Lorenzo and Alice are converting numbers into standard form. Part a) Lorenzo attempts to convert 0.00003517 and writes three times 10 to the power of negative five. What has Lorenzo done wrong? Part b) Alice attempts to convert 0.00625 and writes 6.25 times 10 to the power of three. What has Alice done wrong?
When putting numbers into standard form, it should be in the form 𝑎 times 10 to the power of 𝑏, where 𝑎 is greater than or equal to one, but it has to be less than 10. 𝑏 is the power of 10. When converting numbers that are less than one — so they’re decimals; they’re very small numbers — 𝑏 will need to be negative. 𝑏 will need to be positive for numbers that are greater than one.
Now for the numbers that are equal to one that we’re talking about, 𝑏 will have to be zero because if we’re converting one into standard form, it would just be one times 10 to the zero power. That way, we’re actually not moving the one at all around the decimal point.
Let’s begin by looking at an example. Lorenzo is attempting to convert this number into standard form. So we need to create the number to be greater than or equal to one, but less than 10. So we have to move the digits.
Right now, it’s less than one. So when moving the digits here, we are creating an 𝑎 that satisfies the standard form. So we have to make it to where it’s one or greater but also less than 10. So we have to move our digits to the left. And if we move to the left, we can remember that we need to have a negative 𝑏, because these numbers must be smaller than one. If we had a really large number, we will have to move the digits to the right. So with the larger numbers like we said, numbers greater than one, we would need a positive 𝑏.
So let’s keep our decimal point fixed and move our digits to the left until we have a number that satisfies 𝑎. After moving at one time — so we’ve moved this one space — so right now 𝑏 would just be negative one, cause we’ve only moved it one time. However, this doesn’t satisfy the qualifications for 𝑎. So we move it a second time. And we still don’t. After a third time, we don’t.
And notice that we’re leaving the zeros when there’s multiple of them to the left of the decimal point. We really don’t need that. We only need to write one zero before the decimal point if that’s the only number that’s to the left of the decimal point. But we’ll take those off at the end.
After our fourth try, we are almost there. And finally, after the fifth time, 3.517 is a number that’s greater than or equal to one and less than 10. So again, 𝑎 is 3.517, but 𝑏 is five, but it’s actually negative five because like we said, if we had to move the digits to the left, 𝑏 needs to be negative. Or for numbers that are less than one — and our original number was less than one — 𝑏 will be negative.
So Lorenzo should’ve written 3.517 times 10 to the power of negative five. Unfortunately, Lorenzo only used the first significant figure, three. He should’ve used 3.517 because this is a part of the original number. And it will be essential to be getting the specific number if we were to convert back. He should’ve written 3.517 times 10 to the power of negative five.
Now let’s look at part b). Alice attempts to convert 0.00625 and writes 6.25 times 10 to the power of three. What has Alice done wrong?
Notice, our number is less than one. It’s a small decimal. And her 𝑏 is a positive three. So right away, we should recognize that 𝑏 should be negative. However, let’s go ahead and complete all the steps to get exactly what Alice should’ve gotten. So keep our decimal point fixed, and let’s move the digits to the left. There’s one time, two times, and finally three times.
As we said before, we don’t need all of the zeros that we keep writing out front. We could be writing the numbers like this. And for part a), it will be the same thing. So 𝑎 should be 6.25, which Alice has. She used all of the significant figures. And 𝑏 should be negative three, where Alice had a positive three. Alice has missed the negative sign in front of the three. She should’ve written 6.25 times 10 to the power of negative three.