3.1.1 Rational Numbers
A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers.
- The number 8 is a rational number because it can be written as the fraction 8/1.
- Likewise, 3/4 is a rational number because it can be written as a fraction.
- Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction.
Every whole number is a rational number, because any whole number can be written as a fraction. For example, 4 can be written as 4/1, 65 can be written as 65/1, and 3,867 can be written as 3,867/1.
3.1.2 Irrational Numbers
All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction.
An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers:
π = 3.141592…
= 1.414213…
Although irrational numbers are not often used in daily life, they do exist on the number line. In fact, between 0 and 1 on the number line, there are an infinite number of irrational numbers!
| Example
Which of the following numbers are rational? 1; −6; 3½; − 2; 3; 0; 7.38609 Answer: All of them!
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A rational number can always be written as a fraction a/b, where a and b are integers (b not equal to 0).
So when a and b are positive, that is, when they are natural numbers, then their ratio can always be named. Hence the term rational number.
Now a fraction can always be expressed as a decimal. Either the decimal will terminate — as 1/4 = 0.25; or the decimal will have a predictable pattern — as 1/11 = 0.090909. . .
A rational number, then, can always be expressed as such a decimal.At this point, you might wonder, what is a number that is not rational?
(“Square root of 2”) is not rational. There is no whole number, no fraction, and no decimal whose square is 2.
Try it – do you get 1.414? How would one know that this is not a rational number? Because (1.414)² = 1.999396 — which is approximately 2. No decimal squared will ever produce exactly 2 because if the decimal ends in the digit 1, then its square will also end in 1. If the decimal ends in 2, its square will end in 4. And so on. There is no decimal — no rational number — whose square is exactly 2.000000000. We call √2 an irrational number.
By recalling the Pythagorean Theorem, we can see that these irrational numbers are necessary. For if the sides of an isosceles right triangle are called 1, and then we will have 1² + 1² = 2, so that the hypotenuse is √2.
There really is a length that logically deserves the name, “√2.” Insofar as numbers name the length of lines, then√2 is a number. Which numbers have rational square roots? Only the square roots of the square numbers.
Thus, √1 = 1, which is rational. , √2, √3 are irrational. √4 = 2 –rational. , √5, √6, √7, √8, are irrational. We then come to√9 = 3, which is rational. And so on. Only the square roots of the square numbers are rational.
When the ancient Greeks first realized the fact of irrationals, they called them unnamable or speechless. For if we ask, “In the isosceles right triangle, what ratio has the hypotenuse to the side?” — we cannot say. We can name it only as “Square root of 2.”
The decimal representation of irrationals
When a rational number is expressed as a decimal, then either the decimal will terminate, or there will be a predictable pattern of digits. But when an irrational number is expressed as a decimal, then, clearly, the decimal cannot terminate (for if it did, the number would be rational), and there will not be a predictable pattern of digits.
Famous Irrational Numbers
| Pi is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. The first few digits look like this:
3.1415926535897932384626433832795 (and more …) |
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| The number e (Euler’s Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this:
2.7182818284590452353602874713527 (and more …) |
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| The Golden Ratio is an irrational number. The first few digits look like this:
1.61803398874989484820… (and more …) |
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But √4 = 2 (rational), and √9 = 3 (rational) … So not all roots are irrational. |
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Now, with rational numbers you sometimes see
1/11 = .090909. . .
Here, the three dots (ellipsis) mean “And so on, according to the indicated pattern.”
But if we write ellipsis for an irrational number —
= 1.41421356237. . .then here the three dots mean “And so on, according to the rule for calculating the next digit.”, but there is no pattern!
3.2 REPEATING DECIMALS
The problem that you will be confronted with is the following: “Convert 0.757575 to a common fraction”.
0.757575… =?
If you need to solve the problem, consider the following method:
x = 0.757575… Equate x to the decimal fraction
100x = 75.757575… Multiply the equation with 100. (see below for a detailed explanation)
100x – x = 75 Subtract x, or the value of x from both sides of the equation.
(75.75757575 – .75757575 = 75 on the right hand side)
99x = 75 Simplify the left hand side.
x = 75/99 Divide the equation with the resultant number (99) to get the value of x
Does that make sense?
