It's pretty obvious, this isn't an integer. The ones that never, ever, ever repeat, like pi. So in general, repeatingĭigits are rational. And this technique weĭid, it doesn't only apply to this number. That we were able to represent x, we were able to represent So you could put this in lowestĬommon form, but we don't care about that.
I mean they're both definitely divisible by 2Īnd it looks like by 4. And by doing a little bit ofĪlgebraic manipulation and subtracting one multiple of itįrom another, we're able to express that same exact This number that we started off with, this number that And then, what areĭeal about this? Well, x was this number. Now, if we want to solveįor x, we just divide both sides by 9,900. So if we borrow from the 4, thisīecomes a 3 and then this becomes a 10. Have to do some regrouping there or some borrowing. Regrouping, but we can't borrow yet because we Let's see- The decimal part will cancel out. So on the left-hand side of thisĮquation, 10,000x minus 100x is going to be 9,900x. And if I subtract the bottom oneįrom the top one, what's going to happen? Well the repeating part You always want to write itĪfter the decimal point. Repeating here, but it wouldn't have madeĪs much sense. And then we can just treat themĪs traditional numbers. They're in terms of x, the repeating part disappears. I want to get two numbers that, when I subtract them and Left of the decimal point by five spaces. Go on and on, and on and on, and on after that. The decimal over to the right four spaces. To be equal to? Well every time you multiplyīy a power of 10, you shift the decimal one to the right. And the only reason why I wantġ0,000x is because I want to move the decimal point all the Let's define x as beingĮqual to this number. Of it, can I express this as a fraction? And to do that, let's actuallyĮxpress it as a fraction. So the question is, can IĮxpress this as the ratio of two integers? Or another way to think That means that p can beĮxpressed as the ratio of two integers, m/n. Rational, that means that p can be expressed as the
One that can be expressed as a rational expression The entire number line that we're used to using. We've dealt with so far is the real numbers. To write this line over the 28 to say that it So I could express this numberĪs 3.4028, but the 28 just keep repeating. Question, let's just think about what this represents. Opposites have the same absolute value since they are both at the same distance from 0.The number 3.4028 repeating belong to? And before even answering the If two numbers are at the same distance from 0 as in the case of 10 and -10 they are called opposites. This distance between a number x and 0 is called a number's absolute value. You notice that all integers, as well as all rational numbers, are at a specific distance from 0. It is a rational number because it can be written as:Ī rational number written in a decimal form can either be terminating as in:Īll rational numbers belong to the real numbers. As it can be written without a decimal component it belongs to the integers. The number 4 is an integer as well as a rational number. Integers include all whole numbers and their negative counterpart e.g. Whole numbers are all natural numbers including 0 e.g. Natural numbers are all numbers 1, 2, 3, 4… They are the numbers you usually count and they will continue on into infinity.
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