Key On screen content Narration voice-over Web links Activity Under the Activities heading of the online program Quiz Under the Activities heading of the online program Number/Numeral Systems Throughout history there have been many number/numeral systems used, from the early Babylonian numeral system (3100 BC) through to the modern Binary system used in computing. List of Numeral Systems (http://en.wikipedia.org/wiki/list_of_numeral_systems) The Number System most countries in the modern world use today for counting, calculating and reading and writing numbers is called the Decimal System because it is based on the number 10. Note: Dec in Latin means ten Historical Origin of the Decimal System (http://www2.mae.ufl.edu/~uhk/decimal-system.pdf) Before we look at our number system and how it operates, it is interesting to note that there have been several number systems used in different countries throughout the centuries of civilization. Click on the first link List of Numeral Systems to view some of these systems used in the past and in the present. The following will show examples of eight different numeration systems from ancient times. Most countries nowadays use the Decimal system which is based on the number 10. Click on the second link Historical Origin of the Decimal System to get a brief understanding of how the Decimal system was developed throughout history. Comparing Number Systems
Here we can see a summary of some of the ancient number systems used throughout history and, hopefully, you can see the similarity of the Hindu-Arabic system in the first column to our Decimal System which is used extensively throughout the modern world today. The Decimal System This system comes from and uses the ten Hindu-Arabic digits which are: 0 1 2 3 4 5 6 7 8 9 These numeral digits can be used very easily to represent values, quantities or computations from 0 to 9. e.g., 5 apples, 6 + 2 = 8, etc. Page 2 of 8
For values bigger than 9 we can still use these digits but the PLACE they are in becomes important, so that we are then able to write any numeral from 0 to infinity. As can be seen here, there are ten digits that occur in the Decimal System and they can be used in many simple ways to represent values or quantities from 0 to 9. We can still use the same ten digits to write numbers larger than 9 by having the numbers in different PLACES as we will see in the following. Place Value for Whole Numbers Each PLACE (or column) that the digits 0 to 9 can occupy has a different value, which in turn then gives those digits different values. The following table shows an example of how Place Value looks using just the first seven PLACES for whole numbers in the Decimal System. In this example, we have shown just the first seven places in the Decimal system, but there is really no end to the number of places that can be made. For instance, the next place (the 8 th place) will be the 10 millions place, then the 100 millions in 9 th place, then the billions place, and so on. The number of places will keep going up to infinity, which means that there is no end. Notice again that each place gets bigger in value by a factor of ten as we move to the left, and gets smaller by a factor of ten as we move to the right. For example, the 2 nd place is 10 times the value of the first place (ten is 10 times greater than 1) and the 3 rd place is 10 times the value of the 2 nd place Page 3 of 8
(a hundred is 10 times greater than 10), and so on. Similarly, the 3 rd place is ten times smaller than the 4th place (1 000 is ten times smaller than 10 000), and the 2 nd place is ten times smaller than the 3 rd place (100 is ten times smaller than 1 000), and so on. Later, in Topic 3 Scientific Notation, we will see how each of these places is actually a POWER of ten, using indices. It is important to remember that the place that a digit is in, gives it its value. Place Value Let s now look at how the numeral 8 can change its value by being in a different PLACE in our Decimal Place Value System. Consider the last row in the table above which will show the different values of an 8 in three different places (8, 80 and 80 000) An 8 digit in the 1st Place has a value of just 8 ( 8 ones) However, an 8 in the 2nd Place gives a value of 80 (8 tens) And an 8 in the 5th Place gives a value of 80 000 (8 TEN THOUSANDS) Together these three 8s give a number with a total value of 80 088 (eighty thousand and eighty eight) Go to this links to see a very simple explanation of Place Value: Place Value Introduction (https://www.youtube.com/watch?t=55&v=fshycnqhibw) An 8 digit in the 1st Place has a value of just 8 (8 ones). However, an 8 in the 2nd Place gives a value of 80 (8 tens) and an 8 in the 5th Place gives a value of 80 000 (8 TEN THOUSANDS). Together these three 8s give a number with a total value of 80 088 (eighty thousand and eighty eight). Please notice that in Australia (and some other countries) when writing large numbers, the number is split into groups of three starting from the right hand side. In Australia, we show this separation by using a space. Other countries may use a comma to separate the groups of three numbers. Go to the link provided to see another way of understanding place value. Page 4 of 8