Here is some more detail:
As you may already know, every fraction (technically, a fraction is called a rational number) either terminates – ends with a string of 0’s – or is a repeating decimal (of course, you could think of the string of zeros as just a special kind of repeating decimal).As we have seen above, the decimals neither terminate nor repeat cannot be represented by fractions (with integers on the top and bottom), and are called “irrational” numbers (see the section on irrational numbers). So the question that occurs is how to find out what fraction that repeating decimal is equal to. There is a trick to it. Here’s the trick.
Suppose that you have a repeating decimal, and it looks like
(a)(a)(a)…where (a) is some sequence of repeating digits (technically, (a) is called the “repeated,” i.e., “the thing which is repeated”).
| Example
1/3 = .3333333…, (a) is 3 1/11 = 0.09090909…, (a) is 09 1/7 = o.142857142857…. (a) is 142857 |
3.2.1 Converting repeating decimals to common fraction.
First, you have to count the number of digits in the repeated. When (a) is 3, the number of digits is 1, when (a) is 09, the number of digits is 2, and when (a) is 142857, the number of digits is 6.
Now, multiply your repeating decimal by a power of 10, namely, the power of 10 which is a 1 followed by a number of zeros equal to the number of digits in the repeated. That’s a mouthful, so let’s see how it works in the examples above:
For 0.33333…, the repeated is 3, and that has *one* digit, so multiply by a 1 followed by *one* zero, i.e., by 10 For 0.090909…, the repeated is 09, which has *two* digits, so multiply by a 1 followed by *two* zeros, i.e., by 100 For 0.142857142857…, the repeated is 142857, which has *six*digits, so multiply by a 1 followed by *six* zeros, i.e., by 1,000,000 (one million).
If we multiply the repeating decimal by a power of 10 in this way, we end up with a decimal which has the repeated to the LEFT of the decimal point, and the same repeating decimal we started out with to the RIGHT of the decimal point:
Multiply 0.33333… by 10, and we get 3.33333…
Multiply 0.090909… by 100, and we get 9.090909…
Multiply 0.142857142857… by 1000000, and we get 142857.142857…
But now note that after we multiply by this appropriate power of 10, we get the sum of an integer (which is numerically equal to whatever the repeated was) and the repeating decimal we started out with. If we let x be the repeating decimal we started out with, we find:
If x = .333.., then 10x = 3.333… = 3 + .333… = 3 + x.
That is, we get 10x = 3 + x.
If you remember your algebra, subtracting x from both sides of this equation gives us 9x = 3, so that x = 3/9 = 1/3, after we reduce the fraction to lowest terms.
If x = .0909…, I hope you can see that we get 100x = 9 + x, or 99x = 9, or x = 9/99 = 1/11.
And if x = .142857142857…, do you see that 1000000x = 142857 + x?
Solve that for x, and you get x = 142857/999999 = 1/7 (though the reduction to lowest terms takes a little longer here if you forget that we know that this better be the reduction, since that’s how we got the repeating decimal in the first place!).
If you have followed this argument, then maybe you can see that in general, if the repeating decimal has (a) as the repeated, then the fraction that is represented by that repeating decimal is just (a)/R where R is a number with the same number of digits as (a), but all the digits are 9’s.
Thus,
0.567567… = 567/999 (= 21/37 after reduction)
0.42014201… = 4201/9999 (is already reduced to lowest terms) and so on.
So now that you feel you understand this, let’s look at an interesting case.
If we looked at 0.333333…, who said the repented was 3? Couldn’t it just as well be 33, or 333, or even 333333333333333333333?
The answer is that it could, but (and this takes a little more work) it ends up giving you the same fraction after reduction to lowest terms.
| Exercise
1. Identify the rational numbers by saying the name of each number: Say whether the number is rational or irrational. a) √3 b) √5 c) √4 d) √3/5 e) √4/9 2. Convert the following repeating decimal to the common fraction form 0.090909 3. Convert 0, 123123123… to a common fraction |
Exercise
3.3 SYMBOLS FOR IRRATIONAL NUMBERS ARE LEFT AS IS
3.3.1 Integer
Integers are like whole numbers, but they also include negative numbers … but still no fractions allowed!
