How to Read a Ruler 2 Inches
Having just discussed why we use compass and straightedge in geometry, let's flip that effectually and look at a common question at the more than uncomplicated level: How exercise yous use a ruler to measure or draw a line of a given length? The usual issue here is working with the fractional markings on an inch ruler, simply we'll also bear upon on issues with a metric ruler — and even how the ruler got its name. (It's more interesting than you may think!)
Reading the marks
We can start with an early question and answer (1997) that we regularly used every bit a reference for students asking about this:
Reading a Ruler I need to read a ruler, and I can't. Can you assistance me?
Physician Rob replied:
I promise and so. At that place are ii main kinds of rulers in general use, and other, more obscure kinds. We volition ignore the obscure ones.
Here is an instance of a ruler that combines the two kinds we'll be seeing:
Partial inch rulers
First in that location is the ruler marked in inches, and each inch is subdivided into sixteen parts. The lines on it wait something similar this sketch of a function of a ane-foot ruler: eight 9 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ----------------------------------- D C D B D C D A D C D B D C D The lines have different lengths to assistance you figure out what lengths they represent. The shortest lines (D) stand for an odd number of sixteenths of an inch. The next shortest lines (C) represent an odd number of eighths of an inch. The next shortest lines (B) represent an odd number of quarters of an inch. The next shortest lines (A) correspond an odd number of halves of an inch. The longest lines (8 or 9) stand for whole inches, and are numbered. The lines labeled 8 and nine above mark points on the edge of the ruler that are viii and nine inches from the left-hand end of the ruler. All distances are measured from that aforementioned left-manus cease of the ruler, which could be, but probably isn't, marked "0".
Some rulers accept an actual mark for 0, while others merely start at the edge of the stick.
This traditional set of markings is based, not on 10 as in our decimal arrangement of counting, merely on ii: the denominator is e'er a ability of 2. (Equally we'll see beneath, the markings may become only to eighths, or go farther than 16, but a denominator of 16 is common.) Nosotros'll accept different pictures beneath that may be easier to follow.
The line halfway betwixt them labeled A to a higher place marks a signal on the border of the ruler, which is viii 1/ii inches from the end. That makes sense considering 8 1/2 is halfway between 8 and ix. The next shorter lines labeled B above are halfway between 8 and 8 ane/2, and halfway between eight ane/2 and 9. The former one marks 8 1/iv inches, and the latter marks viii 3/4 inches. These brand sense because viii i/iv is halfway between 8 and 8 1/2, and viii 3/4 is halfway between 8 1/2 and 9. In the words of arithmetic, viii + ([8+1/2]-viii)*(one/2) = [eight+1/4], ^^^^^^^^^ distance from 8 to [8+one/2] and [8+ane/2] + (9-[8+1/2])*(i/two) = [8+three/four]. ^^^^^^^^^^^ distance from [8+1/2] to ix Likewise, the side by side shorter lines labeled C above are halfway between viii and [8+1/iv], between [eight+i/4] and [8+i/2], between [eight+1/2] and [8+3/4], and between [8+3/four] and ix. They must therefore marker the distances [8+i/8], [viii+3/8], [8+5/8], and [8+7/8], respectively. Finally, the shortest lines labeled D above are halfway between adjacent pairs of longer lines, and marking [eight+1/xvi], [8+iii/16], ..., [8+xv/16].
That looks every bit if there were a lot of arithmetic with fractions needed here; it will get simpler in a moment!
When I measure out a altitude, I put the "0" cease of the ruler at one end, then selection the marker on the ruler which is closest to the other stop of the altitude. The nearest inch line to the left gives me the number of whole inches. I then effigy out whether this line is a 1/16 line (shortest), a 1/8 line, a i/4 line, a 1/2 line, or an inch line. That tells me what the denominator of the fraction of an inch will be. From the inch line I count the lines the same length as my called one using odd number: "ane, 3, v, seven, ..." until I find my line. That tells me what the numerator of the fraction of an inch volition exist. I then combine the number of whole inches with the fraction to become the altitude.
This is what replaces the arithmetic to a higher place! We count down in size to the blazon of mark we are looking at (doubling the denominator), and so count odd numbers to discover the numerator.
Metric rulers
The metric system uses decimal numbers rather than fractions, and then here we employ tens.
The other common kind of ruler measures centimeters instead of inches. Each centimeter is divided into x parts (each chosen a millimeter). The lines on information technology look something similar this sketch of a function of such a ruler: thirteen 14 | | | | | ||||||||||| ------------- CAAAABAAAAC The longest lines labeled C represent whole centimeters. The next longest line labeled B represents a one-half centimeter. The shortest lines labeled A represent tenths of centimeters, or millimeters. Since 1/2 = 5/x, the B line besides represents 5 millimeters.
