Number System
The number system is the foundation on which the whole of General Mathematics is built, and for the ADRE Grade IV examination it deserves to be studied before any other chapter. Every question on percentage, average, ratio or simple interest begins with numbers that must be read, written and compared correctly, and a candidate who is at ease with the families of numbers, the place value of digits and the rules of divisibility saves both time and marks. The syllabus of this chapter, as followed in the Master Book: Maths Masti, is not heavy. It asks the candidate to know how numbers are classified, how the value of a digit changes with its position, how to round a number and estimate a sum, and how to tell at a glance whether one number divides another exactly. The paper usually sets these as short, single-step questions, so the work is less about difficult calculation and more about clear ideas. Once this chapter is firm, the remaining topics of the ADRE Grade IV mathematics syllabus feel noticeably lighter, and a small daily habit of number work pays off throughout the paper.
Basic Terms
- Digit: any one of the ten symbols from zero to nine that are used to write numbers.
- Number: a value formed by one or more digits.
- Numeral: the particular way a number is written using digits.
- Natural numbers: the counting numbers one, two, three and so on.
- Whole numbers: all natural numbers taken together with zero.
- Integers: whole numbers together with the negative counting numbers.
- Even number: a number exactly divisible by two.
- Odd number: a number that leaves one as remainder when divided by two.
- Prime number: a number greater than one with only two factors, one and the number itself.
- Composite number: a number greater than one that has more than two factors.
- Co-prime numbers: two numbers whose only common factor is one.
Two is the smallest prime number as well as the only even prime. One is neither prime nor composite because it has only one factor. Four is the smallest composite number, and every composite number can be broken into prime factors.
Families of Numbers
All numbers used in ordinary arithmetic belong to the family of real numbers, and within it the rational and the irrational numbers live side by side. A rational number can be written as a fraction of two integers, so every whole number and every terminating or repeating decimal is rational. An irrational number cannot be written in that form; the square root of two and the value of pi are the standard examples. Among the natural numbers, the even numbers are those whose last digit is zero, two, four, six or eight, and the odd numbers end in one, three, five, seven or nine. A prime number has no factor other than one and itself, and a composite number does. Two numbers are called co-prime when the only factor they share is one, and this idea is used again in the chapters on fractions and on LCM and HCF.
Place Value and Face Value
The face value of a digit is the digit itself, while its place value is the digit multiplied by the value of the position it occupies. In the number six thousand and thirty-four, the digit six has a face value of six but a place value of six thousand, because it stands in the thousands place. In the decimal system every position to the left is ten times the position on its right, so the positions are the ones, tens, hundreds, thousands and so on. For the digit zero the face value and the place value are both zero, and therefore the difference between them is zero. Questions that ask for the difference between the place value and the face value of a marked digit are common, and they are solved simply by reading the position of the digit correctly.
Comparison and Ordering of Numbers
To compare two numbers, first count their digits. A number with more digits is the larger one, provided no unnecessary leading zeros are written. When the number of digits is the same, compare the digits from the left, position by position, and stop at the first position where they differ; the number with the larger digit there is the larger number. To arrange a list of numbers in increasing or decreasing order, apply the same rule to every pair. This skill is tested directly by asking the candidate to pick the smallest or the greatest number from a set, and it is also the hidden step in many comparison questions on fractions and decimals.
Rounding and Estimation
Rounding a number means replacing it with the nearest convenient number of the required place. To round to the nearest ten we look at the units digit; if it is five or more we increase the tens digit by one, otherwise we leave it unchanged, and then write zero in the units place. The same method works for the nearest hundred or thousand by looking at the digit just to the right of the place being rounded. Estimation uses rounding to obtain a quick and roughly correct answer, which is very useful in the examination to check whether a calculated option is reasonable. A good estimate catches an obviously wrong answer before time is wasted on the full calculation.
Divisibility Rules
- By 2: the last digit is even.
- By 3: the sum of all the digits is divisible by three.