Integers are all the positive whole numbers, negative whole numbers, and zero. For example, 43434235; 28; 2; 0; -28; and -3 030 are integers, but numbers like 1/2, 4.00032, 2.5, Pi, and -9.90 are not. We can say that an integer is in the set: {…3,-2,-1,0,1,2,3,…} (the three dots mean you keep going in both directions.) It is often useful to think of the integers as points along a ‘number line.
The terms even and odd only apply to integers; 2.5 is neither even nor odd. Zero, on the other hand, is even since it is 2 times some integer: it’s 2 times 0. To check whether a number is odd, see whether it’s one more than some even number: 7 is odd since it’s one more than 6, which is even. Another way to say this is that zero is even since it can be written in the form 2*n, where n is an integer. Odd numbers can be written in the form 2*n + 1. Again, this lets us talk about whether negative numbers are even and odd: -9 is odd since it’s one more than -10, which is even.
Every positive integer can be factored into the product of prime numbers, and there’s only one way to do it for every number. For instance, 280 = 2x2x2x5x7, and there’s only one way to factor 280 into prime numbers. This is an important theorem: the Fundamental Theorem of Arithmetic.
Rational Numbers 5/1, 1/2, 1.75, -97/3
A rational number is any number that can be written as a ratio of two integers (hence the name! Ratio – nal). In other words, a number is rational if we can write it as a fraction where the numerator and denominator are both integers.
Rational The term “rational” comes from the word “ratio,” because the rational numbers are the ones that can be written in the ratio form p/q where p and q are integers. Irrational, then, just means all the numbers that aren’t rational.
Every integer is a rational number, since each integer n can be written in the form n/1. For example 5 = 5/1 and thus 5 is a rational number. However, numbers like 1/2, 45454737/2424242, and -3/7 are also rational; since they are fractions whose numerator and denominator are integers.
So the set of all rational numbers will contain any of the following numbers 4/5, -8, 1.75 (which is 7/4), -97/3, and so on. Is 0,999 repeating a rational number? Well, a number is rational if it can be written as A/B (A over B): 0,3 = 3/10 and 0,55555….. = 5/9, so these are both rational numbers. In fact all repeating decimals like 0,575757575757… , all integers like 46, and all finite decimals like .472 are rational.
Irrational Numbers √2 , π (pi), e, the Golden Ratio
Irrational numbers are numbers that can be written as approximate decimals but not as fractions.
An irrational number is any real number that is not rational. By real number we mean, loosely, a number that we can conceive of in this world, one with no square roots of negative numbers (such a number is called complex.)
A real number is a number that is somewhere on a number line, so any number on a number line that isn’t a rational number is irrational. The square root of 2 is an irrational number because it can’t be written as a ratio of two integers. Other irrational numbers include the square root of 3, the square root of 5, pi, e, and the golden ratio.
π (pi) is an irrational number because it cannot be expressed as a ratio (fraction) of two integers: it has no exact decimal equivalent, although 3.1415926 is good enough for many applications. The square root of 2 is another irrational number that cannot be written as a fraction.
In mathematics, a name can be used with a very precise meaning that may have little to do with the meaning of the English word. (“Irrational” numbers are NOT numbers that can’t argue logically!)
| Exercise
Symbols for irrational numbers are left as is Define what is meant by an irrational number and give one example |
3.4 ROUNDING PREMATURELY IN CALCULATIONS
When a number is rounded (or rounded off), it is approximated by eliminating the least significant digits. When rounding, you are finding the closest multiple of ten (or one hundred, or other place value) to your number. For example, the number 42 can be rounded down to 40 (this number was rounded to the tens place). Similarly, 285 can be rounded up to 300 (this number was rounded to the hundreds place).
Rounding is used to make a number easier to work with. For example, if you know that there are 496 students in your school, you can say that there are approximately 500 students in your school.
On a number line, you can see how rounding a number approximates its value.
Whole numbers can be rounded to the tens place, hundreds place, thousands place, and so on. When a number is rounded to the tens place, the final form has a zero for the ones digit. When a number is rounded to the hundreds place, the final form has a zero for both the tens digit and the ones digit.
