How to determine which of 0.470, 0.47, and 0.4 is the largest?

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Indranil

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How to determine which one is the largest of the three values 0.470, 0.47 and 0.4?
 
Indranil said:
How to determine which one is the largest of the three values 0.470, 0.47 and 0.4?
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One of the ways one can determine - which number is bigger between two - is to use subtraction. Can you subtract decimal numbers?
 
Indranil said:
How to determine which one is the largest of the three values 0.470, 0.47 and 0.4?
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Another way is to rewrite each number with the same number of decimal places, which is equivalent to using a common denominator for fractions: 0.470, 0.470, 0.400. The zeros I added don't change the value.
 
Seriously? If A and B are two numbers then if A- B is positive, A is greater than B. If A- B is negative then A is smaller than B.
 
JeffM said:
You should be banned. You have asked questions about calculus, but now emphatically deny that you can subtract decimal numbers.
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Please please please don't get me wrong. I know basic substructions but did not know fractional substructions perfectly. I was always confused about that. I know it's a shame but it's the truth.
 
Dr.Peterson said:
Another way is to rewrite each number with the same number of decimal places, which is equivalent to using a common denominator for fractions: 0.470, 0.470, 0.400. The zeros I added don't change the value.
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So here should I consider 0.470 is the largest and 0.400 is the smallest out of 0.470, 0.47 and 0.4?
 
Indranil said:
Please please please don't get me wrong. I know basic substructions but did not know fractional substructions perfectly. I was always confused about that. I know it's a shame but it's the truth.
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OK. The mechanics are simple.

We want to calculate c = b - a, where a, b, or both have one or more digits to the right of the decimal point.

If they are not already the same, first step is to make them the same.

If a = 3.156 and b = 72.1, the two numbers do not have the same number of digits to the right of the decimal point. So add zeroes to the end of the number with fewer digits to the right of the decimal point to force equality of the number of digits.

In other words, express b = 72.100 so it has three digits to the right of the decimal point. Do you understand why 72.1 = 72.100?

Now place the numbers so that the decimal point lines up. Subtract as always and put the decimal point in line.

\(\displaystyle 72.100\)

\(\displaystyle \ \ 3.156\)

\(\displaystyle 68.944\)

Now the rules you have learned about positive and negative numbers and absolute values must be considered as well, but the actual mechanics of determining the numeric value of a difference between numbers with decimal representations is just as simple as what is shown above. Make sure both numbers have the same number of digits to the right of the decimal point, do the subtraction according to the normal rules, and stick a decimal point where the answer will also have that number of digits to the right of the decimal point.
 
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Indranil said:
How to determine which one is the largest of the three values 0.470, 0.47 and 0.4?
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If you don't like decimals … multiply each number by, say, 1000. Their numerical order will be preserved.
0.470 x 1000 = ??
0.47 x 1000 = ??
0.4 x 1000 = ??

(Which one is largest? … or maybe there are 2 that are the larger because they are actually the same number!!)
 
JeffM said:
OK. The mechanics are simple.

We want to calculate c = b - a, where a, b, or both have one or more digits to the right of the decimal point.

If they are not already the same, first step is to make them the same.

If a = 3.156 and b = 72.1, the two numbers do not have the same number of digits to the right of the decimal point. So add zeroes to the end of the number with fewer digits to the right of the decimal point to force equality of the number of digits.

In other words, express b = 72.100 so it has three digits to the right of the decimal point. Do you understand why 72.1 = 72.100?

Now place the numbers so that the decimal point lines up. Subtract as always and put the decimal point in line.

\(\displaystyle 72.100\)

\(\displaystyle \ \ 3.156\)

\(\displaystyle 68.944\)

Now the rules you have learned about positive and negative numbers and absolute values must be considered as well, but the actual mechanics of determining the numeric value of a difference between numbers with decimal representations is just as simple as what is shown above. Make sure both numbers have the same number of digits to the right of the decimal point, do the subtraction according to the normal rules, and stick a decimal point where the answer will also have that number of digits to the right of the decimal point.
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'Do you understand why 72.1 = 72.100?'-No
Could you explain it, please?
 
Indranil said:
'Do you understand why 72.1 = 72.100?'-No Could you explain it, please?
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You're asking about decimal "place values".


\(\displaystyle \quad 72.1 \; = \; 72 \; + \dfrac{1}{10}\)


\(\displaystyle 72.1\)\(\displaystyle 00\)\(\displaystyle ͏\; = \; 72 \; + \dfrac{1}{10}\)\(\displaystyle + \dfrac{0}{100} + \dfrac{0}{1000}\)


Those red fractions are each zero (in other words, the final 00 on the LEFT tells us no hundreths and no thousandths are added). Therefore, the red part does not add any value to 72.1

We are free to write any number of trailing zeros at the very end of a decimal number. All of the following are equal.

