## Number System & Place Value

As you know, we use special symbols to represent numbers. For example, if I have ;1 apple, I can show it with 1; if I have 2 apples, I can show it with 2; if I have 3 apples, I can show it with 3.. and so on. But what if I have more than 9, for example, 10 apples?

I would need to find a symbol for that, and another symbol for 11 apples, and so on. Just imagine wanting to represent 1000 apples or even more. Having a symbol for each number would make both performing operations and keeping track of which number corresponds to which symbol quite difficult, wouldn't it? To overcome this problem, we have developed what we call the base-10 number system. In this system, we have both a wonderful grouping and the ability to represent an infinite number of numbers with only 10 symbols.

symbols ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 )

# Base-10 Number System - Grouping by Tens

Every time I have ten of something, I group them together. For example, when I have 10 ones, I make a group and call it tens; when I have 10 tens, i group them and call it hundreds, when i have 10 hundreds I make a group and call it thousands; when I have 10 thousands, I group them together and call it ten thousands, and so on. Each group is made up of 10 times the previous group. This is why we call it the base-10 number system.

( The assets in the figure have been proportionally scaled down to fit. )

# Digits

I have taught the grouping, I have taught the places, now let's look at how to represent the assets in the base-10 number system.

Let's say I have 10 assets; I can make these 10 assets into a single group. I write this as 1 in the tens place, and since I have no ones, I write 0 in the ones place.

For example, I would represent 16 in this way...

Actually, the numbers shown in each place represent the remainder that cannot be grouped into tens from the next higher place. For example, in 16, I have 1 group of ten and 6 ones. As the ones increase to 7, 8, 9, and 10, I can again form a group of ten.

Let's say I have 99 assets; when one more comes, first the 9 ones are completed, and it becomes a group of ten. Once that group is formed, I now have 10 groups of ten, and I can group them again, and it becomes one group of ten tens, which means I have a group of 100.

# Place Value

We've learned the symbols, we've learned the places, and we've learned that using these places and symbols, we can represent an infinite number of numbers. Ten symbols do all the work. What allows us to represent this infinite number of numbers is the meaning gained by the symbols according to where they are placed. By placing ten symbols in various places, I can give them different meanings; this is what we call place value

## 35

Here, 5 represents five ones. ( 5 )

## 258

Here, 5 represents five tens. ( 50 )

## 547

Here, 5 represents five hundreds. ( 500)

## 5 347

Here, 5 represents five thousands. ( 5000)

exct..

## 23

2 tens and 3 units  20 + 3

500+40+7

## 59 432

50 000 + 9 000 + 400 + 30 + 2

## 324 567

300 000 + 20 000 + 4 000 + 5 000 + 60 + 7

## 3 904 280

3 000 + 900 000 + 4000 + 200 + 80

## 92 980 311

90 000 000 + 2 000 + 900 000 + 80 000 + 300+ 10 + 1

## 195 678 245

100 000 + 90 000 + 5 000 + 600 000 + 70 000 + 8 000 + 200+ 40+ 5

## 5 675 012 978

5 000 000 000 + 600 000 000 + 70 000 000 + 5 000 000 + 10 000 + 2 000 + 900 + 70 + 8

## it is your turn ;

In the octal number system ( 8 ) what is the decimal equivalent of the number 3405?