Decimal numbers can also be rounded; this approximates the number to the nearest tenth, hundredth, thousandth, or other decimal place. When a decimal number is rounded to the tenths place, the final form has no digit in the hundredths place (or any places to the right of that). When a decimal number is rounded to the hundredths place, the final form has no digit in the thousandths place (or any places to the right of that).
When you work out long sums on your calculator, it is very important that you do not round off the numbers in the middle of the sum. You must carry as many digits throughout your calculation as you can. This may seem ridiculous because you have to write down such long numbers.
| Example
That we measure the weight of an object as 3.28 g on a balance believed to be accurate to within ±0.05 gram. The resulting value of 3.28±.05 gram tells us that the true weight of the object could be anywhere between 3.23 g and 3.33 g. The absolute uncertainty here is 0.1 g (±0.05 g), and the relative uncertainty is 1 part in 32.8, or about 3 percent.
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How many significant digits should there be in the reported measurement? Since only the leftmost “3” in “3.28” is certain, you would probably elect to round the value to 3.3 g. So far so good. But what is someone else supposed to make of this figure when they see it in your report? The value “3.3 g” suggests an implied uncertainty of 3.3±0.05 g, meaning that the true value is likely between 3.25 g and 3.35 g. This range is 0.02 g below that associated with the original measurement, and so rounding off has introduced a bias of this amount into the result. Since this is less than half of the ±0.05 g uncertainty in the weighing, it is not a very serious matter in itself. However, if several values that were rounded in this way are combined in a calculation, the rounding-off errors could become significant.
| Exercise
Descriptions are provided of the effect of rounding prematurely in calculations If you need 36 m of copper wire to wind an alternator and you have to produce 19 500 of these – what will happen if you round of prematurely by saying 40 m for 20 000 alternators (at R 17 per meter) |
3.5 ACCURACY AND BEING PRACTICAL
Here one needs to consider to what level of accuracy one wants to solve a problem.
| Example
If you are a warden in a jail block and you have 23 inmates currently incarcerated and 8 more are allocated to you, you will then have 31 persons. If you do your evening roll-call it will not be good enough if your assistant says that there about 30 people there. It will be good enough however if you were organising a bus to transport them and you told the bus company that you needed a bus that can take about thirty to thirty five people.
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| Exercise
The desired degree of accuracy is determined in relation to the practical context: a. If you were a nurse and the doctor said you have to insert the needle deep into the person’s thigh – would you want to be accurate to the nearest mm, cm, or dm? b. If you are recording the mass of maize produced by your farm would you record your answer to the nearest ton, kg or g? c. If you have bollworm infecting your cotton crop, would you want to know the approximate size of the worm in km, m cm or mm?
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3.6 THE FINAL VALUE OF A CALCULATION IS EXPRESSED IN TERMS OF THE REQUIRED UNIT
If a calculation needs to be done to determine how many cc of adrenalin needs to be injected into a patient – the result should be in cc, not ml or cl.
Another situation that one needs to consider when expressing the final value of a calculation in terms of the required unit occurs when after calculating something you land up with a fraction of an integer.
| Example
You are a security guard at a gate and record the following traffic: 08h00 – 09h00 14 cars 09h00 – 10h00 17 cars 10h00 – 11h00 10 cars Your supervisor now asks you “On average, how many cars came through per hour in the past three hours. You pull out your calculator and find that 41 divided by three is 13,6667. Now we all know that 2/3 of a car cannot drive through the gate! So the answer you need to give is 14. |
| Exercise
The final value of a calculation is expressed in terms of the required unit: a) Given the following formula “1cc for every 50kg’s of body mass (or pro rata)” plus 0,1cc for men and 0.85cc for women, for every 5 years the person is over 40. How many cc’s would you administer in each of the following cases: A 35 year old male B 200 kg 4cc A 60 year old women 80kg 1,6cc + (4 x .85cc) = 5cc A 15 year old boy 60 kg b) Add 1m; 1cm and 1 mm – express your answer in cm Concept (SO 3) I understand this concept
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