The fractional ruler used twos because those are piece of cake to count; here we have a special mark at 5 for the aforementioned reason. In effect, the markings use base of operations two and 5 (like Roman numerals).
To mensurate a length, put the left end of the ruler, which could be labeled "0" but probably isn't, at ane finish, and pick the closest marking on the ruler to the other end. Find the closest centimeter marker to the left of your mark. That will tell you the number of whole centimeters (13 in the in a higher place example). The denominator of the fraction of a centimeter is fixed at 10. The numerator is found by counting from the whole centimeter mark yous found higher up, and the medium-length lines B assist yous count past showing you where 5 tenths or half a centimeter is. If you are shut to the "13" marking, you count up as you move to the correct starting with "0" for the "xiii" mark itself. If you lot are shut to the "B" mark, y'all tin can count up equally you move to the right or down as you move to the left, starting with "five" for the B marking itself. If you are close to the "xiv" marking, y'all can count down as you motility to the left, starting with "10" for the "14" mark itself. This will tell you the numerator of the fraction of a centimeter.
Unremarkably we either write the length in millimeters, or utilise decimals rather than fractions. We'll have an illustration of this later.
Naming the fractions
Our adjacent answer is from 1999, and merely fills in some details in the pictures:
Finding Fractions on a Ruler My question is elementary. I want to know how to read all the measurements within 1 inch. We know that every line on a ruler or tape mensurate (whichever) has a fractional meaning. I want to learn how these fractions are arranged. I hope you empathize what I mean by all fractions in an inch. Example: where is 7/eight of an inch compared to 5/16 of an inch on a tape mensurate? I know where the bones measurements are, such equally ane/iv, one/2, and 3/iv of an inch. It's just the other measurements that I don't get.
I first referred to the respond higher up, and then added some more detailed diagrams:
I'll give you a simpler, quick answer that may meet your needs. Hither'south a ruler showing 16ths, with the meaning of each marking indicated: 0 1 | 1/two | | | | | i/4 | 3/four | | | | | | | 1/eight | 3/8 | v/8 | vii/8 | | | | | | | | | | | 1/16 | 3/16 | 5/16 | 7/sixteen | ix/xvi | 11/16 | 13/16 | 15/16 | | | | | | | | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ Some rulers just go to eighths, and it might be easier to start with that: 0 one | 1/2 | | | | | 1/4 | 3/4 | | | | | | | one/viii | three/8 | v/8 | 7/viii | | | | | | | | | | +-------+-------+-------+-------+-------+-------+-------+-------+ 0 one/8 2/8 3/eight 4/8 five/eight half-dozen/eight 7/8 1 Every bit the marks go downwards in size, the denominator of the fraction doubles. The biggest mark betwixt two inches is the half; between that and either inch mark the next largest marking is the quarter inch; and and then on. To find eighths, merely go downwards the scale finding 1/ii, then 1/iv, then 1/8. All the marks that are that size or bigger are eighths: i/eight, two/8 (which is the same as one/4), 3/viii, 4/8 (which is the same equally 1/2), 5/8, then on. If yous want, y'all tin can just count all the marks that are the same size, counting by odd numbers: i/viii, 3/8, 5/8, 7/viii.
As Doctor Rob explained, we can count but odd numbers; or we can just count all the lines of at to the lowest degree the length of the i nosotros are reading. This is because all the fifty-fifty numbers correspond to fractions with a smaller denominator (and longer lines). For example, the 3/4 mark is the third line of at least its length, or we can count 1, 3 looking only at lines of the same length.
Try doing the aforementioned thing on the 16ths ruler above; the only difficult part will be to ignore the smaller 16ths marks while you lot're looking for 8ths. To discover 5/eight, for example, notice which size marks are 8ths, then count one, iii, v/8 until you find it. Now you should exist able to work out 32nds yourself: 0 ane | 1/2 | /2 | | | | 1/four | three/4 | /iv | | | | | | one/8 | 3/8 | 5/8 | seven/8 | /8 | | | | | | | | | | 1/16 | 3/16 | 5/sixteen | 7/sixteen | 9/16 | 11/16 | xiii/16 | 15/16 | /16 | | | | | | | | | | | | | | | | | | 1 | 3 | five | 7 | 9 | 11| xiii| 15| 17| 19| 21| 23| 25| 27| 29| 31| /32 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
The 16th ruler would, of grade, just lack the smallest lines here.
Measuring an object
In 2000, Dr. Ian answered a very like question with details on how to observe the length of an object. The question was this:
Reading a Ruler II What practice the other picayune lines on the ruler stand for?
