x goes with the point (1, 5). Line y = (1/3)x goes with the point (3, 1). Line y = (9/2)x goes through the suggest (2, 9).
And therefore on. Also, the point out I listed are the very first points through whole-number works with these currently go v (after the origin).

 practice a little: What is the very first point through whole-number works with that these lines walk through?a) y = 9x b) y = 243x c) y = 5/6x d) y = 8/3 x e) y = 345/1039 x

Now, deserve to you imagine a line through beginning that does not touch any type of of this points with entirety number coordinates????? It"s hard, yet those sort of lines do exist. They simply avoid touching any of the point out with entirety number coordinates, and their steep is an irrational number!!! an overwhelming to fathom. Of course once you are illustration lines on paper or ~ above computer, friend are minimal in your accuracy and even a line y = Pi*x more than likely to go with a point with whole number coordinates, namely the suggest (7,22). It yes, really wouldn"t walk throuhg the if we can draw incredibly accurately, that would just go close. But due to the fact that it goes close, 22/7 is a pretty approximation come Pi.

The line y = Pi * x without doubt looks like it goes v the allude (7,22) since the graphics regimen cannot attract accurately enough.

## Non-terminating repeating decimals space rational

We talked exactly how terminating decimal numbers room obviously reasonable numbers. How about non-terminating decimal numbers? You could have never ever heard the those, though i hope girlfriend have. They space plentiful, too. Take it for example 1/9 and also convert it right into a decimal number with long division algorithm. What carry out you get? How around 2/9? 3/9? 1/11? 2/13? 7/15? deserve to you find an ext fractions that turn into non-terminating decimal numbers?

Since 0.11111... = 1/9, then the decimal number 0.11111... Is a rational number. In fact, every non-terminating decimal number the REPEATS a details pattern of digits is a reasonable number. Because that example, let"s make up a decimal number 0.135135135135135... That never ever ends. Execute you believe we have the right to write it together a fraction, in the form a/b? This sounds like it would certainly be pure guesswork, however no, over there is a method, a nice and also clever one, in mine opinion.

## How to transform a repeating decimal right into a fraction

Let"s surname our number a = 0.135135135... And multiply that by a strength of 10, then subtract the original a and the new number so that the repeating decimal parts cancel each various other in the subtraction.

 a = 0 135135135... 10a = 1 35135135135... 100a = 13 5135135135... 1000a = 135 135135135...

Okay, utilizing 1000a and also a will work, the decimals will certainly line up! So currently we subtract 1000a and a:

 1000a = 135. 135135135... - a = 0. 135135135... 999a = 135 ,

from i beg your pardon a = 135/999.

## Another example of composing a repeating decimal as a fraction

Sometimes the first couple decimal digits space not part of the repeating pattern. For example, b = 5.65787878787... Is together a number. The very same trick functions though: we multiply b by such strength of ten that the repeating components cancel each other in the subtraction.

 b = 5 65787878787... 10b = 56 578787878... 100b = 565 7878787...

As you deserve to see, the decimal components of b and 100b are identical! so we can subtract them:

 100b = 565. 78787878... − b = − 5. 65787878... 99b = 560. 13

from i beg your pardon b = 560.13/99 = 56013/9900.

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## Irrational numbers

We"ve been discussing terminating decimal numbers and repeating decimal numbers. Assumption: v what? NON-repeating and NON-terminating decimal numbers room the IRRATIONAL NUMBERS.