SPI Mathematics · Number System

Number Types and Place Value

Every number belongs to a family (natural, whole, integer, even, odd), and every digit in it is worth the digit times the value of its place.

All Number System notes

Why this matters

Eight questions in the bank, four of them from the 2022 paper: four EASY and four MODERATE. They ask which family a number belongs to, what a digit is worth in its place, and how many digits or numbers a list holds. Each is quick once you count carefully.

Concept 1 of 3: Natural numbers, whole numbers, integers, even and odd

The number families sit inside each other. Counting numbers come first, adding 0 gives the whole numbers, and adding the negatives gives the integers. Every integer is then either even or odd.

Definition

0 is a whole number but not a natural number.
An even integer can be written as $2m$ and an odd integer as $2m + 1$ , for some integer $m$ .
0 is even. Every integer is even or odd, never both.
$(-1)$ raised to an even power is $1$ ; raised to an odd power it is $-1$ .

Family	What it holds	Examples
Natural numbers	The counting numbers, starting at 1	1, 2, 3, 4, ...
Whole numbers	The natural numbers and 0	0, 1, 2, 3, ... 0 is the smallest whole number. There is no smallest integer.
Integers	The whole numbers and their negatives	..., $-2$ , $-1$ , 0, 1, 2, ...
Even integers	Numbers of the form $2m$	$-4$ , 0, 6, 10
Odd integers	Numbers of the form $2m + 1$	$-3$ , 1, 7, 11
Powers of $-1$	1 for an even power, $-1$ for an odd power	$(-1)^4 = 1$ , $(-1)^7 = -1$

Each family contains the one above it: natural numbers are whole numbers, and whole numbers are integers.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Number System · Number Types and Place Value

For some Integer m every odd integer is of the form

Calling 0 a natural number

The natural numbers start at 1. The set

\{0, 1, 2, 3, \ldots\}

is the whole numbers.

Losing track of alternating signs

In a long sum of powers of

-1

, pair the terms first. If a minus sign sits in front of every even power, then every term becomes

-1

, and the sum is minus the number of terms.

Drill 2 more on natural numbers, whole numbers, integers, even and odd

Concept 2 of 3: Place value and face value

A digit means different amounts in different places. The 3 in 3000 is worth three thousand, but the 3 in 30 is worth only thirty. Its face value, the digit itself, stays 3.

Definition

Face value of a digit is the digit itself.
Place value is the digit times the value of its place: units 1, tens 10, hundreds 100, thousands 1000, and so on.
Name the places from the right: units, tens, hundreds, thousands, ten thousands, hundred thousands (lakhs), millions (ten lakhs).
The smallest number of $n$ digits is 1 followed by $n - 1$ zeros; the largest is $n$ nines.

Place value

\text{place value} = \text{digit} \times \text{value of its place}

Worked example

In 857394, find the place value of 5, and the difference between the place value and the face value of 7.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q9Moderate

Example 2 · Number System · Number Types and Place Value

In the number 5$436$2, the same digit occurs in place of $. The difference between their place values is 699930, then value of $ in this number is____.

Confusing place value with face value

The face value of a digit never changes. Only the place value grows with the place, so read which one the question asks for.

Counting places from the left

Always count places from the right, starting with units. In a seven-digit number the second digit from the left is in the hundred thousands place.

The smallest five-digit number

It is 10000, not 11111 or 00000. A number cannot start with 0, so the first digit is 1 and the rest are zeros.

Drill 2 more on place value and face value

Concept 3 of 3: Counting digits and numbers made from digits

Writing out a long list of numbers uses up digits in blocks: one digit each for 1 to 9, two each for 10 to 99, three each for 100 to 999. And when a few digits are arranged in every order, each digit sits in each place equally often.

Definition

1 to 9: 9 numbers, 9 digits. 10 to 99: 90 numbers, 180 digits. 100 to 999: 900 numbers, 2700 digits.
The count of whole numbers from $m$ to $n$ , both included, is $n - m + 1$ .
Three different non-zero digits make $3 \times 2 \times 1 = 6$ three-digit numbers, and each digit stands in each place twice.
So the sum of those six numbers is $2 \times (\text{sum of the digits}) \times 111$ .

Sum of all numbers from three different non-zero digits

\text{sum} = 2 \times (a + b + c) \times 111 \qquad \text{numbers from } m \text{ to } n = n - m + 1

Worked example

Find the sum of all three-digit numbers that can be formed from 3, 5 and 8, using each digit once.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q1Moderate

Example 3 · Number System · Number Types and Place Value

What will be the sum of adding all the possible three digit numbers formed by 7,2,5 using each digit only once?

Counting a range one short

From 100 to 200 there are 101 numbers, not 100. With both ends included, the count is last minus first plus one.

Using the 111 rule when one digit is 0

A number cannot start with 0. From 0, 3 and 5 only four three-digit numbers exist (305, 350, 503, 530), with sum 1688. The rule

2 \times 8 \times 111 = 1776

wrongly adds 035 and 053.

Drill 1 more on counting digits and numbers made from digits

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Place value and face value
Place value
$\text{place value} = \text{digit} \times \text{value of its place}$
Counting digits and numbers made from digits
Sum of all numbers from three different non-zero digits
$\text{sum} = 2 \times (a + b + c) \times 111 \qquad \text{numbers from } m \text{ to } n = n - m + 1$

Reference tables (1)

Natural numbers, whole numbers, integers, even and odd6 rows

Family	What it holds	Examples
Natural numbers	The counting numbers, starting at 1	1, 2, 3, 4, ...
Whole numbers	The natural numbers and 0	0, 1, 2, 3, ... 0 is the smallest whole number. There is no smallest integer.
Integers	The whole numbers and their negatives	..., $-2$ , $-1$ , 0, 1, 2, ...
Even integers	Numbers of the form $2m$	$-4$ , 0, 6, 10
Odd integers	Numbers of the form $2m + 1$	$-3$ , 1, 7, 11
Powers of $-1$	1 for an even power, $-1$ for an odd power	$(-1)^4 = 1$ , $(-1)^7 = -1$

Each family contains the one above it: natural numbers are whole numbers, and whole numbers are integers.

Watch out for (7)

Calling 0 a natural number
→ Natural numbers, whole numbers, integers, even and odd
Losing track of alternating signs
→ Natural numbers, whole numbers, integers, even and odd
Confusing place value with face value
→ Place value and face value
Counting places from the left
→ Place value and face value
The smallest five-digit number
→ Place value and face value
Counting a range one short
→ Counting digits and numbers made from digits
Using the 111 rule when one digit is 0
→ Counting digits and numbers made from digits

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