What if an alternative system for representing numbers

[PseuTonym]

… would cut the amount of memorizing required to learn the multiplication table from the amount of memorizing required now (call it 100%) to 25% of the usual amount of memorizing?

Would it be worthwhile for children who don’t have an aptitude for memorizing to learn that system? Conversion between the two systems is easily automated, and some students could learn both systems.

For signed quantities (i.e. integers that can be positive or negative or zero), the system is just as concise as the usual base ten system. It doesn’t require writing longer sequences of numerals than usual to represent the same value.

 

[Tarpeian_Rock]

… would cut the amount of memorizing required to learn the multiplication table from the amount of memorizing required now (call it 100%) to 25% of the usual amount of memorizing?

Would it be worthwhile for children who don’t have an aptitude for memorizing to learn that system? Conversion between the two systems is easily automated, and some students could learn both systems.

For signed quantities (i.e. integers that can be positive or negative or zero), the system is just as concise as the usual base ten system. It doesn’t require writing longer sequences of numerals than usual to represent the same value.

I thought calculators long ago did away with memorizing multiplication tables, to say nothing of the capabilities of smart phones.

 

[seagal]

… would cut the amount of memorizing required to learn the multiplication table from the amount of memorizing required now (call it 100%) to 25% of the usual amount of memorizing?

Would it be worthwhile for children who don’t have an aptitude for memorizing to learn that system? Conversion between the two systems is easily automated, and some students could learn both systems.

For signed quantities (i.e. integers that can be positive or negative or zero), the system is just as concise as the usual base ten system. It doesn’t require writing longer sequences of numerals than usual to represent the same value.

Interesting idea. Do you have such a system in mind?

 

[Xystus]

… would cut the amount of memorizing required to learn the multiplication table from the amount of memorizing required now (call it 100%) to 25% of the usual amount of memorizing?

Would it be worthwhile for children who don’t have an aptitude for memorizing to learn that system? Conversion between the two systems is easily automated, and some students could learn both systems.

For signed quantities (i.e. integers that can be positive or negative or zero), the system is just as concise as the usual base ten system. It doesn’t require writing longer sequences of numerals than usual to represent the same value.

multiplication is easily automated too.

 

[Reepicheep]

I thought calculators long ago did away with memorizing multiplication tables, to say nothing of the capabilities of smart phones.

That’s fine for people who have calculators or smartphones. Not everybody has.

Also sometimes it’s helpful to be able to do the math in your head rather than haul out a calculator. Especially for someone who has to do math a lot, being able to work without the calculator could be worth learning a new system.

 

[StephieNorthCo]

That’s fine for people who have calculators or smartphones. Not everybody has.

Also sometimes it’s helpful to be able to do the math in your head rather than haul out a calculator. Especially for someone who has to do math a lot, being able to work without the calculator could be worth learning a new system.

Agreed. I think I’m the last girl on the face of the earth with no cell phone.

I do a LOT of calculations in my head, because I generally get the answer quicker than dragging out a calculator. That goes for log and dB Calc that I routinely do at work (I’m an EE).

I do realize that that is somewhat unusual, though.

A new number system that quarters the effort? That would be awesome.

Blessings,
Stephie

 

[seagal]

Agreed. I think I’m the last girl on the face of the earth with no cell phone.

Second last

 

[Beryllos]

I don’t know what you mean by an alternative system for representing numbers, but let’s think about working in another base. Computers work in binary (base 2), in which there are very few facts to memorize (four addition facts, 0+0=0, 0+1=1, 1+0=1, 1+1=10, and four multiplication facts, 0x0=0, 1x0=0, 0x1=0, 1x1=1). Interestingly, modern computer chips speed up their multiplication and division by effectively using larger bases (processing chunks of two or more bits at a time).

The problem with binary is the conversion to and from decimal. It would be laborious for a child to do this by hand.

It might be more feasible to operate in base 5. The conversion between decimal and base 5 is relatively simple. The number of multiplication facts is roughly one quarter of that in decimal. However, the student would have to memorize a modest set of base-conversion facts, so I am not sure the total burden of memorization would be reduced.

Developing a base 5 multiplication and division system might make a good math exercise for high school students, math hobbyists, or other curious folks.
:newidea:

 

[Joie_de_Vivre]

Agreed. I think I’m the last girl on the face of the earth with no cell phone.

I do a LOT of calculations in my head, because I generally get the answer quicker than dragging out a calculator. That goes for log and dB Calc that I routinely do at work (I’m an EE).

I do realize that that is somewhat unusual, though.

A new number system that quarters the effort? That would be awesome.

Blessings,
Stephie

Base 12, in Base 10 it is divisible by 2, 5 whereas Base 12 is divisible by 2, 3, 4, 6. It is also fairly natural give there are 12 inches in a foot, there are 24 hours in a day, dozens and grosses (a dozen dozen) are normal units, twelve is its own word, etc.

