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.
- The square root of a negative number doesn’t exist: e. g. doesn’t exist.
- The square root of zero is only zero.
- 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