Are surds the only way to get exact numbers?

Hi Willie,

Surds are square roots, third roots, fourth roots, etc.

Please elaborate what you mean by getting exact numbers from surds. In what situation? When what operations are involved?

Best regards,

Thembinkosi

Surds are a convention, a way of denoting certain roots that cannot be written in another way. There are, however, plenty of ways of writing exact values. Trivially, all the whole numbers are exact values, as are all the fractions (rational numbers). Fractions can be written exactly in decimal form if their denominator is a product of only 2s and 5s. Otherwise you could write them in an equivalent form in a different base e.g. if you work in base 3 then what was 0.3333..... in decimal becomes 0.1 in base 3. Surds are used because the (square, cube, fourth,...) roots of some integers and rational numbers give irrational results - ones that can't be written as a fraction.

If you are asked to give an exact answer in your exercise, then yes, you need to leave the answer in surd form. Otherwise, if you find a value of the surd using a calculator, it will be a rounded answer to some requested degree (1dp, 2dp, 3sf, etc).

Hi Willie,

To answer your question we first need to understand what an exact number is.

Let's use an example. Say we had a circle with a diameter of 1 meter and we wanted to find the circumference. This can be done in two ways: we can take out our ruler and try to measure the circle or we could use the formula Circumference = π x diameter.

Say we measure the circumference and find the answer to be 3.14 meters. We can only find an answer to the nearest millimeter because the ruler cannot measure anything smaller than that. If we use the formula we get the circumference to be π meters. We didn't have to round or estimate to get this number. So 3.14 is an estimated value. of the circumference while π is an exact value. So an exact number is one that hasn't been rounded or estimated.

If the diameter of the circle was √2/π the circumference would be √2 or if the diameter was 2/π the circumference would be 2. So both √2 and 2 can be exact numbers.

And so to answer your question: no, surds are not the only way to get exact numbers.

Hi Willie,

Surds are irrational numbers. Surds are numbers like the square root of 2, or the square root of 60, or the cube root of 100. Here's an example of something that isn't a surd: the square root of 25. This is NOT a surd, because you CAN evaluate it. 5 and -5 are the square roots of 25.

By 'exact' numbers do you mean integers? Integers are whole numbers.

However, what I assume you mean is... Is it possible to answer an A-level question, when they ask for 'exact' solutions only, by only using surds? Surds are not the only example of 'exact' solutions. Other irrational solutions you could give are:

Answers in terms of surds.

Answers in terms of pi.

Answers in terms of e.

I hope this helps!

Hi Willie,

No, fractions can be exact numbers as well. The accuracy of our number in a practical sense is limited by what we are using to measure it.

For example measuring a reagent for a chemistry experiment using scales that are only accurate to 2 decimal places, the exact amount of reagent could be a number with 6 decimals places or 9 etc.

Another example is using the sd button on your calculator, when doing a calculation it may give you a fraction instead of a large recurring decimal, 1/3 is 0.333 recurring.

Hope this helps.

Jason

Hello Willie,

Surds do give an exact number. However, when we think of a surd as exact number, it is equivalent to number in the form of a decimal which does not terminate or recur.

An example:

Square Root of 7 = 2.64575131..... This exact number does not recur or terminate. Instead the calculator will round it to several decimal places and we can see no zero`s or repeated consecutive numbers after the decimal as proof of being a recurring or terminated decimal.

I hope this is the answer you are looking for.

Shabiib

No, surds are not the only way to get exact numbers

Surds are a way of representing irrational numbers in an exact form. These numbers cannot be expressed as a fraction. There are other types of exact numbers, such as integers and rational numbers, which can be expressed as a fraction.

Exact form can be as a fraction, surd, interns of a constant e.g. pi when working with circles or a terminating decimal.

Hi Willie, while surds often appear when writing exact numbers, they are not always involved. For example, the exact value of 0.33333... is 1/3. Surds will appear when a square root has been part of the process that generated the number you're dealing with. In general, exact numbers (aside from integers) are often written with fractions and/or indices. I hope this answers your question. Ed

Exact numbers can be in any form as long as you don't round off anything.

Other than surds, exact numbers can be in terms of pi, e or using fractions, logarithms, trigonometry or stopping decimals.