REPRESENTATION OF NUMBERS
Base 10 strategy: the order of the digits of a number is important because the order implies the power of 10 by its position. The first number is the highest and most significant power of ten used. It is important that zero is used if a power of ten is missing following a higher order of ten. The zero is used as a place holder. For example, 1023 implies 1 in the thousands position, but the following zero tells us that there are no hundreds in this number. Then the tens and units (ones) follow.
Expanded form strategy: in this strategy the base 10 number is expanded into individual components. So 1023 would be expanded into 1000+20+3. But in order to carry out arithmetic, the hundreds cannot be ignored and a gap should be left in case the arithmetic requires the hundreds to be represented: 1000+...+20+3. This is different to the base 10 strategy where zero is used as a place holder.
SUBTRACTION
In both strategies, it is usual to align one number below the other by aligning like powers of ten. The difference is that the expanded form strategy consists of a horizontal addition sum for both aligned numbers. The base 10 strategy simply aligns the digits.
EXAMPLE
1023-456
Base 10 layout:
1023
- 456
Expanded form layout:
1000+....+20+3
-........400+50+6
Arithmetic: base 10 strategy:
1023
- 456
It's normal to underline the second number as a separator between the operands and the result. A second underline under the result helps to highlight it.
Starting on the right, we compare the digits in the ones or units position. If the second line digit is bigger than the first line, we cannot subtract straight away and we need to "borrow" from the next highest power of ten, which is the tens column. To do this we reduce the 2 to 1 in the top number and place a little 1 next to the 3 to make it look like 13. Then we can subtract 6 from 13 to give us 7 in the units column of the result. Strike through or erase the 2 in the tens column and change it to 1:
101¹3
- 45 6
7
We can now move a place to the left and continue the subtraction. This time 5 is bigger than the 1 so we have to borrow from the next column to the left. But there's a zero here, so we need to move leftward to the thousands and borrow from that. Conveniently in this case the net result of double-borrowing is to treat the 10 as if it were above the 4 in the second line and simply subtract 1 to make it 9:
9¹1¹3
- 4 5 6
5 6 7
Finally we can subtract 4 from 9 to give us 5.
Arithmetic: expanded form strategy:
1000+.....+20+3
400+50+6
This strategy allows us to be a bit more generous with space, but the principle is similar to base 10. We can actually write 1 in front of 3 to make it 13, which is easier to read. And we can erase 20 to make it into 10 in the next column. When we borrow for the tens, 1000 becomes 900, directly over 400. So the new layout looks like:
900+110+13
- 400+ 50+ 6
500+60+ 7
In the result we simply add across to get 567. The expanded form strategy is less cramped than the base 10 strategy and in many cases may be easier to work with and avoid mistakes. But it involves more writing than base 10, because of the zeroes and the addition signs.