Which of the following is the number of hertz in a gigahertz, shortened GHz? (A) 10 to the power of six hertz, (B) 10 to the power of eight hertz, (C) 10 to the power of nine hertz, (D) 10 to the power of 10 hertz, (E) 10 to the power of 12 hertz.
Okay, so in this question, we’re being asked about the prefix giga-, in this case relating to the unit hertz. Let’s recall that hertz is the unit of frequency and is defined as the number of cycles per second of a wave. A cycle of a wave is a single repeating section of the wave. On a graph, this would be represented by a peak and a trough like this. So for this wave, if the 𝑥-axis is time in seconds and this point here is one second, then we would say that this wave has a frequency of one hertz, because in one second we can see that the wave has one cycle. In this question though, we’re being asked about the conversion from hertz into gigahertz.
Giga- is one of a number of common prefixes that are used in scientific notation to describe large numbers. One way we can answer this question is to recall a list of prefixes and the numbers they represent. And we can then find giga- in this list and read off the answer. Let’s remember though that these prefixes need to be used along with a unit, in this case hertz. For example, it doesn’t make any sense to just talk about a giga-, but a gigahertz is a well-defined quantity. Let’s start our list with the prefix kilo-. And when used with hertz, we have one kilohertz, which is abbreviated as one kHz. This is equal to 1000 hertz because we know that the prefix kilo- means 1000.
We also know that in scientific notation, we can write 1000 as 10 to the power of three. So one kilohertz is equal to 10 to the power of three hertz. Now, 10 to the power of three is actually not one of our possible answers to this question. So let’s carry on with our list of prefixes.
The next prefix in our list is mega-, which when used with hertz gives us a megahertz, abbreviated one MHz. The prefix mega- represents one million. And in scientific notation, we can write one million as 10 to the power of six. So a megahertz is equal to 10 to the power of six hertz. Now, 10 to the power of six hertz is actually one of the possible answers to this question. And we know that it’s not the correct answer because we’ve just seen that 10 to the power of six hertz is equal to a megahertz. So it can’t also be equal to a gigahertz.
So let’s keep going with our list of prefixes. And our next one is giga- and together with hertz gives us one gigahertz, abbreviated one GHz. The prefix giga- means one billion, which we can write in scientific notation as 10 to the power of nine. So a gigahertz is equal to 10 to the power of nine hertz. Remember that a gigahertz is the exact quantity we’re interested in for this question. And 10 to the power of nine hertz is a possible answer to our question. And we now know that this is actually the correct answer to the question since we’ve just seen that one gigahertz is indeed equal to 10 to the power of nine hertz.
Before we finish though, let’s notice a few interesting things about the remaining options from this question. For instance, option (B) was 10 to the power of eight hertz. But our list of prefixes went straight from 10 to the power of six hertz to 10 to the power of nine hertz and skipped over any possible prefix for 10 to the power of eight hertz.
This is because to go down our list of prefixes, we multiply by 1000 or 10 to the power of three. This means we basically add three to get to the next line. So we go from 10 to the power of three to 10 to the power of six to 10 to the power of nine, and so on. So only powers of 10 that are multiples of three have a common prefix used to describe them. This means that our option (B), 10 to the power of eight hertz, is not the correct answer here. But it also has no common prefix to describe it either.
Using this knowledge of how our list works, we can also guess that 10 to the power of 10, option (D), also doesn’t have a common prefix name since 10 in the power is not a multiple of three. We can check this by writing down one more prefix in our list. The next prefix is tera-. So when combined with hertz, we have one terahertz, abbreviated to one THz. A terahertz is equal to 10 to the power of 12 hertz. So we can see that we do indeed go from 10 to the power of nine hertz to 10 to the power of 12 hertz, once again adding three to the power of 10.
So this means we were correct in our guess that 10 to the power of 10 doesn’t have a common prefix. And we also see that the final option, 10 to the power of 12 hertz, is in fact equivalent to one terahertz and not a gigahertz as the question asked. But all of this is extra information since we’ve already found the answer to this question, which is that one gigahertz is equivalent to 10 to the power of nine hertz.