Q Skills Review Dr. C. Stewart Measurement 4: Scientific Notation The Decimal System We are so very familiar with our decimal notation for writing numbers that we usually take it for granted and do not think about what it actually means. For example, we know that 6 means six “ones” or “units”, but when we write 60, then the 6 indicates six tens, or sixty, of the units. The zero must be written to ensure we know that the 6 is in the tens’ place. When we write the digits in a row, we add the values, so that 64 indicates “six tens plus four” of the units. If the quantity we want to indicate is not a whole number then we sometimes use fractions, for example 6⅓ indicates “six units plus one third of a unit”, but this takes us away from the idea of simply writing digits in a row. Let us now consider only those fractions in which the denominator is a power of ten, such as 1 4 9 10 ,1000 , 1000000 . Fractions of this type are called decimal fractions. They are a continuation of the base ten place-value notation, and with this understanding we can simply continue to write down the digits of the numerators in the correct place. We keep track of the value associated with each digit by using a ‘decimal point’ to separate the digits representing whole numbers from those representing parts of a whole. For example, the number 2453.176 represents 2 thousands, 4 hundreds, 5 tens, 3 ones (or units), 1 tenth, 7 hundredths and 6 thousandths. We can expand 2453.176 to show the meaning of this notation as follows: 1 7 6 2453.176= 2000 + 400 +++++ 50 3 10 100 1000 If the numerator of a decimal fraction has more than one digit, then the fraction can be expressed as the sum of decimal fractions with single digit numerators by considering equivalent fractions, 57 50 7 5 7 0 5 7 e.g. 1000 = 1000 + 1000 = 100 + 1000 . This can then be written as 10 + 100 + 1000 to allow us to write it as 0.057 in decimal notation. The ‘0’ after the decimal point is necessary to ensure the 5 and 7 are in the correct places. By convention we very rarely write ‘0’ at the beginning of a number larger than one, and so seeing ‘0’ at the beginning of a number indicates that it has a value less than one. It is important to remember to write this ‘0’, is the convention in Canada: according to the Government of Canada’s terminology and linguistic data bank (www.btb.termiumplus.gc.ca, writing tools, The Canadian Style, section 5.09), “Normally, no number should begin or end with a decimal point. A zero is written before the decimal point of numbers smaller than 1, while in whole numbers the decimal point should either be dropped or be followed by a zero: $0 .64 not $.64; 11 or 11 .0 not 11 .”. This is also an internationally agreed convention (www.bipm.org/en/si/si_brochure, section 5.3.4). One feature of the decimal system is that each digit represents the value of the digit multiplied by a power of ten. We can use this feature to multiply or divide numbers by powers of ten very easily. For example, what is the value of 56.437× 10 2 ? If we look at the meaning of the notation 4 3 7 2 we see that 56.437 means 50++ 6 10 + 100 + 1000 , and so when we multiply by 10= 100 , we get 4 3 7 4 3 7 (50++++ 610 100 1000 ) × 100 =× 50 100 +× 6 100 + 10 ×100 + 100 × 100 +1000 × 100 7 =5000 + 600 +×+ 4 10 3 + 10 The end result is that 56.437× 102 = 5643.7 Isn’t this amazing? 2 [You’re probably wondering what the big deal is, but think about the notations used before the decimal system. How would you do this in Roman Numerals, for example? Apart from difficulties in expressing 0.437, what would 56× 100 = 560 look like? It would be written as LVI× C = DCLX . Because the decimal system is positional (the place where the digits are written indicates their value), then when we multiply (or divide) by a power of ten, we move the digits to a different place, but we use the same digits in the same order!] We often say we are moving the decimal point by the number of places indicated in the exponent of the ten. Although this is technically incorrect, since the decimal point always remains between the whole numbers and the fractional part, it relates easily to what we see written down. In 56.437× 102 = 5643.7 we feel that the decimal point has been moved 2 places to the right. 