The number 1 is such a simple number:
·
It’s the first number when you count positive
integers.
·
It has no effect when multiplying or dividing by
1.
·
It also has no effect as an exponent.
·
Adding or subtracting 1 simply yields the neighboring
integer.
Yet the number 1 can be very powerful in math.
One way to convert units is to multiply by 1. For example,
to convert 4 ft. to inches, you could multiply by 1, writing 1 as a fraction:
12 in. / 1 ft. equals 1. Multiplying 4 ft. by 12 in. / 1 ft. yields 48 in.
(Observe that the feet cancel out.)
Fractions can be added by finding a common denominator, and
the denominators are made common by multiplying by 1. For example, to add 1/4
to 1/3, the least common denominator is 12. Multiply 1/4 by 3/3 (which is 1) to
make 3/12 and multiply 1/3 by 4/4 (also 1) to make 4/12. Now add these together
to get 7/12.
The number 1 is used to turn numbers into their reciprocals.
For example, the reciprocal of 5 is 1/5. The reciprocal of x is 1/x. Any number
times its reciprocal equals 1. For example, 2 times 1/2 makes 1.
We use this principle all the time in algebra. For example,
consider 4 x = 12. In this case, isolate the unknown, x, by multiplying by the
reciprocal of its coefficient. The coefficient of x is 4; the reciprocal of 4
is 1/4. Multiply both sides by 1/4 to make x = 12/4 = 3. (Note that dividing
both sides by 4 is equivalent to multiplying both sides by 1/4.)
The equation x + y0 = 2 wouldn’t have a unique
solution if not for the fact that y0 = 1. (This follows from the
rule that ya/yb = ya–b. Let b = a. Then ya/ya
= 1 = ya–a = y0.) Knowing this, the original equation
becomes x + 1 = 2, or x = 1.
In trigonometry, the number 1 often comes in handy through
trig identities. For example, the equation 1 – sin2x = 0.5 cos x
would be difficult to solve because it includes two different trig functions,
except that 1 can be written as sin2x + cos2x, which
reduces this equation to cos2x = 0.5 cos x. We can then factor it as
cos x (cos x – 0.5) = 0, which means that cos x = 0 or cos x = 0.5, for which
the solutions are 90, 60, or 300 degrees.
In probability, the number 1 represents 100%. If you reach
into a bag with 3 red marbles and 5 blue marbles, you have a 3/8 probability of
selecting red and 5/8 probability of selecting blue. If you add these together,
you get 3/8 + 5/8 = 8/8 = 1. That’s because there is 100% that you’ll select a
marble of some color.
There are many derivations and proofs in higher levels of
math that become eloquent when a term is multiplied by 1 and the 1 is written
in terms of an identity (such as the summation expression for the completeness
theorem in group theory – which itself is another example of the power of one,
in establishing unity for a complete set).
Chris McMullen, author of the Improve Your Math Fluency series of workbooks