72.1
72.10
72.100
72.1000
72.10000
72.100000
72.1000000
72.10000000
72.100000000
et cetera…

None of those added zeros change 72.1
 
Two more examples of decimal place values.


\(\displaystyle 0.123456789 \; = \; 0 \; + \dfrac{1}{10} + \dfrac{2}{100} + \dfrac{3}{1\; 000} + \dfrac{4}{10\; 000} + \dfrac{5}{100\; 000} + \dfrac{6}{1\; 000\; 000} + \dfrac{7}{10\; 000\; 000} + \dfrac{8}{100\; 000\; 000} + \dfrac{9}{1\; 000\; 000\; 000}\)


\(\displaystyle 0.123006009 \; = \; 0 \; + \dfrac{1}{10} + \dfrac{2}{100} + \dfrac{3}{1\; 000} + \dfrac{0}{10\; 000} + \dfrac{0}{100\; 000} + \dfrac{6}{1\; 000\; 000} + \dfrac{0}{10\; 000\; 000} + \dfrac{0}{100\; 000\; 000} + \dfrac{9}{1\; 000\; 000\; 000}\)
 
Indranil said:
How to determine which one is the largest of the three values 0.470, 0.47 and 0.4?
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\(\displaystyle 0.47 = \dfrac{4}{10} + \dfrac{7}{100} > \dfrac{4}{10} \ \because \ \dfrac{7}{100} > 0.\)

\(\displaystyle 0.0470 = \dfrac{4}{10} + \dfrac{7}{100} + \dfrac{0}{1000} = \dfrac{4}{10} + \dfrac{7}{100} = 0.47 \ \because \ \dfrac{0}{1000} = 0.\)

\(\displaystyle \therefore 0.470 = 0.47 > 0.4.\)

If you cannot work it out in decimals, just do the problem in fractions.
 
Dr.Peterson said:
… \(\displaystyle 72\ \frac{100}{1000} = 72\ \frac{1}{10}\)
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@Indranil, Dr. Peterson makes a good point above.

Once you understand decimal place-value, it's easy to switch back and forth between most decimal forms and their equivalent Rational forms.


EG: Write the mixed number \(\displaystyle 9 \frac{23}{10000}\) in decimal form.

We know the fourth decimal place value is 10000ths, so we must position the 23 so that the last digit (3) is in the 10000ths place (the fourth decimal position):

\(\displaystyle 9 \frac{23}{10000} \; = \; 9.0023\)


EG: Write 6.275 as a Rational number

We see the last decimal-digit (5) is in the thousandths place, so we write

\(\displaystyle 6.275 \; = \; 6 \frac{275}{1000}\)

275 and 1000 are each multiples of 25, so we reduce the fraction to lowest terms by dividing both numerator and denominator by 25.

\(\displaystyle \dfrac{275 \; ÷ \; 25}{1000 \; ÷ \; 25} \; = \; \dfrac{11}{40}\)

Therefore

\(\displaystyle 6.275 = 6 \frac{11}{40}\)
 
Dr.Peterson said:
It can also be understood as simplifying the fraction:
\(\displaystyle 72.100=72\ \frac{100}{1000}=72\ \frac{1}{10}=72.1\)
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Does the decimal point(.) means 'addition' or 'multiplication' here?
like 72.1
72 + 1/10 or 72 X 1/10? which one make sense?
 
Have you never taken an arithmetic class? The "decimal point" always means "add". Sometimes a raised dot is used to mean multiplication, as in \(\displaystyle 3\cdot 6= 18\) but that is not a "decimal point" and that usage is deprecated.
 
1½ is called a "mixed number" because it's a combination of a Whole number (1) and a fraction (½).

We say 1½ outloud as "one and one-half" or "one and a half". Here, the word "and" really means "plus". We don't write + and we don't say "plus" but all mixed numbers are the sum of a Whole number and a fraction.

1½ means 1+½

All mixed numbers can be written in decimal form.

1½ = 1.5

Here, the decimal point means plus.

1.5 is 1+0.5

The fractional part of this mixed number (0.5) shows digit 5 in the tenths place.

0.5 = 5/10

5/10 reduces to 1/2

1½ = 1+½ = 1+0.5 = 1.5


Note: some fractions have a repeating decimal form.

1/3 = 0.3333… (the dots indicate digit 3 repeats forever)

4/7 = 0.57142857142857142857142857…

Another way to write a repeating decimal is to draw a bar over the repeating part.

\(\displaystyle \frac{1}{3} = 0.\overline{3}\\
\;\\
\frac{4}{7}= 0.57\overline{142857}\)


:cool:
 
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