Reciprocals
Numbers · Reciprocals
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 2 Reciprocals
Lesson objectives
By the end of this topic, you should be able to:
- Find the reciprocal of a number by division.
- Find reciprocals of numbers from mathematical tables.
- Find the reciprocal of decimals and fractions.
- Use reciprocals in computation.
Reciprocals
The reciprocal of a number is 1 divided by that number. Multiply any number by its reciprocal and the answer is always 1.
a) Finding a reciprocal
For a whole number, write 1 over it. The reciprocal of 4 is 1/4, or 0.25.
For a fraction, simply invert it. The reciprocal of 3/5 is 5/3.
For a mixed number, convert to an improper fraction first. So 2½ becomes 5/2, and its reciprocal is 2/5.
For a decimal, divide 1 by it. The reciprocal of 0.4 is 2.5.
b) The defining property
A number times its reciprocal gives 1. That is the definition, and it is also the check.
If your reciprocal does not multiply back to 1, it is wrong.
c) Zero has no reciprocal
Nothing multiplied by zero gives 1, so zero has no reciprocal.
Division by zero is undefined for the same reason.
d) Reciprocals from tables
Reciprocal tables give values for numbers from 1 to 10.
For numbers outside that range, adjust by powers of ten.
The reciprocal of 40 is one tenth of the reciprocal of 4, so 0.025.
The reciprocal of 0.4 is ten times that of 4, so 2.5.
Move the decimal point the opposite way to the number itself. That is the whole trick.
e) Using reciprocals in computation
Dividing by a number is the same as multiplying by its reciprocal.
Before calculators, this made long division far easier, and it is why reciprocal tables exist.
It matters still. In algebra, dividing by a fraction is done by multiplying by its reciprocal.
f) A useful pattern
A number greater than 1 has a reciprocal less than 1.
A number less than 1 has a reciprocal greater than 1.
The reciprocal of 1 is 1 itself. So is the reciprocal of −1.
A quick sanity check. The reciprocal of 0.5 must be larger than 0.5. A smaller answer is wrong.
g) Where this is used
Dividing fractions. Formulas involving rates, such as time from speed. Lens and resistance formulas in physics, which add reciprocals directly.
Words to know
- Reciprocal -- 1 divided by a number; its multiplicative inverse.
- Multiplicative inverse -- another name for the reciprocal.
- Improper fraction -- one whose numerator is at least as large as its denominator.
- Mean differences -- correction columns in mathematical tables.
- Rate -- a quantity per unit of another quantity, such as tanks per hour.
:::checkpoint Check yourself
- Find the reciprocal of 8.
- Find the reciprocal of 2/7.
- Find the reciprocal of 0.25.
- Why has zero no reciprocal? :::
Bridge to practice
The exercises begin with the definition and the special cases, move through reciprocals of fractions, mixed numbers and decimals, and finish with tables and work-rate problems. For every answer, check that a number below 1 has produced a reciprocal above 1, and vice versa.