472.3 400 70 2 3 1 Consider the following example: 472.3÷= 10 2 = + ++× 100 100 100 100 10 100 7 2 3 =4 +++ = 4.732 10 100 1000 472.3 1 But 472.3÷= 102 = 472.3 ×= 472.310 × −2 , and so 472.3× 10−2 = 4.723 . Here it looks 102 10 2 like the decimal point has been moved 2 places to the left. In summary: When we multiply by a positive power of ten, 10 n , we see the decimal point is now n places to the right. When we divide by a positive power of ten, 10 n , or multiply by a negative power of ten, 10 −n , we see the decimal point is now n places to the left. Scientific Notation Very large or very small numbers can be difficult to read and so we sometimes use words like million or trillion to help us. Another technique is to give prefixes to the units, for example milligram (mg) or gigabyte (GB). A useful notation for writing very large or very small numbers is “scientific notation”. A number in scientific notation has the form: a ×10 n where 1≤a < 10 , and n is an integer This means that in the number a, there is exactly one digit before the decimal point (but this cannot be ‘0’), and the power of ten (10 n) is used to ensure the correct value of each digit. Here are some examples: 472.3= 4.723 × 10 2 , 0.00345= 3.45 × 10 −3 , 0.16= 1.6 × 10 −1 . How would we write the value 2.7 in scientific notation? We need to find a value for n such that 2.7= 2.7 × 10 n . We know that 2.7= 2.7 × 1 , and that 100 = 1 , and so we can write 2.7= 2.7 × 10 0 . There different ways to enter numbers in scientific notation on a calculator, for example, some have a button marked EXP or EE which you press before entering the power of ten. Also, calculators differ in the way they present numbers in scientific notation: some miss out the ‘×10’, for example, showing 3.6× 10 7 as 3.6 07 or 3.6 E07; others may use a different notation. It is very important to get to know your own calculator. 3 In addition to knowing how to enter numbers in scientific notation for use with a calculator or computer, it will help your understanding of this notation if you learn how to calculate without using a machine. Multiplying and Dividing When multiplying or dividing numbers given in scientific notation it is easiest to deal with the number part and the powers of ten separately. Example 1: (610×23) ××( 710 5) =×× 6710 235 × 10 [re-order the multiplications] (23+ 5 ) 28 =×42 10 =× 4210 [using laws of exponents] 1 28 29 =××4.2 10 10 =× 4.2 10 [write in correct scientific notation] Example 2: 17 17 2 8 ×10 2× 10 810×17 ÷× 410 − 8 = = [write the division as a fraction and cancel] ()() −8 − 8 1 4 ×10 10 (17−( − 8 )) =2 × 10 [using laws of exponents] =2 × 10 25 Adding and Subtracting When adding or subtracting numbers given in scientific notation we must first write the numbers with the same power of ten - we cannot use laws of exponents to combine the powers of ten when adding or subtracting numbers. Once the power of ten is the same we simply add or subtract the number part (and re-write in correct scientific notation if necessary). Example 3: 13 12 13 13 (a) (2.34× 10) +×( 3.110) =( 2.34 × 10) +×( 0.3110 ) [get the same power of ten – the largest] 13 =()2.34 + 0.31 × 10 [the power of ten is a common factor] 13 =2.65 × 10 OR – 13 12 12 12 (b) (2.34× 10) +×( 3.110) =( 23.4 × 10) +×( 3.110 ) [the same power of ten – the smallest] 12 =()23.4 + 3.1 × 10 [the power of ten is a common factor] 12 13 =26.5 × 10 = 2.65 × 10 [write in correct scientific notation] Method (a), writing both numbers with the largest power of ten, is slightly shorter, but some people find it easier to use the smaller power of ten, method (b). The important thing is to write both numbers in a form with the same power of ten – you can choose whichever method you find easier. Look at the following examples and decide which you find easier when there are negative powers of ten involved. 4 Example 4: −5 − 8 − 5 −− 35 (a) (7.594× 10) −×( 3.8 10) =( 7.594 × 10) −××( 3.8 10 10 ) [‘split up’ one power of ten ...] −5 − 5 =(7.594 × 10) −( 0.0038 × 10 ) [..