Archive for November, 2010

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Lesson 3: Curiosities and Jokes (IV)

November 19, 2010

The number Pi

The mathematical constant pi (π) is defined as the ratio of a circle’s circumference to its diameter. It is also the ratio between a circle’s area and its radius squared. It is approximately equal to 3.14159, but has been calculated to over one trillion digits.

A Pythagoras’s student

The ancient Greek mathematician Pythagoras believed that all numbers were rational (could be written as a fraction), but one of his students Hippasus proved (using geometry, it is thought) that you could not represent the square root of 2 as a fraction, and so it was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus’ “irrational numbers” and so Hippasus was thrown overboard and drowned!

  

Jokes

We will finish with

Popular Pi Jokes

  • Q: What do you get when you take the sun and divide its circumference by its diameter?
  • A: Pi in the sky.
  • Mathematician: Pi r squared”
  • Baker: No! Pies are round, cakes are square!

Click on The “Pi” Joke and you will watch a funny animation.

 Here you are pi song

If you are in the mood you can listen the complete version here .

See you!!

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Lesson 3: Decimal Numbers (III).

November 15, 2010

4. Approximating and estimating decimal numbers

 4.1 . Approximating decimal numbers

Rules for rounding decimals:

1. Retain the correct number of decimal places (e.g. 3 for thousandths, 0 for whole numbers).

2. If the next decimal place value is 5 or more, increase the value in the last retained decimal place by 1.

Some uses of rounding are:

  • Checking to see if you have enough money to buy what you want.
  • Getting a rough idea of the correct answer to a problem

Example:

36,74691 rounded to the nearest whole number is 37.

12,34690 rounded to the nearest tenth is 12.3.

89,46917 rounded to the nearest tenth is 89.5.

 50.02139 rounded to the nearest hundredth is 50.02.

Practise rounding on this web.

Practise on this website.

Rule for truncating decimals:

Retain the correct number of decimal places (e.g. 3 for thousandths, 0 for whole numbers).

Example:

Consider the number

5,6341432543653654

To truncate these numbers to 4 decimal digits (to the nearest ten thousandth), we only consider the 4 digits to the right of the decimal point.

The result would be:5,6341  

Note that in some cases, truncating would yield the same result as rounding, but truncation does not round up or round down the digits; it merely cuts off at the specified digit.

4.2. Estimating the sum or the difference of two decimals 

To estimate a sum by rounding:

  • Round each decimal term that will be added or subtracted.
  • Add or subtract the rounded terms.

Now is time to practise. Click on

4.3. Absolute error

 The difference between the exact mathematical value of an operation or number , and the calculated approximation value , considered always positive, (with the absolute value taken) is called the absolute error,

 Δx = |x0  − x|

 In Spanish, we write it:

Ea= |valor exacto  − valor aproximado| 

5. Word problems with decimal numbers

We will finish the lesson with our habituals, Word Problems.

You can practise on the following websites:

Mathgoodies: This is a web when you have solved examples and several exercises.

Kwiznet: Problems with solutions

IXL: Problems you must solve but you can check your solutions and getp explanations.

Here you are a video on solving problems

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Lesson 3: Operations With Decimals (II)

November 13, 2010

2. Operations with decimals

I suppose you remember how to operate with decimals but if you have any problem here you are an abstract on how to add up, subtract, multiply or divide decimal numbers.

2.1. Adding and subtracting decimals

To add or subtract decimal numbers:

  • Write the numbers in a column so the decimal points are directly lined up.
  • When one number has more decimal places than another, use 0’s to give them the same number of decimal places.
  • Add or subtract each column starting at the right side following the rules for adding or subtracting whole numbers and placing the decimal point in the same column as above.

You can practise on the web.

2.2. Multiplying decimals

To multiply decimals:

  • Multiply like if they were whole numbers
  • To decide how many digits to leave to the right of the decimal point, add the numbers of digits to the right of the decimal point in both factors.

You can practise on the web.

2.3. Dividing decimals

The procedure for the division of decimals is very similar to the division of whole numbers. We are going to study different cases that can appear:

DIVISOR WHOLE NUMBER

  • Dividend decimal number. If the divisor is a whole number and the dividend is decimal, proceed with the division as you normally would except put the decimal point in the quotient when you divide the first decimal digit in the dividend.
  • Dividend whole number. If the divisor and the dividend are whole numbers, to obtain decimal digits in the quotient we convert the dividend into decimal: add a decimal point followed by as many zeros as decimal digits we want in the quotient.

