Hello, technics!

Give two relevant examples of difficulties learners may face when attempting calculations in denary

and the use of place value in denary.

Assuming we are involved in Arithmetic (and calculators are not used),

. . there are difficulties in almost every operation.

When adding (or subtracting) decimals, the decimal points must be aligned.

While previous (whole number) problems are always right-justified,

. . this is not true with numbers containing decimals.

Example: $\displaystyle 7.5 + 1.32 + 9 + 0.004$

Code:

The problem is NOT set up like this:
7.5
1.3 2
9
0.0 0 4
-------
Instead, it must be set up like this
7.5
1.3 2
9.
0.0 0 4
-------
Often, the "gaps" are filled with zeros.
7.5 0 0
1.3 2 0
9.0 0 0
0.0 0 4
-------
Then we can add and get: 17.824

When multiplying, the decimal points are not aligned.

But the decimal point in the product must be carefully placed.

Example: $\displaystyle 1.9 \times 2.47$ Code:

The problem is set up like this:
2.4 7
1.9
-------
Multiply "as usual":
2.4 7
1.9
-------
2 2 2 3
2 4 7
-------
4 6 9 3

Now the decimal point must be placed in the product.

The first number has two decimal places (counting from the right).

The second number had one decimal place.

. . We want the *sum*: $\displaystyle 2 + 1 \,= \,3$ decimal places.

Hence, the answer will have *three* decimal places: $\displaystyle 4.693$

There is an *intuitive* way to place the decimal point.

The product has $\displaystyle 4693$.

The answer could be: $\displaystyle 46930,\;4693,\;469.3,\;46.93,\;4.693,\;0.4693,\;$ etc.

__Estimate__ the size of the answer.

2.47 is very close to 2.5 . . . and 1.9 is very close to 2.

Our problem is roughly: $\displaystyle 2.5 \times 2 \,=\,5$

The only choice which approximates 5 is: $\displaystyle 4.693$

I hope this is the type of stuff you were asking for.

If not, please clarify . . .