- By 4: the number formed by the last two digits is divisible by four.
- By 5: the last digit is zero or five.
- By 6: the number is divisible by both two and three.
- By 8: the number formed by the last three digits is divisible by eight.
- By 9: the sum of all the digits is divisible by nine.
- By 10: the last digit is zero.
- By 11: the difference between the sum of the digits in the odd places and the sum of the digits in the even places is zero or a multiple of eleven.
These rules prevent needless long division and are quicker to apply than they look. Questions often ask for the smallest digit that must be placed in a blank so that the number becomes divisible by three, by nine or by eleven, and such questions yield at once when the rule is applied carefully.
Fractions and Decimals
A fraction is a part of a whole written as one number over another. The upper number is the numerator and the lower is the denominator, and the denominator must never be zero. A decimal is only another way of writing a fraction whose denominator is ten, one hundred or one thousand. To change a fraction into a decimal we divide the numerator by the denominator, and to change a decimal into a fraction we write it with the correct power of ten in the denominator and then reduce it to its lowest terms. One half is zero point five, one quarter is zero point two five, and three quarters is zero point seven five. Because the ADRE Grade IV syllabus joins fractions and decimals with the number system, a candidate should be fluent in moving between the two forms.
Squares, Cubes and Roots
The square of a number is the number multiplied by itself, and the cube is the number multiplied by itself twice. The squares of the numbers from one to twenty and the cubes of the numbers from one to ten should be known by heart, because they appear again in mensuration and in simplification. The square root of a perfect square is the number which, multiplied by itself, gives that square, and the cube root works in the same way. Recognising the common perfect squares and cubes makes several kinds of question almost instant.
How This Chapter Appears in the Examination
ADRE Grade IV, APSC, Assam Police and other state papers set the number system in short and direct questions. The common forms are choosing the prime or the composite number from a list, finding the place value of a marked digit, picking the smallest and the greatest number from a set, applying one of the divisibility tests, and converting between fractions and decimals. Because each question tests a single clear idea, this chapter offers dependable marks to a well-prepared candidate. Its greatest value, however, is indirect, because the mental arithmetic practised here is used again and again in average, percentage, ratio and simplification.
Common Mistakes to Avoid
- Calling one a prime number, although it has only one factor.
- Comparing numbers with a different number of digits by looking only at the first digit.
- Quoting the face value of a digit when the question asks for its place value.
- Rounding up a number when the deciding digit is less than five.
- Assuming that every odd number is a prime number.
How to Prepare This Topic
- Learn the families of numbers and the small facts about prime, composite and co-prime numbers.
- Master place value and face value using five-digit and six-digit numbers.
- Practise comparison, ordering and rounding on random lists of numbers.
- Memorise and apply the divisibility rules from two to eleven.
- Finish with the conversion of fractions into decimals and decimals into fractions.
Frequently Asked Questions
Is one a prime number?
No. A prime number must have exactly two factors, one and itself, but the number one has only one factor. One is therefore neither prime nor composite, and the smallest prime number is two.
What is the difference between place value and face value?
The face value of a digit is the digit itself, while the place value is the digit multiplied by the value of its position. In five thousand and sixty, the face value of five is five but its place value is five thousand. For the digit zero the two are equal.
Which prime number is even, and why?
Two is the only even prime number. Every other even number is divisible by two as well as by one and itself, so it has more than two factors and is composite. This small fact is asked directly in the examination.
How do I check divisibility by three and by nine quickly?
Add all the digits of the number. If the digit sum is divisible by three, the number is divisible by three, and if the digit sum is divisible by nine, the number is divisible by nine. For example, the digit sum of one thousand two hundred and thirty-three is nine, so the number is divisible by both three and nine.
How should I prepare the number system for ADRE Grade IV?
Begin with the families of numbers and the small tested facts, then work on place value, comparison and rounding, and afterwards memorise the divisibility rules from two to eleven. End with the conversion between fractions and decimals, and revise the whole chapter once a week with a set of twenty short questions.