Doctor Ian beginning showed a metric ruler:
A metric ruler divides a meter into 100 centimeters, 0 i two 99 100 |___|___|___ ... ___|___| and divides each centimeter into millimeters, 0 i 2 | | | ||||||||||||||||||||| ^ ^ ten mm 19 mm i cm 1.nine cm 0.one m 0.nineteen one thousand which means that the distance between any of the smallest lines is 1/chiliad of a meter.
This version lacks the longer line at 5; but nosotros can run into here that we are counting millimeters (mm), which are tenths of a centimeter, so that 19 mm is the same as i.9 cm.
After showing fractional rulers every bit nosotros've seen to a higher place, he took it further, showing exactly how to read a length:
And so let's say I desire to measure out something with a ruler. I ready it down adjacent to the ruler, 0 1 2 iii | | | | | | | | | | | | | | | | | | | | | | | | |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_| ############################################ ############################################ ############################################ ############################################ ^_______________^ 2 to 3 in ^_______^ 2-1/2 to 3 in ^___^ 2-1/two to 2-iii/4 in ^_^ ii-5/8 to two-iii/4 in and I find the 2 closest inch-marks that are on either side of it. In the picture above, these are the 2- and 3-inch marks. So nosotros know that the thing is betwixt ii and three inches long.
If all we intendance virtually is the nearest inch, we're done! It'southward between 2 and 3 inches, and closer to 3. Just we desire to be equally precise as possible:
Side by side, we look at the 2-i/ii inch marking. The thing extends past it, so nosotros know that the thing is between 2-ane/2 and 3 inches long. Side by side, we look at the mark halfway between those, the 2-3/4 inch mark. The thing doesn't extend quite that far, so we know that the affair is between 2-1/ii and two-3/4 inches long. And then on, until we've identified the two closest marks that are on either side of the stop of the thing we're measuring. At this point, if it'due south very close to either mark, we tin just telephone call that the measurement. Or if it's right in between, nosotros can take the average. In the example to a higher place, that would be: 5/8 + iii/4 2 and ----------- inches 2 v/viii + 6/8 = 2 and ----------- inches 2 11/eight = 2 and ----------- inches 2 = ii and 11/16 inches Note that you can measure something to the nearest 16th of an inch, even though the ruler is just marked to the nearest 8th of an inch.
An alternative way to practise this final adding is just to see that it's virtually ane/16 more five/eight, which is 10/16, and so we estimate it as 11/16.
Tin you just count lines?
The next question, in 2005, chosen for a still more detailed explanation:
Reading a Ruler III My daughter came home with a worksheet that had a ruler on it and information technology has letters on the ruler in which she is supposed to say the place (value) the letter represents on the ruler. Now it's been a while since I accept done this. I am looking for simple, bones instructions to teach a kid how to read a ruler. Of course I tin instantly tell you where a 1/2 is; however, the balance is very vague to me. I somehow remember that nosotros used to count the lines--where the line falls on the ruler is the acme number of the fraction and how many lines in betwixt 0 to ane inch represents the bottom number (the whole). Is this correct? I like your examples on how to read a ruler. However, if you can but count where the line is and put that in the numerator and count how many lines make the whole and put that in the denominator that would be helpful to make information technology so that anybody truly understands how to read the ruler.
Doctor Ian assumed that Mary had read the answers above, and needed more:
Let'southward await at an instance. Suppose I'thou measuring something that's something and 7/16 of an inch long. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ----------------------------------- xxxxxxxxxxxxxxxx
Here we encounter but the final inch of the object; the "something" would be the characterization on the inch mark at the left.
In practice, I'd look at the 1/2 marking, and say "That's too big." So I'd look at the 1/four marker, and say "That'due south too small." So now I know it's between 1/4 and ane/2, i.e., between ane/4 and 2/4, which is between 2/eight and 4/8. What'due south halfway between them? 3/eight. So I expect at 3/8, and say "That'due south too small-scale." So now I know it's halfway between 3/8 and 1/2, i.east., between three/8 and 4/eight, which is between half-dozen/sixteen and eight/16. What's halfway between them? 7/16. And that works. So I'thou done.
Another way would be to simply see that it is one pocket-size mark (1/xvi) less than 1/2, and decrease; only that requires work with fractions. We'll go to the "just count" approach Mary wants soon!