DIVISOR DECIMAL NUMBER

  • Dividend whole number. If the divisor is a decimal number and the dividend is a whole number, make the divisor into a whole number by multiplying both it and the dividend by the same number (such as 10, 100, 1000 etc.). An easy way to do this is to move the decimal point to the right end of the divisor and add as many zeros to the dividend as decimal digits has the divisor.
  • Dividend decimal number. If the divisor and the dividend are decimal numbers, make the divisor into a whole number by multiplying both it and the dividend by the same number (such as 10, 100, 1000 etc.). An easy way to do this is to move the decimal point to the right end of the divisor and move the decimal point of the dividend the same number of places. If it is necessary, add zeros to the dividend.

You can practise on the web., but be careful because in English speaking countries the division is written in other way. You must write the quotient on the divisor not under.

3. Square root

In this lesson, we will learn to work out the square root with pencil and paper. However, we studied on this topic in lesson 1. Remember the following definitions.

The perfect square root or exact square root of a number a is other number b, such as if it is squared b2 , we obtain the number a.

REMEMBER:

  1. The square root of a negative number doesn’t exist: e. g. doesn’t exist.
  2. The square root of zero is only zero.
  3. Every positive number has two square root with the same absolute value and opposite signs.

The whole square root of  a is the greatest integer  b whose square is less than a, this is  b2<a . We work out the remainder  by subtracting:

Radicand- Root2 = Remainder

3.1. The algorithm to find the square root of a natural number

To find the square root of a natural number -using a piece of paper and a pencil- you have to take the steps we will see in the link below (click on the picture). If you want to practise the algorithm, click on the right arrow at the bottom of the screen.

3.2. The algorithm of the square root with decimal digits

 Only the natural numbers that are square perfects have another natural number as a square root. In the rest of the cases, in order to find its square root more accurately and exactly, you have to get decimal digits.

To work out the square root with decimal digits we follow the same method as for natural numbers, with a slight modification.

You can see it in the link below (click on the picture). If you want to practise the algotithm, click on the right arrow at the bottom of the screen.

WHAT IS AN ALGORITHM?

 An algorithm is the set of calculations and procedures to work out an operation. For example, when we were children we were taught to sum up using a pencil and a piece of paper, we were told to align the numbers in columns starting from the right with the units of the same range below one another, to sum up the units, to write down the outcome below – if it is larger than 9, the extra digit is carried into the next column – and to add it to the tens…

Each step of an algorithm has a reason for being, but we can implement an algorithm and work out an operation correctly and not know why we have to do it that way.

You can know why the algorithm of the square root works on

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Lesson 3: Decimal Numbers (I)

November 11, 2010

Vocabulary on Decimals

Lesson 3: Decimal Numbers (Notes)

Worksheet on Changig Decimals to Fractions

Word Problems on Decimals

I Have to know by the End of this Lesson

1. Decimal Numbers

Decimal numbers such as 3.762 are used in situations which call for more precision than whole numbers provide.

As with whole numbers, a digit in a decimal number has a value which depends on the place of the digit. The places to the left of the decimal point are ones, tens, hundreds, and so on, just as with whole numbers. This table shows the decimal place value for various positions:

Note that adding extra zeros to the right of the last decimal digit does not change the value of the decimal number.

Place (underlined) Name of Position
1.234567 Ones (units) position
1.234567 Tenths
1.234567 Hundredths
1.234567 Thousandths
1.234567 Ten thousandths
1.234567 Hundred Thousandths
1.234567 Millionths

 Whole Number Portion

The whole number portion of a decimal number are those digits to the left of the decimal place.

Decimal  Portion

The decimal portion of a decimal number are those digits to the right of the decimal place.

Example:

In the number 23.65, the whole number portion is 23 and decimal portion is 65.

In the number 0.024, the whole number portion is 0 and the decimal portion is 024.

1.1. Types of decimal numbers

Any number can be written in “decimal form”.

There are three different types of decimal number: exact, recurring and other decimals.

  • An exact or terminating decimal is one that does not go on forever, so you can write down all its digits.

Example: 0.125

  • A recurring decimalis a decimal number which does has infinite decimal figures, but where some of the digits are repeated over and over again. We can distinguish two types:
    • Pure recurring decimals are those decimals in which all the digits after the decimal point form a pattern that is repeated indefinitely.

