3.3 Subtraction of Whole Numbers

1. $5+c=9\Rightarrow c=4$, so $9-5=4$.
2. $1+c=7\Rightarrow c=6$, so $7-1=6$.
3. $14+c=27\Rightarrow c=13$, so $27-14=13$.

1. Counter Example: $8-5=3$, but $5-8$ is not a whole number.
2. Counter Example: $10-(6-2)=6$, but $(10-6)-2=2$.

Line 'em up.

Subtract the ones. $3-1=2$

Subtract the tens. $6-4=2$

And finally the hundreds. $2-0=2$.

And we're done!

$$263-41=222$$

Line 'em up.

We can't subtract $4-7$, so we borrow a group of 10 from the 3 groups in the next column over. $$10+4-7=7$$

That sets up our next column to be $2-8$, which again we can't do. So we borrow one from the 100's

And finally $1-9$ isn't defined, so we borrow that final group of 10, from the 1000's this time, and finish the difference.

And we find $$1234-987=247$$

We begin by lining up the places.

First up, $4_{\text{five}}-2_{\text{five}}$, no issues here.

We hit our fist hiccup here, $3_{\text{five}}-4_{\text{five}}$ isn't defined. So we need to borrow a 25 from the next column over.

Now we're ready to subtract $13_{\text{five}}-4_{\text{five}}$. Remember that $13_{\text{five}}=1(5)+3(1)=8$, so we're really subtacting $8-4=4$

Can't subtract $0_{\text{five}}-2_{\text{five}}$ so we borrow a 125, $10_{\text{five}}-2_{\text{five}}=3_{\text{five}}$

And finally $1_{\text{five}}-0_{\text{five}}=1_{\text{five}}$

and were done!

$$2134_{\text{five}}-242_{\text{five}}=1342_{\text{five}}$$

The subtraction isn't neccessarily easy to see. But if we were to add 1 to both numbers:

$$ 19-7=20-8 $$

And I think that $20-8$ is much easier to see that the difference is $12$. And because we used an equal addition, the difference here will be the same as the original.

Same difference, easier subtraction.

Starting from $31$ we can go to $40$ by adding $9$.

From $40$ we can jump to $100$ by adding $60$.

$$ 9+60=69 $$

So $100-31=69$.

We're going to start and $524$ and count up to easy to use spots. First, it would be nice to count up to a multiple of 10.

Next, we'll count up from $530$ to a nice even multiple of 100.

Now we'll add hundreds to get as close as we can to the $765$

Now we'll finish it off by adding the last bit.

And we've counted up! The difference of these two numbers is the total of all the counting up we did.

So we have $$765-524=241$$

Count up:

$$11\to20\text{ is }9$$ $$20\to50\text{ is }30$$ $$9+30=39$$

So Noah gets $39 back.

Recall that the addition table for base 5 is

When we try to start the difference

We realize we need to borrow once, so we'll go ahead and do that part..

Now we're ready to start subtracting. $3_{\text{five}}-2_{\text{five}}$ is no issue.

But $11_{\text{five}}-4_{\text{five}}$ requres a bit of thought. But go back to the addition table. Find $11_{\text{five}}$ in the center and a $4_{\text{five}}$ in the same row. You see that uniquely identifies a column.

This entry corresponds to $4_{\text{five}}+2_{\text{five}}=11_{\text{five}}$, but for our purposes, using the definition of subtraction, that relationship is better written as $2_{\text{five}}=11_{\text{five}}-4_{\text{five}}$. So by finding this entry in the table we now know the difference without breaking about the calculation into base 10.

Recall that the addition table for base 7 is

When we try to start the difference

We realize we need to borrow in the ones place, so we'll go ahead and do that part.

Now we're ready to start subtracting. $13_{\text{seven}}-5_{\text{seven}}$ is no issue.

But $1_{\text{seven}}-4_{\text{seven}}$ requires a borrow from the next column over.

Now we can use the table for $11_{\text{seven}}-4_{\text{seven}}$. Find $11_{\text{seven}}$ in the center and a $4_{\text{seven}}$ in the same row.

This entry corresponds to $4_{\text{seven}}+4_{\text{seven}}=11_{\text{seven}}$, so using subtraction we get $4_{\text{seven}}=11_{\text{seven}}-4_{\text{seven}}$.

We still have $0_{\text{seven}}-3_{\text{seven}}$, so we borrow from the next column and finish.

So the difference is

$$ 6123_{\text{seven}}-345_{\text{seven}}=5445_{\text{seven}} $$