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- 1. I.E.S. MARÍA BELLIDO - BAILÉN BILINGUAL SECTION – MARÍA ESTHER DE LA ROSA UNIT 2. DECIMAL NUMBERS AND SEXAGESIMAL SYSTEM 1. DEFINITION OF DECIMAL NUMBER "Decimal Number" usually means there is a Decimal Point. The digits to the left of the point are in the ones, tens, hundreds, thousands (and so on infinitively) place. Digits to the right of the decimal point are tenths, hundredths, thousandths (and so on ) place. 17.591 = 1 tens is 10, 7 units is 7, 5 tenths is 0.5, 9 hundredths is 0’09 and 1 thousandths is 0’001 2. READING A DECIMAL NUMBER You can read a decimal number in two ways: •Read the number after the decimal point as a whole number and give it the name of its last decimal place. Examples: 0’234 is read as two hundred thirty-four thousandths 3’12 is read as three (units) and twelve hundredths •Another way to read a decimal is: Examples: 0’234 is point, two-three-four 3’12 is three, point, one-two 23’4 Ξ twenty three, point, four
- 2. 3. CONVERTING FRACTIONS TO DECIMALS. We can convert fractions to decimals dividing the numerator by the denominator.There are three different types of decimal number: •An exact or terminating decimal is one which does not go on forever, so you can write down all its digits. For example: 0.125 •Recurring decimal is a decimal number which does go on forever, but where some of the digits are repeated over and over again. For example: 0.1252525252525252525... is a recurring decimal, where '25' is repeated forever. 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. •Other decimals are those which go on forever and don't have digits which repeat. For example pi = 3.141592653589793238462643... 4. CONVERTING DECIMALS TO FRACTIONS. In decimal form, a rational number is either an exact or a recurring decimal. The reverse is also true: exact and recurring decimals can be written as fractions. For example, 0.175 =175/1000 = 7/40. Also, 0.2222222222... is rational since it is a recurring decimal = 2/9. You can tell if a fraction will be an exact or a recurring decimal as follows: fractions with denominators that have only prime factors of 2 and 5 will be exact decimals. Others will be recurring decimals. • To convert an exact decimal to fraction, write the decimal number without a decimal point as a numerator , with a denominator beginning with a denominator beginning with one and having as many zeros as there are numbers after the decimal. Example: 2′ 345 = 2345 / 1000 •To convert a Recurring Decimal to a Fraction: The trick is to use a little algebra. Example Convert 0.142857142857... into a fraction. Let x = 0.142857142857... We want to move the decimal point to the right, so that the first "block" of repeated digits appears before the decimal point. Remember that multiplying by 10 moves the decimal point 1 position to the right. So in this example, we need to move the decimal point 6 places to the right (so we multiply both sides by 1 000 000): 1000000x = 142857.142857142857... Now we can subtract our original number, x, from both sides to get rid of everything after the decimal point on the right: 1000000x - x = 142857 So 999999x = 142857 x = 142857/999999 = 1/7 (cancelling)
- 3. 5. OPERATIONS WITH DECIMALS: a) ADDING OR SUBTRACTING DECIMALS To add or subtract decimals, follow these steps: •Write down the numbers, one under the other, with the decimal points lined up •Put in zeros so the numbers have the same length •Then add or subtract normally, remembering to put the decimal point in the answer b) MULTIPLYING DECIMALS Just follow these steps: •Multiply normally, ignoring the decimal points. •Then put the decimal point in the answer: it will have as many decimal places as the two original numbers combined. (just count up how many numbers are after the decimal point in both numbers) Example: Multiply 0.03 by 1. 3 × 11 = 33 0.03 has 2 decimal places, and 1.1 has 1 decimal place, so the answer has 3 decimal places: 0.033 Explanation: Because when you multiply without the decimal point (which makes it easy), you are really shifting the decimal point to the right to get it out of the way.Then we do the (now easy) multiplication: 3. × 11. = 33. But remember, we did 3 Moves of the decimal point, so we need to undo that: c) DIVIDING A DECIMAL NUMBER: BY A WHOLE NUMBER: To divide a decimal number by a whole number use long division, and just remember to line up the decimal points. BY A DECIMAL NUMBER: The trick is to convert the number you are dividing by to a whole number first, by shifting the decimal point of both numbers to the right: Now you are dividing by a whole number, and can continue as normal.
- 4. 6. SEXAGESIMAL SYSTEM Sexagesimal is a numeral system in which each unit is divided into 6 0 units of lower order, that is to say, it is a base-60 number system.T h e s e x a g e s i m a l s y s t e m w a s u s e d b y t h e Sumerians and Babylonians. It is currently used to measure time and angles. 1 h 60 min 3600 s 1º 60' 3600'' 7. CONVERTING SEXAGESIMAL INTO DECIMAL FORM. Example: Convert 3 hours, 36 minutes, 42 seconds to seconds. 8. CONVERTING DECIMAL INTO SEXAGESIMAL FORM To convert to major units, divide: 7,520'' To convert to minor units, multiply:
- 5. 9. OPERATIONS: a) Addition: 1 . Place the hours under the hours (or the degrees under the degrees), the minutes under the minutes and the seconds under the seconds and add together. 2 . If the seconds total more than 60, they are divided by 60, the remainder will remain in the seconds column and the quotient is added to the minutes column. 3 . Repeat the same process for the minutes. b) Subtraction: 1 . Place the hours under the hours (or the degrees under the degrees), the minutes under the minutes and seconds under seconds and subtract. 2 . If it is not possible to subtract the seconds, convert a minute of the minuend into 60 seconds and add it to the minuend seconds. Then, the subtraction of the seconds will be possible. 3 . Repeat the same process for the minutes.
- 6. c) Multiplication by a number: 1 . Multiply the seconds, minutes and hours (or degrees) by number. 2 . If the seconds exceed 60, divide that number by 60, the remainder will remain in the the seconds column and the quotient is added to the minutes column. 3 . Repeat the same process for the minutes. d) Division by a number: Example: Divide 37º 48' 25'' by 5. 1 . Divide the hours (or degrees) by the number. 2 . The quotient becomes the degrees and the remainder becomes the minutes when multiplied by 60. 3 . Add these minutes to the minutes column and repeat the same process for the minutes.
- 7. 4 . Add these seconds to the seconds column and then divide the seconds by the number.