The whole system is based on successively breaking things into halves, so I just go with the catamenia. That is, yous tin imagine a ruler where the markings simply go downwards to 1/2 an inch: | | | | | | | | | | | | | | ----------------------------------- 0 i/ii 2/2 = 1 If we decide that's not fine enough, nosotros double the denominators, | | | | | | | | | | | | | | ----------------------------------- 0 two/four 4/4 = 1 and that gives us room for more numerators: | | | | | | | | | | | | | | | | | | | | ----------------------------------- 0 1/4 2/4 three/4 four/4 And and then on, through 8ths and 16ths and even 32nds, if nosotros can make our markings finely enough. Of course, this is all only impossible to deal with unless you lot're pretty facile at converting between equivalent fractions with powers of 2 in the denominator, east.grand., 1/ii = 2/4 = four/8 = 8/16 = xvi/32 That's the basic skill you need to make the organization work.
What you don't do this often enough to develop that skill?
The idea of counting all the marks to figure out what the denominator should be would work in theory, merely in practice it would exist pretty ho-hum. But with some practice, you could exercise something sort of equivalent but quicker. That is, yous could offset by identifying what size mark is adjacent to the end of the thing you're measuring: | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ----------------------------------- xxxxxxxxxxxxx^ |_________ This is the size I desire. Next, you could look at the one/2 mark, and back up halfway to the 1/4 marker. If that's the right size, y'all tin can now just count quarters. If it's still too small, back up halfway to the 1/viii mark. If that's the correct size (in the case in a higher place, information technology is), you tin only count eighths. And and so on. Note that when you're 'counting quarters' or 'counting eighths', you take to count any marks that are at least every bit large as the one you're using; merely you can ignore whatever that are smaller. And then that's pretty straightforward, but again information technology require at least a little facility with powers of 2 in the denominator.
In this example, we've identified the mark as an eighth, so we tin just count marks this big or longer: 1, 2, 3; or, every bit nosotros've said to a higher place, only count the bodily eighth marks by twos: ane, three.
Why is it called a ruler? Isn't that a king?
Several of u.s. are interested not only in numbers, but also in words. I find the following fascinating:
History of the Discussion I was merely wondering, why is a ruler chosen a ruler?
Doctor Rick answered:
Do you sometimes await at a discussion you run across every day, and suddenly it seems strange? I do that. I, likewise, wonder how a word can mean two things that seem very different. The history of words (called etymology) has lots of strange stories that sort of brand sense when you retrieve about them. A good place to commencement answering questions about words is a dictionary. Looking in my dictionary (Random Business firm Webster'south Higher Dictionary), the etymology listing under "rule" says this: [1175-1225; (n.) ME riule, reule < OF riule < L regula direct stick, pattern, der. of regere to fix the line of, direct (see -ULE). In other words: The word was outset used around 1200. Before that (in "Heart English language", it was spelled with an "i" or an "e". Why? Because it was spelled that way in Former French. The French took the Latin word "regula", dropped the g and changed the a to a silent east. (The French did that sort of thing a lot, dropping letters and not pronouncing a lot of the messages they kept!)
English does that too …
The Latin give-and-take "regula" meant a straight stick. That's a ruler: a device for making straight lines. Whether the Romans put marks on their rulers to measure lengths, I don't know, only apparently the most important matter about it was that information technology was straight. Why do I say that? Because information technology comes from the word regere, which meant "to guide or straight, to make straight." When yous use a ruler to brand a line, it keeps the line from going astray in either direction; it guides your pencil the way you desire it to go.
We still (sometimes) talk almost "ruled" paper, meaning information technology has straight lines drawn on it.
Now that we know that the word from which we get "rule" or "ruler" had "a straight stick" as its primary pregnant, our question gets turned around: How did "rule", pregnant "a straight stick", come to hateful "tell others what to practise", or "a law that we take to follow"? But maybe you tin can already see the answer. A ruler sets the standard; it's a pattern to be followed. If yous put a ruler next to a line, y'all can tell right away if the line is crooked. A rule or regulation (observe where *this* word comes from?) sets a standard, so everyone knows if you "break the rule". Someone who sets the rules for others to follow is called a ruler.
Did y'all observe that the original meaning of ruler was just "straightedge", without reference to having marks on it?
There are lots of other words related to rule, ruler and regulation. Something that is "regular" follows the rules; something irregular breaks the rules. The Romans put the prefix de- in front of regere and it became derigere: to straighten or direct--we go the word direct (and the word dirigible, something that can be steered) from derigere. And detect the "rect" office of directly: all sorts of words, such as rectangle, and even right (angle), come from Latin rectus, which means "made straight" from regere, to brand directly. There's more on that here: Left Angles http://mathforum.org/library/drmath/view/58420.html If you constitute that interesting, I hope you can take a Latin class in a few years. You'll acquire a lot well-nigh words that fashion. If you lost interest after the first paragraph, ... anyway, I'grand washed now.
Adjacent fourth dimension, permit's look at protractors!
Source: https://www.themathdoctors.org/using-a-ruler/
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