The Power of One in Math

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 y^a/y^b = y^a–b. Let b = a. Then y^a/y^a = 1 = y^a–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