 

[EasterJoy]

Memorizing the multiplication tables isn’t a very large part of the overall time required to master mathematics. Switching to either a new numerical system or else using two different systems simultaneously, meantime. The metric system of expressing measurements, meanwhile, is much easier than the English system of measurement, but look at how much resistance there has been to switching over.

I don’t see people going for it, but particularly not people who feel they have a shaky grasp on math. I see the people who like math in the first place thinking it is a great idea, but I don’t see them convincing everyone else.

 

[mommy_k]

I don’t see people going for it, but particularly not people who feel they have a shaky grasp on math. I see the people who like math in the first place thinking it is a great idea, but I don’t see them convincing everyone else.

Yes…as they say…there 3 types of people in this world…
People who are math inclined, and people that are not.

 

[EasterJoy]

Yes…as they say…there 3 types of people in this world…
People who are math inclined, and people that are not.

(And heaven help the poor ones who are not when those who are so inclined find a way to explain math that “will make it so much easier, trust me!”)

 

[StephieNorthCo]

(And heaven help the poor ones who are not when those who are so inclined find a way to explain math that “will make it so much easier, trust me!”)

Yeah… I used to be guilty of that. Gave up, though. I’m resolved to the 4 different kind of people in this would. After all, there’s room for all 5 types.

Blessings,
Stephie

 

[Beryllos]

There are 10 kinds of people in the world:
Those who understand binary numbers,
And those who don’t.

 

[Sarcelle]

There are 10 kinds of people in the world:
Those who understand binary numbers,
And those who don’t.

Haha!



Now translate that into hex.

 

[StephieNorthCo]

Haha!



Now translate that into hex.

Still 10. Shakespeare really used hex arithmetic in his quote:
2B or not 2B, that is the question.
Of course, the answer is obviously FF.

Blessings,
Stephie

 

[psalm90]

I don’t know what you mean by an alternative system for representing numbers, but let’s think about working in another base. Computers work in binary (base 2), in which there are very few facts to memorize (four addition facts, 0+0=0, 0+1=1, 1+0=1, 1+1=10, and four multiplication facts, 0x0=0, 1x0=0, 0x1=0, 1x1=1). Interestingly, modern computer chips speed up their multiplication and division by effectively using larger bases (processing chunks of two or more bits at a time).

The problem with binary is the conversion to and from decimal. It would be laborious for a child to do this by hand.

It might be more feasible to operate in base 5. The conversion between decimal and base 5 is relatively simple. The number of multiplication facts is roughly one quarter of that in decimal. However, the student would have to memorize a modest set of base-conversion facts, so I am not sure the total burden of memorization would be reduced.

Developing a base 5 multiplication and division system might make a good math exercise for high school students, math hobbyists, or other curious folks.
:newidea:

Base 60 is good, too.

 

[PseuTonym]

Interesting idea. Do you have such a system in mind?

Yes. For convenience, we can use the following symbols:
L for the number negative one
S for the number negative two
E for the number negative three
P for the number negative four

(There are better choices for the symbols, but we need symbols that everybody here can type.)

Our inventory of digits will include the above four symbols along with the following six symbols having their usual meanings: 0, 1, 2, 3, 4, 5
(Again, that is just for convenience here and now.)

Thus, the system has the following symbols: 0, 1, L, 2, S, 3, E, 4, P to be understood in that sequence as having the values 0, 1, -1, 2, -2, 3, -3, 4, -4

We can multiply using the usual algorithm taught in school because the distributive property is true not only for positive integers, but for any integers.

Now, the extreme values for the multiplication table are 5 times 5, P times 5, and P times P.

P times 5 = (-4) times 5, = (negative 20), which can be written as S0.
P times P = (-4) times (-4) = 16 in ordinary decimal notation = (20 + (-4)) = 2P.

 

[PseuTonym]

Thus, the system has the following symbols: 0, 1, L, 2, S, 3, E, 4, P to be understood in that sequence as having the values 0, 1, -1, 2, -2, 3, -3, 4, -4

I apologize for the above error of omission. The digit 5 should have been included.
The system has the following symbols: 0, 1, L, 2, S, 3, E, 4, P, 5 to be understood in that sequence as having the values 0, 1, -1, 2, -2, 3, -3, 4, -4, 5.

 

[PseuTonym]

P times P = (-4) times (-4) = 16 in ordinary decimal notation = (20 + (-4)) = 2P.

Observe that 4 is in the inventory of digits, but 6 isn’t. Thus, people who learned the alternative system would learn (4 times 4) = 2P, just as you have learned (4 times 4) = 16.

If you had learned (4 times 4) = 2P, then you would be able to reason as follows:
P times P
= (L times 4) times (L times 4)
= (L times L) times (4 times 4)
= 1 times 2P
= 2P

Written like that, it looks complicated, but the essence of it is quite simple:
(-4) times (-4) equals 4 times 4.

If children in future have the system explained to them in a clear manner, then they will have less memorizing to do than is required with the usual base ten system that uses zero and all other digits having strictly positive value.