Using the Decimal System to Create Numbers The value (or size) of the number in the example in the bottom row of the table is written as: 9 573 842 (nine million, five hundred and seventy three thousand, eight hundred and forty two). Go to Activity One (under Activities in the right-hand side of the screen) and have a go at creating some numbers yourself, remembering to separate the numbers into groups of three. Looking at the example in the bottom row of the table, we have created a number with a value of 9 573 842. We can see that this rather large number consists of a 2 in the first place (or ones place) giving a value of just 2; a 4 in the second place (or tens place) giving a value of 40 when a 4 is put in this 2 nd place it no longer has a value of 4 but has a value of 4 lots of 10, which is 40; an 8 in the third place (or hundreds place) giving a value of 800 when a 8 is put in this 3rd nd place it no longer has a value of 8 but has a value of 800; and so on, until all numbers are done. Have a go at Activity One (under the Activities heading in the right-hand side of the screen). Expanded Notation Expanded Notation is simply a way of writing a number by adding together the value of each digit eg. 7 392 = 300 + 90 + 2 These values can then be further expanded to include the place value of each digit: eg. 7 392 = 300 + 90 + 2 = (7 x 1 000) + (3 x 100) + (9 x 10) + (2 x 1) Writing a number in expanded form (https://www.khanacademy.org/math/pre-algebra/order-of-operations/place_value/v/place-value-3) Page 5 of 8
By analysing numbers this way we can get a good understanding of place value and an appreciation of how place value works in our Decimal Number System. Now have a go at Activity Two to practise analysing some bigger numbers according to their place value. Expanded Notation is simply a way of writing a number by adding together the value of each digit. In the example of 7 392, you can see that 7 is not just a 7, but because of the place it is in (it is in the fourth place), it is 7 000. The 3 is not just three, but because of the place it is in (it is in the third place), it is 300. Similarly, the nine is not just 9, but in fact 90 because it is in the second place and the 2 is just 2 because it is in the first place, the ones place. In the second stage of expanding this number it is a simple matter of multiplying each digit by its place value. 7 000 can then be written as 7 lots of 1000. 300 can then be written as 3 lots of 100. 90 can be written as 9 lots of 10 and the 2 can be written as 2 lots of 1 as it is in the ones place. Have a look at the video in the link provided to see another way of looking and expanded notation and how it relates to place value. Make sure you ignore the commas separating the groups of three numbers. Remember that spaces, not commas, between the groups of three numbers are used in Australia. By analysing numbers this way we can get a good understanding of place value and an appreciation of how place value works in our Decimal Number System. Have a go at Activity Two (under the Activities heading in the right-hand side of the screen) to practise analysing some bigger numbers according to their place value. Bigger Numbers Writing numbers to show positive values or quantities of tens of millions, hundreds of millions, billions, tens of billions, and so on follows the same system of "Place Value" as was explained in the previously, and shown in the table below. Have a look at this video to help you understand the size of some of these large numbers: Page 6 of 8
What is the difference between a million, billion, and a trillion?? (https://www.youtube.com/watch?v=wnrpcheye6o) Now that you have had a look at how our number system works for smaller numbers, we will show you how larger numbers can be examined to see how they represent larger quantities. Please remember that, when using positive numbers, as we move left towards infinity in our Place Value chart, the places (and therefore the values of any digits in those places) become bigger. Conversely, as we move right towards zero in our Place Value chart, the places (and therefore the values of any digits in those places) become smaller. As will be seen in Topic 3, it is much easier to write some of these bigger numbers using Scientific Notation rather than writing endless numbers of zeros. Reading Big Numbers Consider this number: 367 894 230 814 926 It s difficult to read because the place values of the first three numbers is not apparent by just looking at it. It is easy to read if we name each of the groups of three digits: Now have a go at Activity Three to practise some of these bigger numbers. With the large amounts of places involved in some of these bigger numbers, the difficulty in reading the numbers is in establishing what the place values of the first three digits are. An easy way to remember to read and write these numbers is to look at them in groups of three, starting from the right hand side, going from the smaller place values to the bigger place values. The first group of three from the right hand side is always the hundreds, tens and ones; the next group is always the thousands, then millions, billions and trillions. Once we have established what the place value of the extreme left hand group of three digits is (trillions in this case), then we can read the numbers normally from left to right, which in this case is three hundred and sixty seven trillion, eight hundred Page 7 of 8
and ninety four billion, two hundred and thirty million, eight hundred and fourteen thousand, nine hundred and twenty six. Have a go at Activity Three (under the Activities heading in the right-hand side of the screen). Quiz Now that you have completed Topic 1.1 about how our decimal number system works, test your understanding of the topic by completing the Quiz (Under Activities in the right-hand-side of the screen). Are you satisfied with your understanding of this topic? If not, we suggest that you go through this topic again and then have another go at the quiz. If you re happy, you should now move onto Topic 1.2 which is still about Place Value in our Decimal number system, but will include decimal or fractional numbers (numbers less than 1). After you have worked through this topic, have a go at the Quiz (under the Activities heading) to test your understanding of the concepts covered. The correct answers have also been provided. If you did not score well, go through the topic again and have another go at the Quiz. If you are happy with your score, move onto the next topic, which is an extension of this topic looking at Place Value using decimal or fractional numbers less than 1. Page 8 of 8