Example: 2, 343434……… is a pure recurring decimal, where all the digits after the decimal point, 34, are repeating.

    • Mixed recurring decimals are those decimals the first decimal(s) after the decimal point are not repeated, but the subsequent numbers form a pattern, which is repeated indefinitely.

Example: 0.1252525252525252525… is a mixed recurring decimal, where ’25’ is repeated forever and 1 is not repeated.

Sometimes recurring decimals are written with a bar over the digits which are repeated, or with dots over the first and last digits that are repeated. In Spanish we use an arc.

Example:

  • Other decimals are those which go on forever and don’t have digits which repeat.

Examples:

√2=1,414213562…, π =3,141592653589….

1.2. Changing fractions to decimals

A fraction can be changed into a decimal by dividing the numerator by the denominator. You can use a calculator, or divide.

The result could be:

1. A whole number, this is the case when the numerator is a multiple of the denominator. For example: 72/9=8.

2. A terminating decimal, this is the case when, after simplifying the fraction, the denominator only has 2 and 5 as a factors. For example: 197/40=4,925.

3. A pure recurring decimal, this is the case when, after simplifying the fraction, the denominator has only factors different from 2 and 5. For example: 4/11=0,36363636…

4. A mixed recurring decimal, this is the case when, after simplifying the fraction, the denominator has 2 or/and 5 as a factors and other factors different from them. For example: 87/66=1.3181818…

Here you are a video on changing fractions to decimals. You will find they have a tricky method to do this without working out the calculator.

And the reverse?

The reverse is also true: exact and recurring decimals can be written as fractions.

1.3. Converting decimals to fractions

CONVERTING AN EXACT DECIMAL TO A FRACTION

This is an easy question. To do this:

  • Write as a numerator the number with no decimal point.
  • Write as a denominator 1 followed as many zeros as digits has the decimal portion.

Example:

Convert 0,175 into a fraction.

175/1000              Work it out if you don’t believe it !

CONVERTING A RECURRING DECIMAL TO A FRACTION

We know that recurring decimals can be written as fractions. The trick is to use a little algebra. But it is difficult to insert equations on the blog so I will give you the trick:

1. Write in the numerator the difference between the number without the decimal point and the whole number part.

2. Write in the denominator as many nines as decimal digits has the pattern

This trick is valid for any pure recurring decimal number. If the number is mixed recurring decimal then the trick is similar. But “the other decimals”, like π, can’t be changed to fractions.

Here you are a video on changing decimal numbers to fractions. It is the easier case: changing exact decimals to fractions.

1.4. Relation between rational numbers and decimal numbers

We have just seen, in the section above, how we can get four different types of numbers when we divide a fraction (a whole number, a terminating decimal or a pure or mixed recurring decimal).

Therefore, we could say that any fraction can be expressed as a whole number, a terminating decimal or a recurring decimal. We can also say that the opposite is true, i.e. that any whole number, terminating decimal or recurring decimal can be expressed as a fraction. From now on we shall refer to these numbers as rational numbers.

We call fractional numbers to that fractions that can not be converted into whole numbers like 1/3,  4/6,  9/10 .

To sum up:

ANY NUMBER THAT CAN BE EXPRESSED AS A FRACTION IS KNOWN AS A RATIONAL NUMBER

We have just seen how we can get four different types of numbers when we divide a fraction. All of these numbers are RATIONAL. We are going to refer to this group of numbers with the letter Q.

We can classify the set Q of RATIONAL NUMBERS as follows:

 

1.5. Comparing Decimal Numbers

Symbols are used to show how the size of one number compares to another. These symbols are < (less than), > (greater than), and = (equals).

To compare the size of decimal numbers:

1. We compare the whole number portions. The larger decimal number is the one with the lager whole number portion. If the whole number parts are both equal, we compare the decimal portions of the numbers.

2. The leftmost decimal digit is the most significant digit. Compare the pairs of digits in each decimal place, starting with the most significant digit until you find a pair that is different. The number with the larger digit is the larger number.

Note that the number with the most digits is not necessarily the largest.

Example:

Compare 0.402 and 0.412.

1. The numbers 0.402 and 0.412 have the same number of digits, and their whole number parts are both 0.

2. We compare the next most significant digit of each number, the digit in the tenths place, 4 in each case. Since they are equal, we go on to the hundredths place, and in this case, 0<1, so 0.402<0.412.

Time to work ! Compare decimals on this web.