Using the and connective.
In “apple words” we have
start with 7 apples
subtract 3 apples
result 4 apples
or as a single “sentence”,
Starting with 7 apples and subtracting 3 apples gives the result 4 apples
or without the apples,
Start with 7 and subtract 3, gives the result 4
Using the minus sign in place of subtract.
We can simply replace subtract by minus in the “subtract 3 apples” expression as they have the same meaning.
And we can use the minus sign, ” − “, instead of the word.
So we have converted subtract into the minus sign, and, correspondingly, add into the plus sign, ” + “, and used the word and to link the terms together.
Beginning with the example on the previous post,
start with 9, add 2, subtract 3, subtract 5, add 4, subtract 2 (final state 5)
we get the expression on the left
(start with) 9 and +2 and -3 and -5 and +4 and -2
and since the and connectives are all of the connectives we can dispense with them and write the expression as
(start with) 9 +2 −3 −5 +4 −2 with result or final state 5
We can even cross out the start with at the beginning as it is always there,
and simplify using the arithmetical equals, or ” = ” :
9 +2 −3 −5 +4 −2 = 5
Multiplication and division, with numbers.
start with 5, add 6, multiply by 4, multiply by 2, subtract 21, result is 32
5 and +6, multby 4, multby 2, and -21
5 +6*4*2 -21 result 32
At this point there appears to be a problem:
Do we multiply out the factors, (multbys), before the sums are calculated, or afterwards?
The convention is to multiply the factors in each group of multbys first, and then combine (and) the add and subtract items.
For the above calculation the result develops as follows:
5 +6*4*2 -21 = 32
5 +6*8 -21 = 32
5 +48 -21 = 32
And with two groups of mults:
5 +6*3 +1 -2*4 +12 = 28
5 +18 +1 -2*4 +12 = 28
5 +18 +1 -8 +12 = 28
And now we have to do ALGEBRA.
Algebra – expressions, terms and factors.
A term in algebra is not just a number, preceded by a “+” or a “−”, it is a number (which may be 1 and can be ignored) followed if needed by a sequence of letters (variables) and expressions. Each of the expressions is enclosed in brackets (parentheses).
An algebraic expression is (startwith) 0, followed by a list of terms.
Simple example :
(startwith) 0 +8a +bx
The (startwith) number is always zero and we can ignore it, leaving only the terms.
The first term has a “+” or a “−”, and we can ignore the first “+”, as 8a +bx
If the terms are rearranged then the missing “+” needs to be put back in,
as in bx +8a
4 +3x −7xy
-4 +3x −7xy
and with included expressions like (x +5):
4 +3x(x +5)
2 -5x -6(x +2)(x +9)
The value of an expression is the numerical result of the expression when all the individual variables are replaced by numbers and combined.
An equation is two expressions with an “equals” sign between them.
If the value of the left side of an equation is equal to the value of the right side then the equation is satisfied. The number on the left equals the number on the right.
x +3 = 4x -6
The equation is satisfied when x = 3, and nowhere else.
3x -a = 2x where x = 6. Now find “a” to satisfy the equation.
The distributive property of multiplication over addition.
This is the most important feature of arithmetic and algebra, where the additive form of an expression is converted into a multiplicative form, and vice versa.
Four examples will show the methods:
a(x +y) = ax +ay
ax +bx = (a +b)x
(a +b)(x +y) = ax +ay +bx +by
(a −b)(x −y) = ax −ay −bx +by
This last is a bit obscure, but a two dimensional array will help.
More tricky details, for exponents and for division in algebra.
Exponents: Simple examples
2 +3x2 +3x
5 +4x +6x +2
The two examples above are the conventional forms of expressions with exponents.
The two below are fully bracketed expressions showing the linked pairs of parentheses, and showing the signs and numbers for the exponents. “^” is the sign for “exponent”.
2 +(3 *(x ^2)) +(3 ^x))
5 +(4 *x) +(6 ^(x +2))
These are both one-line expressions.
Division: Simple examples
The two examples above are the conventional forms of expressions with division.
The two below are fully bracketed expressions showing the linked pairs of parentheses, and showing the signs and numbers for division. “/” is the sign for “divide by”.
6 +(7 /x)
(6 +(7 *y)) /(x +2)
These are also both one-line expressions.
It will appear that with the “apple” construction the formulae for arithmetic and algebra are identical.